Musical flue instruments such as the pipe organ and flute mainly consist of the acoustic pipe resonance and the jet impinging against the pipe edge. The edge tone is used to be considered as the energy source coupling to the pipe resonance. However, jet-drive models describing the complex jet/pipe interaction were proposed in the late 1960s. Such models were more developed and then improved to the discrete-vortex model and vortex-layer model by introducing fluid-dynamical viewpoint, particularly vortex sound theory on acoustic energy generation and dissipation. Generally, the discrete-vortex model is well applied to thick jets, while the jet-drive model and the vortex-layer model are valid to thin jets used in most flue instruments. The acoustically induced vortex (acoustic vortex) is observed near the amplitude saturation with the aid of flow visualization and is regarded as the final sound dissipation agent. On the other hand, vortex layers consisting of very small vortices along both sides of the jet are visualized by the phase-locked PIV and considered to generate the acceleration unbalance between both vortex layers that induces the jet wavy motion coupled with the pipe resonance. Vortices from the jet visualized by direct numerical simulations are briefly discussed.
- edge tone
- pipe tone
- jet wave
- jet-drive model
- discrete-vortex model
- vortex-layer model
- vortex sound theory
- flow visualization
- acoustically induced vortex
- direct numerical simulation
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 self-sustained 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
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 with large amplitudes. Any synchronization is then required, and it is realized by the suitable
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
Then, if we introduce a thesis, “the vortex itself is the true sound source,” of the vortex sound theory  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
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
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, 5, 6]. A thick (or semi-infinite) jet usually drives a wall-mounted cavity to produce its resonance called the cavity tone [7, 8, 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 . 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
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, 16, 17]. In the context of musical instruments, the
Howe  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
This discrete-vortex model of Howe is successfully applied to analyze and evaluate both cavity-tone generation  and tone generation in flue instruments [10, 14] when the jet is thick and the condition
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
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 Section 4, and some examples of jet vortices are also introduced from experiments and simulations. Conclusions are given in Section 5.
2. Models on sound generation in flue instruments
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 . The pipe length
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 vortex-like 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 .
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
where is the air density, the temporally varying cross section of the jet entering the pipe from the edge, and the pipe cross section. Eq. (2) defines the jet pressure of the jet momentum model, which was first proposed by Rayleigh  and utilized by many researchers afterwards [1, 2, 10, 11, 12, 13, 21, 22, 23, 24, 25, 26, 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  by deriving the so-called jet momentum equation and then simplified by Fletcher . They assumed a small control volume with length 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 . 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 , this phase difference , which gives the phase lag of the pressure drive, is given by:
where is the angular frequency and the effective mouth length including the open end correction. Since in usual cases, is quite small. However, as mentioned above, becomes appreciably large when the high blowing pressure is applied. See [1, 2, 10, 24, 25, 26, 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
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
where denotes the transverse displacement of the jet at distance
The envelope of positive and negative peak displacements is yielded from Eq. (4) as follows:
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 (mm), (mm−1), =2(rad/s), (mm−1), (m/s). The slowdown of is assumed so that
The jet displacements from
It is possible to directly estimate from Eq. (4) by applying it to the experimental data. However, such an approach needs exact information about
Digitally superposed jet waves are shown in frames (a), (b), and (c) for different blowing pressures in Figure 4, where the
In Figure 4(d) the estimated envelope is shown by dots on a template, that is, curves of jet envelope function for various values assuming that = 1. Figure 4(d) suggests a close fit of the dotted line to an envelope curve with = 0.24 mm−1 when the magnitude ratio of the dotted line to that curve is about 1.3. This ratio determines = 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:
The amplification factor tends to decrease and saturate to a given value as the oscillation of each mode shifts toward higher blowing velocities, although the data on the second mode are not sufficient.
The averaged amplification factor is roughly estimated as 0.24 mm−1.
The mouth-field strength , 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 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 .
In order to confirm the validity of our digital superposition explained above, was determined from measurements of the acoustic particle velocity with a hot-wire anemometer (its sensing part is 1-mm long and 5 m in diameter) and of the sounding frequency (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 jet flow for the measurement of
Comparing the values of 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 and 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, 33, 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 .
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.  and Verge et al. , 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 [)] and () whose temporal waveforms are rectangular pulses with the same amplitude . These sources are supposed to be placed at a distance from the edge tip at the lower and upper sides of the edge.
where is the air density,
In the limit of thin jets , where denotes the jet thickness at the flue exit, (the jet thickness at the edge) is assumed .
