Phenomenon Of Sound Generation By Airflow Biology Essay

Abstract:

A phenomenon of sound generation by airflow through corrugated pipes is not only the working principle of a children’s toy called “Hummer”, “Voice of the Dragon” or “Magic Whistle”, but it can also be a severe problem in certain domestic appliances involving corrugated tubes (ex: vacuum cleaners), and in other fields of engineering such as “singing risers” in oil and gas industries, chemical industries HVAC, aircrafts and automobiles. This review paper focuses on the research work carried out till date to study the sound generation mechanism and its reduction methodology if any in corrugated pipes with air flow. This paper summarizes the various theoretical, experimental and computational work carried out in acoustics of corrugated pipes.

1. Introduction and literature survey

Air flow through a short or long length of corrugated pipes can cause the pipes to emit loud and clear “tonal” sounds. This is known as “whistling”. The production of these tones is interesting in a sense that the same flow through a smooth pipe of similar geometry will not produce clear tones. Pipes with transversally corrugated walls are used in many industrial and domestic applications because they offer globally flexibility combined with locally rigidity. Some of the engineering application of corrugated pipes are “Flexible risers” in oil and gas industry, vacuum cleaners, HVAC control systems of heating ducts in buildings, aircraft cabin conditioning system, liquid propulsion system in rockets and compact heat exchangers. Corrugated pipes are also used as musical toys such as “Hummer” or “Voice of the Dragon”. Whistling induced by flow through such pipes can lead to serious environmental and structural problems.

“Singing” of corrugated pipes was first observed by Burstyn1, and who used a helical pressing and found that the pitch of the note was a harmonic of the fundamental and was dependent on the air velocity. He presented a short technical note with a title “a new pipe” at the meeting of German Society for Technical Physics. He suggested that the corrugations act as “numerous lips” which produce the sound. A more detailed experimental study on this subject was reported by Cermak2. He experimented with 1 m and 0.5 m long tubes of narrow (10 mm outer diameter) bore, and reported that the fundamental tone was difficult to excite, but when extracted as the difference between consecutive higher partials, it was found to be lower than the resonance frequencies expected from smooth pipe of same geometry. He ascribed it tentatively that the “air flow” rather than the “tube shape” is responsible for the reduced resonant frequency of the smooth tube. He also observed a “jump in tones” phenomenon in which a tone will remain constant over a flow velocity interval, and then jump to the next harmonic once a certain velocity is exceeded. Cermak also calculated a “bump frequency” based on flow velocity and corrugation pitch and found this to be close to the generated tones. In a further paper3, he studied the sound produced by a tube obstructed with washer and the produced sound appears to have been a kind of “bird call”. The possibility of resonance exists in an arrangement similar to corrugated pipes was also noted by Mason4.

The effect of transverse harmonic corrugations in the bed of an open water channel is a classical problem which was solved by Kelvin5. With the assumption that the amplitude of the disturbance is infinitely small, he showed that on the free surface a pattern of stationary waves is formed, and this result is confirmed by common observation. Later, Binnie6 studied the consequence of corrugating the sides of the water channel; and when an experiment was made for the first time, it was unexpectedly found that, if the velocity and the depth were properly adjusted, a continuous train of progressive waves appeared and moved steadily upstream into the reservoir from which the channel drew its supply. That a stream confined by fixed boundaries can be disturbed by self-induced travelling waves is similar to the earlier phenomenon observed when the air flow through corrugated pipe. The stream velocity at which self induced waves appeared for single corrugation and multiple corrugation proved the hypothesis that waves were dependent upon the pitch was plausible as noted by Binnie6.

After investigated the effect of the corrugated walls on the water which produced the self induced wave, Binnie7 also carried out the experiments with air on the metal bellows. Binnie likened the flow through the corrugated tubes as converse of an exhaust silencer because unlike the silencer the tubes produce and strengthened the sound waves. More powerful notes were obtained during his experiments when the metal bellows inside the pipe were replace by helically finned tubes and also in the case of when air was passed outside the corrugated or finned tubes. He noted that the phase velocities of corrugated pipes were less than the velocity of sound and this effect was explained by applying the theory of wave propagation in periodic structures. The pressure distribution in this arrangement was examined and found that the ratio of average frequency at which an air particle passed the perturbances, to the frequency of the sound emitted was usually 1.6.

