**1. Introduction**

Musical wind instruments have a mechanism converting the direct energy of the fluid flow into the alternative energy of the sound. Such a system is called the selfsustained oscillation system. The fluid flow that drives the instruments may be regarded as the aerodynamical sound source or aeroacoustical source. Wind instruments are a very extensive subject of research over the vibration theory, acoustics, and fluid dynamics. The interaction between the resonance of the instrument [called generically *flue instruments* such as an organ pipe, flute, and recorder in this chapter (see **Figure 1**)] and the jet as the aeroacoustical source will be adequately described in this chapter.

Fluid flow brings about vortices and then generates the sound as well. However, one of the essential characteristics of wind instruments is the resonance, which is an acoustic mechanism amplifying very small perturbations to periodic disturbances

pipes (see **Figure 1**). The flute may have such asymmetry depending on the player. This jet-edge divergence is called the *offset*, which is one of important parameters to

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

Also, we should relevantly notice largely different flow-acoustic interactions involved in various vortex-related sound generations. A thin jet and a sharp edge are used for the edge tone [4–6]. A thick (or semi-infinite) jet usually drives a wallmounted cavity to produce its resonance called the cavity tone [7–9]. A thin jet drives a sharp edge (called the labium) of the resonant pipe to produce an organ pipe tone [1, 2, 10, 11]. A thin jet drives a thick edge with an angle of about 60° in the flute [1]. A thin jet issuing from a flue with the chamfer drives a sharp edge in the recorder [12, 13]. In addition, jet velocity widely extends from a few meters per second to about 50 m/s for these tone productions. Flow condition is laminar or turbulent. The Strouhal number *St* = *fd*/*U*<sup>0</sup> (*f* the sounding frequency; *d* the flue-toedge distance; and *U*<sup>0</sup> the jet velocity at the flue exit) extends from about 0.05 to 5. Generally, a thin edge tends to enhance higher harmonics. As *St* has higher values, the jet flow, which drives the resonant pipe, tends to break down into *discrete*

Although the vortex is essential in flow-excited sound generation, it may operate

Although the jet-drive model has been proven to be effective for an explanation of sound generation by the thin jet, there remain rooms for improvement in applying the vortex sound theory for another explanation of sound generation by the thin jet in flue instruments because small vortices may be produced along the boundaries by the mixing process between the jet flow and the surrounding still air. The boundary layer consisting of small vortices is called the *vortex layer*, which can act as the source of an accelerating force to oscillate the jet. Based on such a viewpoint, the

In Section 2, the jet-drive, discrete-vortex, and vortex-layer models are described. Acoustically induced vortices (simply, acoustic vortices) on sound dissipation are discussed with the aid of flow visualization in Section 3. The jet vortex layer on sound generation in an organ pipe is visualized by the particle image velocimetry (PIV), and the microstructure of the vortex layer is demonstrated in

as an important source of acoustic energy dissipation in various flow-acoustic interactions [3, 15–17]. In the context of musical instruments, the *acoustically induced vortex* shedding at the edge is a key damping mechanism to determine the final amplitude of the steady-state flue instrument tones [16, 17]. Hence, sound dissipation and generation in flow-acoustic interactions are widely dominated by

Howe [18] assumes that a compact vortex core with relatively large size appearing alternately just above and below the pipe edge is created by the interaction between the jet velocity vector *U* and the cross-flow velocity (acoustic reciprocating velocity) vector *u* at the mouth opening formed between the flue and the edge. This vortex core with the vorticity *ω* ð Þ ¼ **∇** � *U* is then considered to drive the air column in the pipe. The sound excitation by this periodic vortex shedding at the edge is controlled by the product of the aeroacoustic source term div ð Þ *ω* � *U* and the potential function representing the irrotational cross-flow *u* at the mouth. This discrete-vortex model of Howe is successfully applied to analyze and evaluate both cavity-tone generation [9] and tone generation in flue instruments [10, 14] when the jet is thick and the condition *d*/*h* < 2 (*d* the width of the mouth opening or the flue-to-edge distance and *h* the jet thickness) is satisfied. On the other hand, when the condition *d*/*h* > 2 is satisfied for thin jets, a jet-drive model on the basis of the intrinsic jet instability [19, 20] is applied instead of the discretevortex model [21]. This jet-drive model has been developed in the field of acoustics

adjust the tone color of flue instruments.

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

*vortices* [10, 14].

[1, 2, 11, 20, 22–27].

**45**

vortex-layer model was proposed recently [28].

the vortex shedding at the edge [3].