The power generated by the source is calculated by assuming that the source is in phase with the acoustic volume flow through the mouth opening, where denotes the particle displacement over the mouth opening . This is supposed to be a local two-dimensional incompressible flow. The above in-phase relation between the pressure source and the acoustic volume flow 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
In addition to the thin jet assumption , we have to suppose that the jet does not break down into discrete vortices. This is only reasonable for the first hydrodynamic mode (). 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  proposed a discrete-vortex 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 [in Eq. (9) corresponds to one-dimensional (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  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 and the potential function representing the irrotational cross-flow into and out of the mouth as expressed by the following integral :
where the vorticity is defined as and the velocity is the superposition of the jet mean flow velocity directing against the edge and the time-dependent cross-flow velocity that is specified by reciprocating potential flow through the mouth opening (see Figure 6). That is, is given by:
According to , the power density supplied from the acoustic field to the vortical field around the edge is given by:
Therefore, if , it may be said that the energy of the acoustic field is absorbed by the vortical field. As a result, an
This equation determines the rate of dissipation of acoustic energy, where in Eq. (10) is now simply denoted by . Also, denotes a volume enclosing the vorticity formed in the flow field. This 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.
where the vorticity is simply given as . If the time average is positive, the vorticity production from the jet flow supplies the acoustic power to the resonant pipe.
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  described both shear layers in terms of discrete vortices (see Figure 7). He used the jet with (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 was set to be 8 mm (). The orifice width was 28 mm, and the edge thickness 2 mm.
Meissner  found experimental results as follows: In stage I (, 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 (8.3), 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 .
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 . 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, may be approximated as that of an infinite street .
where is the distance between successive vortices in the lower and the upper line vortices and is the distance between both lines which is equal to the jet thickness. It is also noted that the circulation of the vortex increases linearly with the time according to .
In the case of flue instruments with thick jets () , a new vortex is formed at the inner shear layer (on the resonant pipe side) each time the acoustic velocity 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 later (acoustic pressure in the resonator takes the maximum). In the steady state of oscillation, the circulation of the
The acoustic power generated by the vortices is calculated by Eq. (14). The average of it over the oscillation period is as follows:
where the volume integration is taken over the source region of volume . The vorticity field takes into account the contribution of each vortex at the shear layers :
where defines a two-dimensional (2-D) point (), the position of the
It is first necessary to know the position and the circulation of vortices in order to calculate from Eqs. (20) and (21). It was done by time-domain simulations in . For the sake of paper space, see the details described in .
2.3 Flow visualization and some discussion
2.3.1 Jet-wave drive vs. discrete-vortex drive
Dequand et al.  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 flue exit, a common flue-to-edge distance
Frames in the left column [(a), (c), and (e)] show flow conditions at the phase of (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 (the instant from the negative to the positive). Note that the positive
The jet-wave drive illustrated in Figure 8(a) reveals that the jet enters into the pipe at the instant when the acoustic pressure
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
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 in Eq. (20) is upward (positive
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.,
where 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):
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 . At that instant, the initial transverse displacement of the jet at the flue exit is supposed to be non-zero and related with the acoustic velocity at the flue exit as follows [38, 40]:
which yields the acoustic pressure at the pipe entrance according to Eq. (7) in which 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 denotes the time delay of the jet wave when it travels from the flue exit to the edge (.
The acoustic pressure drives the pipe and yields its resonance. As a result, at the starting point is fed back through the input admittance of the pipe. The Fourier transform of is thus given by:
where is the Fourier transform of and is assumed by neglecting the jet spreading for simplicity . The feedback loop gain is thus defined as
Hence, the phase condition for the self-sustained (feedback) oscillation is:
That is, the time delay of the jet wave must satisfy:
where is abandoned because is always positive and (= 0, 1, 2, ….) denotes the hydrodynamic mode number. Usually, sound generation in flue pipes occurs for . Since at the pipe resonance (see Figure 9), we finally have:
for the first mode . 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 , respectively. As explained in Section 2.3.1 the convection of these two vortices may create the acoustic power defined by Eq. (20) for sound generation in flue pipes. Thus, the time delay of vortex convection in the discrete-vortex model is:
Although is easily derived from Figure 9(b), it will be desirable to consider 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
2.3.5 Aspect ratio d/h of the jet
The aspect ratio
Dequand et al.  calculated as a function of
2.4 Vortex-layer model
Howe  and Dequand et al.  proposed the discrete-vortex model driven by thick jets (
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
Helmholtz  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  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 . 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 (
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 . The delay in reaching thermal equilibrium is called relaxation, and its time constant is called the relaxation time . The source of molecular absorption in air is oscillatory relaxation of oxygen. The relaxation frequency (defined as ) 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 ). The acoustic dissipation in moist air is significantly greater than the classical absorption given by .