The papers by Crawford8, Silverman et al9, Cadwell10, Nakamura et al11,12 and Serafin et al13-15 were inspired by the musical toy called “voice of the dragon” and report further information on the frequencies heard and associated flow velocities as well as required turbulence levels and characterizing Strouhal numbers. Although Crawford8 did not refer the work of Burstyn1 or Cermak2,3 in his paper, he also introduced a term “bump frequency,” (the frequency at which air bumps into the corrugations) and stated that the pipe start singing when the bump frequency equals a natural harmonic of the pipe provided also that the flow velocity is sufficiently high to induce turbulent flow, the same observations reported by the other researchers1-3. Crawford8 concluded that Reynolds number of minimum 2000 was required, which is a threshold value for generating turbulence in smooth pipe, to cause the pipe to sing. But he also observed that for some pipes, it start singing for much smaller value of Reynolds number. Based on his experiments, Crawford8 proposed a theory for predicting the singing in a corrugated tube based on frictionless flow, turbulence and Bernoulli’s principle. By testing tubes of different designs and geometry, Cadwell10 also finds that a Reynolds number based on the corrugation length, must exceed 500 for narrow bore pipes to sing. Cadwell10 interpretated that this as a minimum turbulence level of this scale is required but this prediction of Reynolds number indicates that the flow is in laminar region with very small scale turbulence. Hence the work of Crawford8 and Cadwell10 implied that the flow may not require to be fully turbulent for the corrugated pipe to start singing.

Petrie et al16 studied the noise problem encountered in a vacuum cleaners which has flexible corrugated pipes. They observed that, a low velocity air traveling through the flexible tube can generate a noise at the duct outlet in the region of 140 dB. They demonstrated that roughening the cavities of corrugated pipe, i.e, placing an obstruction closes to the flow separation region (close to the cavity) or introducing smooth pipe at upstream of corrugated region would reduce the noise. They reasoned that, the avoidance of noise generation appears to be centered around the changing of the turbulence structure in the system closer to the corrugated wall cavity.

The acoustic resonant modes of a corrugated tube open at both ends and rotating in a plane were examined experimentally by Silverman et.al9. He carried out the experiment in a rotating apparatus based on Bernoulli’s principle to measure the relation between rotational velocity and resonance frequency of the tube. He observed the similarity between the sound generation in rotating the corrugated pipe to the electromagnetic waves generation in a Smith-Purcell17 light source where by the image charge of an electron beam races grooves in metal surface placed in an optical cavity.

Nakamura et al11,12 carried out wind tunnel measurements to study the acoustic oscillation in a corrugated tube open at both ends, with air flowing through the tube. They concluded that the sounds that the tube emits are the natural harmonics of the tube, and resonant modes are excited by the impinging shear layer instability that occurs in the flow over corrugations. The resulting instability is characterized by a self-excited resonant oscillation occurring in the fluid-acoustic coupled system. They suggest that the acoustic oscillation in a corrugated tube is driven by vortical sound sources in accordance with theories of Powel18 and Howe19.

Experimental work similar to Nakamura et al11,12 was also reported by Hammache et al20, where they studied the sound generation mechanism of various corrugated geometries in a wind tunnel using hot wire probes and microphones. They carried out the parametric study to figure out the minimum number of corrugation required to cause the tube to emit sound. One of the surprising results they reported was, presence of even one bump (two bumps required to form the cavity) placed at the downstream end, the place where pressure node set in is enough to cause the tube to sing. The level of sound pressure observed for the one bump in smooth pipe is comparable to the sound produced by the full corrugated pipe of same length and cavity geometry. Hammache et al20 mapped the pressure and velocity standing waves patterns formed during the resonant condition at the second fundamental mode since they also found that the fundamental does not sing. They showed that the pressure node is a place where sound generation occurs which was later confirmed by Tonon et al21,22 and Nakiboğlu et al23. In their experiment, Hammache et al20 noted the presence of vortical structure during pipe singing condition (resonant mode) which was also studied in detail by Peters24 and Kriesels et al25. Hammache et al20 also looked in to the sound suppression mechanism in corrugated pipe and they also explored the application of active control methods.

Taylor et al26,27 studied the acoustics of sound generation in music toy called “Magic whistle” and this new music toy tube geometries smaller in comparison with “Voice of the dragon. The musical notes in magic whistle can be produced easily by simply blowing the air through the tube unlike the Voice of the Dragon. Blowing through the Voice of the dragon is very difficult because the tube diameter is very large. Some of the interesting observation they made was increasing the height of the corrugation would lower the fundamental frequency, as will the increasing the corrugation width with respect to the spacing in some cases. Impedance measurement and insertion loss measurement carried out by the Taylor et al26,27 reveals that, the presence of corrugation affects the acoustic impedance and insertion loss value and corrugation height is the parameter which affects the both values very significantly. The authors also reported that for some industrial grade pipe geometries the singing frequency is more than the smooth tube frequency which was unusual and opposite of the other observations reported in the literatures.