#### **Figure 1.**

*An organ flue pipe as a typical example of the musical flue instruments and its important parameters. d, the flue-to-edge distance (or cutup, jet length); h, the jet thickness (or height of the flue exit); R, the pipe inner radius. The origin of the coordinate system is located at the center of the flue exit surface.*

with large amplitudes. Any synchronization is then required, and it is realized by the suitable *phase relation* between the flow (or the jet) driving wind (or flue) instruments and the acoustic wave propagating in the instruments. For example, in the case of flue instruments, if the air flow enters into the pipe at the instant when the acoustic pressure near the edge takes a relatively large positive value, acoustic power (given by the product of the alternating volume flow and the acoustic pressure) becomes positive, and the sound is sustained.

However, a big dissatisfaction to the above viewpoint is the assumption of the existence of the sound at the starting point. Therefore, exactly saying, acoustical theory above is not sound generation theory but sound *regeneration* theory. The viewpoint of positive feedback between the jet and the pipe and the time-domain formulation based on the pipe reflection function [1] are both sound regeneration theory [2]. Musical instrument acoustics has treated such regeneration theories and phenomena as chief objects of research. This is because the resonance is acoustically essential, and we may consider that the resonance controls fluid movement as the energy source. It will be open to the charge of being imperfect combustion that sound existence is presupposed at the starting point when we try to answer how flue instruments produce their sounds.

Then, if we introduce a thesis, "the vortex itself is the true sound source," of the vortex sound theory [3] to flue instruments, is the problem solved? Flue instruments do not seem to be such an obedient subject. Certainly, the vortex sound theory is satisfactorily valid to the edge tone, where the jet-edge system has no pipe that gives the resonance or the acoustic feedback; instead the fluid-dynamical feedback between the edge and the flue (flow issuing slit) is a main mechanism of sound generation.

Moreover, there are a few non-negligible differences other than the acoustic resonance between the edge tone and the pipe tone (or flue tone). First is the amplitude magnitude when the jet oscillates against the edge. The oscillation amplitude of the edge-tone jet is as small as two to three times the jet thickness. On the other hand, the pipe-tone jet in an organ pipe often exceeds 10 times the jet thickness. The edge in an organ pipe (or flue instruments) is just a partition wall which separates the inside from the outside of the pipe. It may be said that the direct jet-edge interaction time is quite short compared with a tonal period in flue instruments. Large vortices *visible* behind the pipe edge are, so to speak, odds and ends of the jet driving the pipe. We should pay more attention toward *invisible* (for our naked eyes) vortices carried along the jet to the edge.

Second is the difference in the jet-edge configuration. The configuration is usually symmetrical in the edge tone. In other words, the jet center surface corresponds to the edge tip. Alternate small vortices continuously appear above and beneath the edge. On the other hand, the edge is usually displaced upward in organ pipes (see **Figure 1**). The flute may have such asymmetry depending on the player. This jet-edge divergence is called the *offset*, which is one of important parameters to adjust the tone color of flue instruments.

Also, we should relevantly notice largely different flow-acoustic interactions involved in various vortex-related sound generations. A thin jet and a sharp edge are used for the edge tone [4–6]. A thick (or semi-infinite) jet usually drives a wallmounted cavity to produce its resonance called the cavity tone [7–9]. A thin jet drives a sharp edge (called the labium) of the resonant pipe to produce an organ pipe tone [1, 2, 10, 11]. A thin jet drives a thick edge with an angle of about 60° in the flute [1]. A thin jet issuing from a flue with the chamfer drives a sharp edge in the recorder [12, 13]. In addition, jet velocity widely extends from a few meters per second to about 50 m/s for these tone productions. Flow condition is laminar or turbulent. The Strouhal number *St* = *fd*/*U*<sup>0</sup> (*f* the sounding frequency; *d* the flue-toedge distance; and *U*<sup>0</sup> the jet velocity at the flue exit) extends from about 0.05 to 5. Generally, a thin edge tends to enhance higher harmonics. As *St* has higher values, the jet flow, which drives the resonant pipe, tends to break down into *discrete vortices* [10, 14].

Although the vortex is essential in flow-excited sound generation, it may operate as an important source of acoustic energy dissipation in various flow-acoustic interactions [3, 15–17]. In the context of musical instruments, the *acoustically induced vortex* shedding at the edge is a key damping mechanism to determine the final amplitude of the steady-state flue instrument tones [16, 17]. Hence, sound dissipation and generation in flow-acoustic interactions are widely dominated by the vortex shedding at the edge [3].

Howe [18] assumes that a compact vortex core with relatively large size appearing alternately just above and below the pipe edge is created by the interaction between the jet velocity vector *U* and the cross-flow velocity (acoustic reciprocating velocity) vector *u* at the mouth opening formed between the flue and the edge. This vortex core with the vorticity *ω* ð Þ ¼ **∇** � *U* is then considered to drive the air column in the pipe. The sound excitation by this periodic vortex shedding at the edge is controlled by the product of the aeroacoustic source term div ð Þ *ω* � *U* and the potential function representing the irrotational cross-flow *u* at the mouth.