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 (see Figure 3.10 in ). This characterizes the thickness of the wall boundary layer and is given by
where is the skin depth and the equation of the right-hand side is for the air with the kinematic viscosity = 1.5 × 10−5 m2 sec−1 .
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
where denotes the viscosity of the air, the specific heat at constant pressure, the thermal conductivity, and the ratio of specific heats. The ratio of thermal loss to viscous loss is given by for the air.
The attenuation constant in total is given by adding in Eq. (36):
where is approximately evaluated in and . The conversion is done by the relation based on the exponential decay. For example, the modern flute with indicates for . 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 . This large but still small enough magnitude of 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 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 (denotes the wave number and ) and the source strength of radiation at each open end is the same, we have the value of
where denotes the time average of the power lost from the bore by sound radiation and the time average of the power stored in the bore. For
where indicates the attenuation constant due to sound radiation. Also, the total defined by Eq. (38) is equal to . Since
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 . 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 150 Pa
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 . 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 is in the negative
Although the jet deflection shown in Figure 12(c) is negative, the jet might be moving upward [the phase of may be around in Figure 9(a)] and then as well as 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 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, 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 . Instead of that, we observed a steadily deflecting jet, particularly its penetration into the pipe as captured in Figure 13 in . 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 . 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].
3.2.2 Acoustic power balance between vortex layers and acoustic vortices
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 acoustic vortices in the sense of time average:
where is given by Eq. (14) with concerning the vortex layer, is the power lost from the bore that is given by the total attenuation constant of Eq. (40), and is given by Eq. (13) with concerning the acoustic vortex. A more exact description of derived from the unbalance between the upper and lower vortex layers will be given in Section 4.1.
3.2.3 Acoustic losses due to vortex shedding at the edge
In the framework of the jet-drive model, Dequand et al.  assumed that the separation of the acoustic flow occurs at the edge by following Verge et al. . This acoustic flow separation causes a free jet .
Although they neglect the effects of the separation of the jet flow and their viewpoint is different from the modeling illustrated in Figure 13, it seems to be worth taking into consideration. The effects of vortices can be represented by a fluctuating pressure across the mouth [12, 21]:
where (= 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 .
where it is assumed that the dissipation occurs during the entire period
Therefore, the power dissipation given by Eq. (43) may be considered as an upper limit approximation, and by neglecting 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 of and an appropriate division of
This interesting non-dimensional relation was almost confirmed by the experiment on thin jets () for the four different flue-edge geometries (see Figure 11 in ). The maximum of non-dimensional amplitude reached for the edge with an angle of is 20% higher than that obtained for the edge with an angle of . This difference in amplitude can reflect the difference between the flute and the recorder. The recorder with a sharper edge probably brings about stronger losses due to vortex shedding at the edge.
4. Vortices on sound generation
In this section let us consider what the cause of the jet oscillation is for thin jets (). Fletcher’s displacement model of Eq. (4) [1, 31] has no definite physical basis, and Coltman’s velocity model [22, 23] lacks in quantitative analysis. The present author proposed an acceleration model based on the pressure difference between the upper and lower surfaces of the jet . Although this model could not involve the effects of the jet instability , it could successfully predict the possibility of underwater organ pipes . Therefore, another acceleration model based on the vorticity generation is greatly expected .
4.1 Vortex layer along the jet visualized by PIV
A great advantage of PIV is to yield global and quantitative information on the flow-acoustic interaction. The PIV was already successfully applied to the experimental research of the edge-tone generation , where the complicated jet-edge interaction was investigated to accurately localize the vortex cores (dipole sources) just before the edge. Also, it was applied to measure the flow velocity and acoustic particle velocity . Measurements of both quantities are required to consider sound generation based on the vortex sound theory.
4.1.1 Measurement requirements
Since the vortex sound theory hypothesizes an irrotational potential flow for
Both requirements of (1) a potential flow and (2) a sinusoidal flow for
Since , it is very difficult to measure the distribution of over the mouth area using the PIV when the pipe is driven by the air jet. Therefore, both measurements of and should be separately carried out. Of course, cannot be measured without using the jet. On the other hand, can be measured by resonating the pipe externally, for example, by using an inverse exponential horn [28, 49]. A larger cross section of this horn is firmly fitted to the loudspeaker diaphragm, and a smaller cross section is coupled to the pipe end with a distance larger than the end correction to maintain the resonance pattern of the pressure distribution along the air column. The loudspeaker is driven by an oscillator to generate a sinusoidal wave in the pipe with the same frequency and amplitude as those when the jet drives the pipe. The organ pipe is thus driven by this loudspeaker horn system when is measured.