Serafin et al13-15 combined the rotating corrugated tube model of Silverman et al9 and digital wave guide model of Smith28 and synthesized the corrugated tube resonances in virtual music production. They implemented a computer algorithm based on Doppler shift effect. The simulation Doppler shift for virtual acoustic environment was studied by Takala et al29 and Savioja et al30. The simulation of Doppler shift effect for Leslie horn was proposed by Smith et al31, which uses time varying delay lines32. Since there is a strong similarity between the rotation of Leslie horn and corrugated tube, Serafin et al13 used the same algorithm which was used earlier for simulation of Leslie horn. The virtual corrugated tube model based on Doppler shift effect has been implemented as an extension to the real time environment Max/MSP platform33. This virtual model controls the virtual choir of tubes using the fiddle and pitch tracker34 of Max/MSP audio. Hence, this work by Serafin et al13-15 shows that, the sound produced by corrugated tubes not just having detrimental effect such as noise in vacuum cleaners, but it can be used to produce music and in real time audio analysis.

A recent contribution to the acoustics of corrugated pipes is presented by Elliot35. His experiment on various corrugation pipe geometries confirmed that the resonant frequency of self excited oscillation is indeed less than the resonant frequency of equivalent smooth pipe. He noted that the fundamental mode is difficult to excite and disturbances to flow at the entrance of corrugated tube is little importance in the sound generation process. The observation of effect of inlet condition on resonant frequency also reported by Petri et al16 and Hammach et al20. Elliot35 proposed a sound generation mechanism in corrugate tube based on the noise due to single cavity oscillations.

Sound generation of flow over single cavity is well understood phenomenon and numerous literature available on this subject36-56.The observed flow-acoustic coupling in corrugated tube is similar to single cavity flow-acoustic coupling which was observed by Bruggeman57-60. This flow acoustic coupling involves an acoustic dipole source which causes an unsteady pressure differential across a free shear layer, giving rise to vortices that are convected with mean flow and interact with sold pipe surface leads to structural vibration. Analyzing the shear layer movement in the cavity of corrugation, Elliot35 recognized that, the real part of the admittance (which is reciprocal of the input impedance) presented by the cavities to the pipe flow must be negative for any sound generation to be occurred. A theoretical model for the sound propagation in corrugated pipes by Cummings is also detailed in the same reference of Elliot35. Cummings assumes that, the effect of the cavities to be similar to compressible springs presenting a reactance to the acoustic field along the cylindrical boundary. By assuming an idealized cavity shape, a formula predicting the corrugated pipe’s acoustic resonances was proposed by Cummings35.

Kopiev et al61 studied the effect of external sound field introduced to the corrugated tube with flow. They observed the strong decay of sound for Strouhal numbers below 0.4 and sound amplification at Strouhal numbers above 0.4. Testing of corrugated pipes with more marked differences between cavity length and pitch indicates that the cavity length is a better parameter to use to define the Strouhal number21,62.

The structural problem encountered in flexible risers of offshore natural gas and oil production platforms due to whistling63 forced the acoustical engineers to understand the problem deeply and how to solve them effectively. Tonon et al21, 22,64, Niagobu et al 24, 65 and Dequand et al66 modeled the corrugated pipe as series of side branch resonator. Radavich et al67 used the computational approach to solve acoustic problem in side branch resonator. Tonon et al21 utilized the Cummings acoustic models on the multiple side branch resonators to predict the low frequency resonance modes and used the plane wave acoustic model to predict the high frequency resonance modes and compared with experiments. They proposed a model for the whistling behavior based on energy balance approach and this proposed model predicts the resonance frequencies accurately but the amplitude of sound pressure with lesser accuracy. But this is the only model (energy balance approach)64 available in literature, which would predict the acoustic pressure amplitude. They also observed that the sound generation regions are at pressure nodes which are converse to the experimental work carried out by Kristiansen et al68 where he predicted that the source of noise is at the pressure anti node region. Experimentally observed Strouhal numbers during whistling for corrugated pipes are in the region of 0.32 to 0.5, but for the side branch resonator it is in the region of 0.5 to 0.623,65. For multiple side branch resonators it was showed that the pipe termination geometry is one of the design parameter. Another interesting result the authors23 noted on side branch resonator is, the shape of the upstream edge will play an important role in pressure amplitude. It has been observed that the rounding of upstream edge increases the pressure fluctuation amplitude to 5 times and shape of the downstream edge of the cavity did not having an effect on the pressure amplitude but this observation is may not be applicable to corrugated tube11,12. They23 also reported the effect of cavity depth, gradient in side branch depth on resonant frequency and pressure amplitudes. For multiple branch resonators, the author observed the hysteresis at very high rate of change in flow velocity, which was in consistent with the observation earlier made by Petrie et al16 for flow through corrugated pipes.