This discrete-vortex model of Howe is successfully applied to analyze and evaluate both cavity-tone generation [9] and tone generation in flue instruments [10, 14] when the jet is thick and the condition *d*/*h* < 2 (*d* the width of the mouth opening or the flue-to-edge distance and *h* the jet thickness) is satisfied. On the other hand, when the condition *d*/*h* > 2 is satisfied for thin jets, a jet-drive model on the basis of the intrinsic jet instability [19, 20] is applied instead of the discretevortex model [21]. This jet-drive model has been developed in the field of acoustics [1, 2, 11, 20, 22–27].

Although the jet-drive model has been proven to be effective for an explanation of sound generation by the thin jet, there remain rooms for improvement in applying the vortex sound theory for another explanation of sound generation by the thin jet in flue instruments because small vortices may be produced along the boundaries by the mixing process between the jet flow and the surrounding still air. The boundary layer consisting of small vortices is called the *vortex layer*, which can act as the source of an accelerating force to oscillate the jet. Based on such a viewpoint, the vortex-layer model was proposed recently [28].

In Section 2, the jet-drive, discrete-vortex, and vortex-layer models are described. Acoustically induced vortices (simply, acoustic vortices) on sound dissipation are discussed with the aid of flow visualization in Section 3. The jet vortex layer on sound generation in an organ pipe is visualized by the particle image velocimetry (PIV), and the microstructure of the vortex layer is demonstrated in

with large amplitudes. Any synchronization is then required, and it is realized by the suitable *phase relation* between the flow (or the jet) driving wind (or flue) instruments and the acoustic wave propagating in the instruments. For example, in the case of flue instruments, if the air flow enters into the pipe at the instant when the acoustic pressure near the edge takes a relatively large positive value, acoustic power (given by the product of the alternating volume flow and the acoustic

*An organ flue pipe as a typical example of the musical flue instruments and its important parameters. d, the flue-to-edge distance (or cutup, jet length); h, the jet thickness (or height of the flue exit); R, the pipe inner*

*radius. The origin of the coordinate system is located at the center of the flue exit surface.*

However, a big dissatisfaction to the above viewpoint is the assumption of the existence of the sound at the starting point. Therefore, exactly saying, acoustical theory above is not sound generation theory but sound *regeneration* theory. The viewpoint of positive feedback between the jet and the pipe and the time-domain formulation based on the pipe reflection function [1] are both sound regeneration theory [2]. Musical instrument acoustics has treated such regeneration theories and phenomena as chief objects of research. This is because the resonance is acoustically essential, and we may consider that the resonance controls fluid movement as the energy source. It will be open to the charge of being imperfect combustion that sound existence is presupposed at the starting point when we try to answer how flue

Then, if we introduce a thesis, "the vortex itself is the true sound source," of the vortex sound theory [3] to flue instruments, is the problem solved? Flue instruments do not seem to be such an obedient subject. Certainly, the vortex sound theory is satisfactorily valid to the edge tone, where the jet-edge system has no pipe that gives the resonance or the acoustic feedback; instead the fluid-dynamical feedback between the edge and the flue (flow issuing slit) is a main mechanism of

Moreover, there are a few non-negligible differences other than the acoustic resonance between the edge tone and the pipe tone (or flue tone). First is the amplitude magnitude when the jet oscillates against the edge. The oscillation amplitude of the edge-tone jet is as small as two to three times the jet thickness. On the other hand, the pipe-tone jet in an organ pipe often exceeds 10 times the jet thickness. The edge in an organ pipe (or flue instruments) is just a partition wall which separates the inside from the outside of the pipe. It may be said that the direct jet-edge interaction time is quite short compared with a tonal period in flue instruments. Large vortices *visible* behind the pipe edge are, so to speak, odds and ends of the jet driving the pipe. We should pay more attention toward *invisible* (for

Second is the difference in the jet-edge configuration. The configuration is usually symmetrical in the edge tone. In other words, the jet center surface corresponds to the edge tip. Alternate small vortices continuously appear above and beneath the edge. On the other hand, the edge is usually displaced upward in organ

pressure) becomes positive, and the sound is sustained.

our naked eyes) vortices carried along the jet to the edge.

instruments produce their sounds.

*Vortex Dynamics Theories and Applications*

sound generation.

**44**

**Figure 1.**

Section 4, and some examples of jet vortices are also introduced from experiments and simulations. Conclusions are given in Section 5.