Also, in order to experimentally examine the generation of the vortex sound based on Eq. (14), both measurements of and must be carried out at the same condition as exactly as possible. That is, these vectors must be measured at the same phase of the generating sound and at the same measurement area by using the same organ pipe. However, since and cannot be measured simultaneously, the jet drive and the loudspeaker horn drive must be switched as quickly as possible while maintaining the same sounding condition and the same measurement condition. The phase-locked PIV measurement on and (see Figure 14) is thus essentially important to evaluate Eq. (14). Since , should be first measured using the horn drive at a given phase of the sound, and the is measured at the same phase by quickly switching the horn drive to the jet drive.
4.1.2 Measurement procedures
PIV measurement of and was carried out twice (Trials 1 and 2) in . The fundamental frequencies of the pipe tone were 192.0 Hz and 192.1 Hz, respectively (the cutoff frequency of the horn was designed to be about 150 Hz). The averaged sound levels were 59.0 dB and 59.3 dB, respectively. When the averaged sound level was 57.8 dB (the
The phase lock of the PIV system is easily implemented if the external trigger signal is produced to activate the laser and the CCD camera. This is because the PIV system can set the trigger delay almost arbitrarily through the software embedded in the trigger signal production system shown in Figure 14. The trigger delay was set to be (1/12)
In the experiment a metallic organ pipe, which was made by a German organ builder, was measured . Its cross-sectional structure (in
4.1.3 Calculation of the acoustic generation formula
The PIV can derive the vorticity map from the jet velocity distribution. The vorticity at a field point is calculated from the 2-D velocities at four discrete points surrounding the point of interest. Therefore, the aeroacoustical source term and the acoustic power generation term can be calculated from the measurement of velocity fields and (see  on their measurement results, which are spared in this chapter).
The vorticity map given at Trial 1 is illustrated in Figure 15(a). Since the 2-D velocity was measured in
The vorticity is formed along the upper and lower boundaries of the jet. The upper layer possesses the positive vorticity (the counterclockwise rotation of small vortices) and the lower layer the negative vorticity [cf. Figure 10(a)]. These layers may be called
The resulting aeroacoustical source term is displayed in Figure 15(b), where the upper and lower layers of the vorticity yield the positive
4.1.4 Generation of the acoustic power from the vortical field
Acoustic generation term defined by Eq. (14) is shown in Figure 15(c). Since indicates the outflow at Phases 1 and 4, and inflow at Phases 7 and 10, takes the opposite sign along the vortex layer between these phases. This sign inversion occurs near Phases 0 and 6 . The maximum magnitude of appears near 1–2 millimeters downstream from the flue at Phases 3 and 8 as about . The jet crosses the edge from the inside at Phase 3 and from the outside at Phase 8 as inferred from Figure 15(c). Although the magnitude of is relatively small near the edge, it should be noted that originally has very large values in acoustical sense.
Since the volume integral defined by Eq. (14) is not easily executed, the acoustic power generation from the vortex layer is estimated from the following surface integral by assuming the 2-D property (see Figure 11 in ) of and :
This surface integral, which may be called the
Significant double-peak structure of is clearly demonstrated in Figure 16(a). A larger peak is indicated at Phases 2, 3, and 4 when the jet crosses the edge from the inside and moves to the outside (cf. Figure 15). On the other hand, a smaller peak is shown at Phases 10 and 11 when the jet enters deeply into the pipe. It should be noted that the jet crosses the edge from the outside at Phase 8 and almost null vortex sound power is generated at Phase 8. Hence, this smaller peak occurs in a little phase delay from the impingement of the jet against the edge. The temporal average of estimated from the 2-D vortex sound power in Figure 16(a) will take a definitely positive value. Therefore, it may be recognized that the acoustic power is generated from the jet vortex layers. Although Figure 16(b) shows the characteristics similar to those of Figure 16(a), the value at Phase 3 seems to be too large and erroneous because of the instant when the jet impinges against the edge .