The paper by Blackburn et al69-71 studied the effect of corrugation height on flow in wavy walled pipe using CFD- Direct Numerical Simulation (DNS). In this study, an acoustical aspect of corrugated pipe is not considered, but the friction values near the corrugation is computed. Popsecu et al72 used the commercial CFD software Fluent to study the wave propagation in low Mach number flow using Large Eddy Simulation (LES) approach, and in fact the first people who used commercial CFD software is Taylor et al but they used steady “k- ɛ” model which less accurate than the unsteady LES or DNS model. Popsecu et al72 demonstrated that the acoustic source in corrugated pipes is dipole which was earlier proposed by Elliot’s35. They72also able to capture the pulsating vortex near the cavity and standing wave pattern in the corrugated pipe.

Nakiboğlu et al73 carried out the numerical investigation to understand the effect of the ratio of cavity depth (H) to pipe diameter (D) in acoustics of corrugated pipe. Combining the total enthalpy differences across the cavity, which was calculated from the numerical simulations, and vortex sound model, an equivalent time averaged acoustic source power is determined as a function of the Strouhal number for different cavity geometries and pulsation amplitudes. This model73 accurately predicts the increase in peak whistling Strouhal number observed earlier by Binnie7 upon increase of the cavity depth to pipe diameter ratio.

Another interesting paper which is not exactly dealing with flow through corrugated pipe but similar phenomenon of generation of acoustic waves in pipe are studied by Bonneau et al74. In that paper, the authors experimentally showed that contrary to ordinary fluids low Mach number granular pipe flows are linearly unstable towards the emission of acoustic waves. Exponential amplification of acoustic waves in the direction opposite to the main flow was observed and the same observation has also been made by Binnie6 and Kristiansen et al68 on flow through corrugated pipes. Bonneau et al74 study shows that a small change in the roughness inside of the pipe is sufficient for sound generation.

Tonon et al75 studied the reason for fundamental mode of corrugated pipe is not excited . The one of the explanation of missing fundamental mode is the visco-thermal losses high at the lowest resonant mode. Nakiboglu et al76 studied the effect of pipe length, flow profile on the whistling of corrugated pipe and found that the peak-whistling Strouhal number, where the maximum amplitude in pressure fluctuations is registered, is independent of the pipe length.

The main focus of this review paper is to compile the research work carried in acoustics of corrugated pipes till date. The review summarizes in detail, the theories which are used to model the corrugated pipes, experimental work carried out to understand the sound generation mechanism and numerical, computational work performed on the corrugated pipes to understand the flow physics. Also noise reduction mechanism proposed in the literatures will be discussed.

The organization of this paper is as follows. The theories which are used to predict the resonant behavior and sound pressure amplitude are given in section 2. The experimental work performed on the corrugated pipe to understand the acoustics of corrugated pipe is explained in section 3. The computational and numerical work on corrugated pipe is given in section 4. Section 5, discuss about the noise reduction attempt carried out in corrugate pipe. The conclusion and references for this review are given in section 6 and 7. In this work, only acoustics part of the problem is reviewed and flow-structure interaction is not considered.

2. Acoustics of Corrugated pipes: The theory

This chapter reviews the theories available in literatures, which are used to describe the sound generation mechanism in corrugated tubes. The musical toys made of corrugation are shown in Fig. 1. Corrugated pipes which are used in vacuum cleaners, aerospace, automobile industries and flexible risers in offshore oil and gas industries shown in Fig. 2. The literature review reveals that most of the researchers explained the sound generation mechanism in corrugated pipes predominately using the following three theories. They are (1) Bernoulli’s principle, (2) theory of or vortex sound which is also called as Cummings Acoustic Model (CAM), and (3) energy balance acoustic model (EBM). The following sections will describe each model and associated equation used by researchers to predict the resonant frequencies in corrugated pipes.

2.1. “Bernoulli’s Principle” for Measuring Resonant Frequencies in Corrugated Pipes

Application of Bernoulli’s principle for acoustics of corrugated pipes is first proposed by Crawford8 and later it has been confirmed by Silverman et al9 and Serafin et al15 with their experiments. This theory works well for the “singing” of rotating musical toys such as “Voice of the dragon” and “Hummer”. This theory can be used for non rotating type of corrugation as well provided one should know about the friction coefficient due to the corrugation. The main focus of the theory is relating the rotational velocity with pipe resonance. It is not possible to explain the flow physics such as vortex shedding closer to corrugation using this theory.