4.1.5 Dominant area for the acoustic power generation and receptivity problem
The maps of the vorticity and aeroacoustical source term definitely indicate much larger magnitudes at the flue side as shown in Figure 15(a) and (b), respectively. On the other hand, the acoustic flow velocity takes much larger magnitudes at the edge side (see Figures 5(a) and 7 in ). It should be then discussed which side is more dominant for the acoustic power generation.
The area for the surface integral of Eq. (47) is now set to be 2mm and mm. Then, this area is divided into two at 7, 8, and 9 mm. Hence, we have six sub-areas with the same extent. The calculation result is demonstrated in Figure 12 of . A very sharp contrast is displayed between area 5 (2mm) and area 6 (9mm): Area 5 yields larger negative values of at Phases 3, 4, and 5; area 6 yields much larger positive values of at the same phases. Hence, it may be concluded that such a small area as area 6 (very close to the edge) is most responsible for the acoustic power generation whose instantaneous contributions are given from Phases 2 to 5.
Also, the phase relation between the jet displacement, the acoustic velocity, and the acoustic pressure at the edge can be considered based on the PIV measurement results. The result is the same as Figure 9(a) (see Figure 13 in ). The dominant sound generation in our PIV experiment occurs with a phase lag of about from the jet impingement against the edge . This seems to verify that our experiment satisfies the requirements for the volume-flow model.
Coltman  discussed the activating force for the jet wavy motion. This is the most difficult problem in the flue instrument acoustics and is defined as the problem of the
Since this can also activate the jet motion in the edge-tone generation [3, 4, 5, 6, 18], the vortex-layer formation may be regarded as the fluid-dynamical mechanism common to the edge-tone generation and the pipe-tone generation. This fluid-dynamical model, which is a leading candidate to solve the receptivity problem, may be referred to as the
4.2 Vortices from the jet visualized by direct numerical simulations
Sound generation in flue instruments is the revelation of the fluid
For the sake of page limitation, the description here is restricted to an essential point given by Eq. (24), which manifests the importance of the acoustic velocity at the flue exit. By reformulating Fletcher’s displacement model given by Eq. (4), Onogi et al.  proposed another formula that decomposed the jet oscillation into hydrodynamic and acoustic displacements, which were simulated on the basis of the 3-D compressible Navier–Stokes equations. They supposed the non-zero initial amplitude at the flue exit and the variable oscillation center with the flow direction for the jet displacement, although Coltman  strongly denied Fletcher’s displacement model. Their simulation results (see Figures 7, 9, and 10 and Table IV in ) seem to confirm the non-zero amplitude at the flue exit, and the acoustic feedback effects on the jet wave may be given at its starting point.
Vortices on sound generation are clearly revealed in edge tones (with thin jets, without any resonators) and cavity tones (with thick, almost semi-infinite, jets, with cavity resonators). Although visible, relatively large vortices are seen in flue instruments driven by thick jets, these are in rare cases. Usually flue instruments are driven by thin jets [(jet length
The jet-wave drive (or the volume-flow drive) and the vortex-layer drive by thin jets assure sound generation in good manner when the jet enters into the pipe at the instant when the acoustic pressure is maximum. In the discrete-vortex drive by thick jets, the acoustic cross-flow (particle velocity) takes positive and negative values during the passage of the lower and upper vortices from the flue exit to the pipe edge, respectively. These vortex configurations can create sound power during the former and latter halves of an oscillation period.
On the other hand, acoustically induced vortices universally appear as the final dissipation agent. Their role in acoustic energy balance near the saturated state in flue instruments should be reconfirmed in more detail to exactly judge whether the acoustic vortex is generated just at the saturated state or just before the saturated state (at the pre-saturated state).
The receptivity problem is a key point to elucidate the sounding mechanism in flue instruments from the fluid-dynamical viewpoint. The initial amplitudes of acoustic quantities at the flue exit are regarded as the starting point for the acoustic feedback effects upon the jet wave. The vortex-layer model above will then be expected to solve this problem with the aid of direct aeroacoustical simulations.
The present author expresses his appreciation to three European scientists: Dr. Judit Angster of Fraunhofer-Institute fur Bauphysik, in Stuttgart, for her long-term support to carry out the PIV measurement; Prof. Avraham Hirschberg of Technishe Universiteit Eindhoven for his kind offer of the picture used as Figure 12(c) and helpful comments to the author’s journal papers from the aeroacoustical viewpoint; and Prof. Andreas Bamberger of Freiburg University for his effective comments and suggestions on the PIV. Also, the author thanks Keita Arimoto and Takayasu Ebihara of Yamaha Corporation, in Hamamatsu, Japan, for their sincere support and appropriate comments to the manuscript.