For an ideal open-ended tube, the resonant modes are given by the following expression:

(1)

for n = 1, 2, 3…. where c is the speed of sound and L is the tube length. Since the air is moving in and out of the tube, the effective length L′ should be used instead of L. The effective length of

given by3:

(2)

where r is the radius of the tube.

If both ends of the tube are open, the rotating corrugated tube resembles a centrifugal pump. When whirling, the air is sucked in through the end closer to the hand and pushed out through the outer end. In order to make the vibrational modes resonate and thus produce pitch, some of the airflow energy is converted to excitation energy.

At a large scale there exists a vortical flow centered in the stationary end of the tube and normal to the axis of the tube with tangential air velocity given by,

where ωm is the angular velocity and s is the position along the tube (from 0 to L).

Along the tube, the rotationally induced pressure difference between the two ends produces an axial flow with velocity υ. In order to relate the angular velocity ωm to the pressure difference p, assuming the flow to be incompressible and smooth, we can use Bernoulli’s principle:

(3)

where p(s) is the pressure and υ(s) is the axial velocity at position s. With the same assumptions, the axial velocity through the tube is uniform, i.e. υ(s) = const. for 0 ≤s ≤ L. Therefore the pressure difference between the two extremities of the tube (stationary= 0 and rotating = L) is given by,

It is now necessary to explain the role of the corrugations. This is done by relating axial air velocity υn to distance between corrugations d as follows:

(4)

where α1is a proportionality constant. Equation 4 is justified by the fact that moving past the corrugations the air is perturbed at a frequency proportional to the axial air velocity and inversely proportional to the corrugation spacing. When fn coincides with a resonant frequency of the tube, the sound is amplified.

To relate the axial velocity υ to the tangential air velocity V(s) Bernoulli’s equation given in equation 3 is used, which gives υn = ωmnL = V (L), where ωmn is the angular velocity of mode n. This equation states that, in case of frictionless flow, the axial velocity along the tube is the same as the tangential velocity at the rotating end. However, since the flow is not perfectly frictionless, it is assumed that the axial air velocity is proportional to the tangential air velocity. i.e,

(5)

where α2 is another proportionality constant (experimentally, however, α2 = 1). Combining equations 4 and 5, one gets,

(6)

which states that the modes of the rotating tube are directly proportional to the angular velocity and inversely proportional to the distance between corrugations. Since the flow inside the tube is not perfectly frictionless, it can be shown that friction generates a non uniform flow and the effect of friction is to give α1= 2. From equation 4 it can be shown that,

(7)

The previous equation states that knowing the axial air speed and the distance between corrugations, it is possible to calculate the frequency at which the tube resonates.

2.2. Cummings Acoustics Model (CAM) for Corrugated Pipes

In case of zero mean flow with M=0, by solving the Helmholtz’s equation in polar coordinates, one can show that the real part of the wall admittance (Re (β)) of smooth pipe is a parameter responsible for sound pressure amplification or decay 35,61. Cummings35 in his model, assumed the effect of the cavities in corrugated tubes to be similar to compressible springs presenting a reactance to the acoustic field along the cylindrical boundary.

A simple model of corrugated flow presented by Cummings35 is as follows. Let the corrugation be rectangular cylinders, having a cavity length lc and cavity depth dc, and a corrugation pitch pc as shown in Fig. 3. Let a basic uniform flow, for 0 < r < R be separated from one at rest, for R < r < R+dc by a shear layer of zero thickness. We shall then assume that the fluctuations do not vary across the pipe cross section but allow a small radial particle velocity at the shear layer by considering a control value indicated by the dotted lines, whose cross section has a perimeter P=2πR and n area A=πR2, it can be shown that in the interior of the pipe, the Euler equation have a quasi 1-D form,

(8)

(9)

Where 𝜐′ is the space averaged radial velocity. Combining these two equation result in a single equation of the form,

(10)

Taking ξ’ to be the particle displacement and assuming the pressure and radial particle displacement to be continuous across the shear layer, gives

(11)

Where υω′ is the radial velocity within the corrugations. Taking,

(12)

results in,

(13)

moreover since we have,

(14)

then for a prescribed space averaged wall admittance, given by

(15)

we have the dispersion relation,

(16)

Alternatively we can write this as quadratic

(17)

Which has solutions κ = κ+ and κ = κ-

(18)

for resonance we must again have,

(19)

for integer n, where here Le is the effective length of the pipe. Thus the resonance frequencies of the smooth hard walled pipe are the discreet values,

(20)

For low frequencies, Cummings35 assumes that the walls have spring like, purely imaginary impedance (involving no energy loss or gain) and sets,

(21)

Where S = 2πRlc is the area presented by the cavity to the tube and V is the cavity volume given by,

(22)

then we have,

(23)

Upon substitution, and recalling that P/A= 2/R , we obtain the result

(24)

if we assume that,

(25)

Since the phase speed of the upstream and downstream travelling waves is c± = c0/κ±, we see the effect of the corrugations is to slow down the wave. Finally in such a limit the resonant frequencies of the pipe are given by,

(26)

From equation 26, we can see that the resonant frequencies can be effectively predicted if we know the flow speed and geometry of the corrugations. Cummings35 model cannot be used to predict the pressure amplitude of the sound wave.

Neglecting the Mach number dependency due to convective effects we can see that the equation 26 is same as equation 1 but with some additional terms. If we assume that the reactance has the effect of decreasing the speed of sound c0 then the equation 26, can be re written in simple form as,

(27)

where, ceff is the new speed of sound due to the presence of corrugation, V is the volume of the corrugation, S is the cross sectional area of the pipe and l is the corrugation pitch.

Based on the new speed of sound ceff, one can calculate the resonant frequencies in corrugated pipe as,

(28)

Though the equation 28 explains the reason for reduced resonant frequency in corrugated pipe compared to smooth pipe, this equation is actually a modified form of Binnie’s7 equation. The experimental work of the Binnie7 suggests that corrugated pipes can be treated as periodic structures with flow. The general theory of waves in media of this type has been described in the monograph by Brillouin77, who showed that it can be traced back to Newton. Using the lumped impedance theory of Stewart78 and Lindsay et al79, Binnie7 modeled the corrugated pipe as tubes with closed branches. As per the Bennie’s model, the resonant frequency of corrugated pipe is,

(29)

where V is the volume of the corrugation, S is the area of the pipe and d is the pitch of the corrugation. The same equation can also be derived by assuming that the corrugated wall is characterized by the effective acoustic compliance Y with ImY < 0. Then the sound velocity in the pipe of radius R is determined by80,

(30)

Where ρ and c are the density of the medium and speed of sound and k is the wave number. For example, for a pipe with small, closely spaced, axisymmetric, rectangular grooves, we obtain

(31)

where a and bare the depth and width of a groove, respectively, and L is the spatial period of corrugation. Then, the velocity of sound is expressed as

(32)

This velocity linearly decreases with increasing groove depth. In the general case, the effective compliance of a corrugated surface cannot be determined by simple physical methods, and, therefore, the deceleration of a sound wave should be calculated using the general theory of wave propagation in periodic structures77,81,82. Lapin83,84 showed that the velocity of sound for a pipe with a small sinusoidal roughness (corrugation),

(33)

This velocity decreases with increasing roughness amplitude according to a square law. The difference in the amplitude dependences in formulas (32) and (33) is determined by the difference in the corrugation slopes.

2. 3. Energy Balance Model (EBM) for Predicting the Sound Pressure Amplitude in Corrugated Pipes

The model based on energy balance approach64 is used to predict the acoustic pressure amplitude of the corrugated system. As per this model, in order to obtain the dimensionless amplitude (where is the maximum acoustic pressure amplitude in the corrugated pipes, ρ0 is the density o the medium, c0 is the speed of sound and U is the mean flow velocity) of the acoustic field inside the corrugated pipes, the time-averaged acoustic source power has to be balanced by the time-averaged acoustic losses,

(34)

The following sections explains the sound generation mechanism in single cavity, corrugated tube and application of EBM to corrugated tubes.

2.3.1. Impinging Shear Layer Instability and Mechanism of Sound Generation in Corrugated Tube

It is well known that when a fluid flow impinging upon a solid surface or other configurations, the free shear layer (vortex shedding) can exhibit a self-sustained oscillation, known as impinging shear layer instability or, more simply the edge tone. Wind-tunnel experiments carried out by Nakamura et al12, showed that the sound in corrugated pipe is excited by the impinging-shear-layer instability. The resulting instability is characterized by a self-excited resonant oscillation occurring in the fluid-acoustic coupled system.

In the case of the flow past cavities as shown in Fig. 4, the shear layer separated from the upstream corner of the cavity can be unstable in the presence of a sharp downstream corner. The acoustic oscillation in a corrugated tube is driven by vortical sound sources due to impinging shear layer instability. The frequency of the impinging-shear-layer instability increases with increasing flow velocity. Accordingly, when the flow velocity is close to the vortex-resonance velocity where the shear-layer frequency coincides with one of the natural harmonics of the tube, the acoustic oscillation of the tube could be resonantly excited. The measurements by Nakumura et al12 on corrugated pipe shows the sound frequency plotted against the flow velocity in Fig. 5. As can be seen, the sound frequency is constant and locked in to one of the natural harmonics of the pipe with increasing flow velocity and it jumps to the next at a certain threshold value of the flow velocity.

The closer look of frequency lock-in shown in Fig. 5, indicates that the shear layer instability is itself strongly influenced by the acoustic osculation occurring in the fluid acoustic coupled system. A block diagram illustrating the flow-acoustic coupling is shown in Fig. 6 was proposed by Nakamura et al12 The similar behavior of flow acoustic coupling was observed by Bruggeman et al25 on the study of aero acoustic pulsation in gas transport system and also reported in other literatures85,86.

From the block diagram illustrating the onset of fluid acoustic instability shown in Fig. 6, we could observe that the system under consideration consists of two subsystems; (1) The acoustic subsystem with natural frequency equal to nf (n=1, 2...), and (2) the fluid subsystem with vortex shedding frequency fv as its natural frequency. These two subsystems are strongly coupled in such a way that the acoustic subsystem is driven by the vortex-induced pressure while the fluid subsystem in influenced and controlled by the acoustic pressure. Thus, the acoustic oscillation in the corrugated tube should be a self-excited oscillation occurring in this fluid acoustic coupled system. The essential feature of this instability is that the natural frequency of the fluid system is proportional to the fluid velocity. Therefore the imposed acoustic oscillation can set the fluid subsystem into resonance when the vortex resonance velocity is approached. The vortex induced acoustic pressure source can be explained using the theory of vortex sound. The next subsection illustrates the calculation of acoustic source power due to vortex shedding in the cavity.

2.3.2. Theory of Vortex Sound and Source Pressure

A formal relationship between vortex shedding and sound generation has been established for free field conditions by Powell18 and generalized by Howe19. Using a Helmholtz decomposition of the flow field to define the acoustic field,

(35)

where is a steady scalar potential, φ′ is the unsteady scalar potential and ψ the stream function. The acoustic field υ′ is defined as the unsteady irrotational part of the velocity field:

(36)

Neglecting friction and heat transfer and assuming a homentropic flow condition, the momentum equation can be written as per Crocco is,

(37)

where, is the total enthalpy and is the vorticity.

At low Mach numbers and high Reynolds numbers, we could neglect the convective effects on the propagation of sound waves. With this approximation one finds

(38)

This corresponds to the assumption that the Coriolis force density where ρo is the fluid density, acts as source of sound. The time-averaged acoustic source power can be estimated using the following estimation as per Howe,

(39)

where V is the volume in which the vorticity is not vanishing and the brackets <….> indicate time averaging. The vorticity in a flow field is related to the forces acting on the flow, therefore it is related to the sound produced. Equation 38 gives the source power value for single cavity, so, in order to use this for corrugated pipe model, we have to add the value for number of corrugation present. If n number of corrugation present then,

(40)

2.3.3. EBM Model

As shown in equation 34, in order to satisfy the energy balance of the corrugated pipe, the time averaged acoustic source power has to be equal to the time averaged acoustic losses61. The losses are due to the radiation acoustic waves, the visco-thermal dissipation (due to heat transfer and friction), and the vortex shedding at the open termination of the main pipe. If we assume that the losses are independent of each other, then they are given as,

The time averaged acoustic power radiated loss:

(41)

The time averaged visco thermal loses to the main pipe:

(42)

The time averaged visco thermal losses to the corrugation:

(43)

The time averaged acoustic losses due to vortex shedding at the outlet of the pipe87:

(44)

where, I is the acoustic intensity, Ssb is the area of the corrugation, and Sp is the cross sectional area of the main pipe. From the above equations we can see that the energy balance model can be used to predict the acoustic sound pressure amplitude of corrugated tubes. Tonon et al22,64 successfully applied this energy balance model to the multiple side branch resonators which mimics the corrugated pipes model. The result of the predicted and measured sound pressure amplitude is given in Table 1. This is the only model available in the literature, which explains the prediction of sound pressure amplitude. Though this model predicts the pressure amplitude reasonably, one must use suitable assumption for ratio of acoustic velocity to the mean flow velocity.

Some of other theories such as Elliot’s vortex sheet model, Plane wave acoustic model (PAWM), Serafin et al’s15 singing tube model of virtual corrugated tube for audio analysis and Debut et al’s88 non-linear phenomenological model for corrugated tube are also reported in literatures. Brief description of each of this theory is given in next subsections.

2.4. Theory of Vortex Sheet Model

From Fig. 5, we infer that for the given harmonic mode, the frequency is constant but flow velocity increases. We know that the Strouhal number89 is the parameter which connects the resonant frequency of the corrugated pipe, main pipe flow and corrugation pitch. Using the vortex sheet theory, one could obtain the range of Strouhal number for the given resonant mode before the pipe moves to higher harmonics.

In the theory of vortex sheet/ sound interactions, one can assume that the effect of corrugations in terms of the acoustic wall admittance of an equivalent smooth cylindrical pipe. The vortex sheet which forms across each cavity of corrugated pipe, changes the effective acoustic admittance of the pipe wall so that t it can now have a real part.

The simplest corrugation configuration is one of length lc, and infinite depth, the basic flow being that of a uniform stream of speed U separated from a fluid at rest by two semi-infinite flat plates located at y=0 for x< 0 and x > lc respectively, as shown in Fig. 7. Consequently there is a vortex sheet across the cavity at y=0 emanating from the leading edge of the cavity. We therefore consider the scattering problem due to a wave of frequency ω=coκ. For small disturbances the vortex sheet is displaced to y= η(x, t) = ἠ(x) e-iωt. On physical ground the vortex sheet must vanish at the trailing edge of the cavity.

Integrating the wall pressure over the cavity length lc<<l, yields,

(45)

where, vaires over the scale of the pipe length. However the vortex sheet and the normal velocity υ′ will vary over the shorter scale of the cavity. Consequently upon integration over the cavity we have

(46)

Where λi is the wavelength of the instability wave and υ0* is a complex constant. This implies that the wall admittance over the slot, is a complex constant. Indeed analysis of the trailing edge problem suggests that β is the form,

(47)

where β0> 0 is real. In order for the real part of β to be negative, we require

(48)

Now since

(49)

The frequencies for the first domain of radiation lie in the range

(50)

Analysis suggests that

(51)

2.5. Plane Wave Acoustic Model (PWAM)

Plane Wave Acoustic model (PWAM) of corrugated tube has been established by applying the continuity of mass flow and of pressure at each bifurcation and the perfect reflection condition at the side branch terminations. The resulting system of 4N equations with4N+ 2 unknowns, where N is the number of side branches composing the multiple side branch system, is closed mathematically by imposing two boundary conditions. The two boundary conditions are an unflanged open pipe termination at the outlet and flanged open pipe termination at the inlet. The results of successful application of this model for multiple side branch resonator by Tonon et al22 and its comparison with CAM is shown in Table 2. They22 found that higher frequency resonance modes can accurately be predicted by using a plane wave model (PWAM) and lower frequency resonance modes can be predicted by using CAM model for side branch resonators.

2.6. Singing Tube Model of Corrugated Tube for Virtual Audio Analysis

The complete singing tube model used for audio analysis developed by Serafin et al15 is summarized in Fig. 8. This model is derivative of Bernoulli’s principle model explained in section 2.1 which is used for rotating corrugated tube singing dynamics, with addition of Doppler shift effect for virtual audio analysis. The basic cylindrical tube is modeled as loaded waveguide, the corrugation inside the tube is modeled using Bernoulli’s principle and Doppler shift effect of tone generation is modeled using basic flow physics.

The loaded wave guide model for the basic tube is shown in Fig. 9 and the Doppler shift effect which affects the sonorities (or tones) is given by,

(52)

where ωs is the radian frequency emitted by the source at rest, ωl is the frequency received by the listener, υls denotes the speed of the listener in the direction of the source, υsl denotes the speed of the source in the direction of the listener, and c denotes sound speed

2.7. The Coupled Non-Linear Acoustic Model for Corrugated Tube

In this phenomenological non linear model88,90-92, the acoustic response of the corrugated pipe is simulated by connecting the lossless medium moving with a constant velocity with a source based on discrete distribution of Van-der-Pol oscillators arranged along the pipe.

A simple model93 which utilizing the concept of mechanical oscillator to represent shear layer instability is described by the following equation,

(53)

where, ps is the acoustic pressure in the cavity, p pressure in the neck of the cavity, ωr is angular frequency of the resonator and ηr is the reduced damping. The alternative form of the equation using Von der Pol type is given by94,

(54)

where, ω is angular frequency, v is diffusion parameter, A and B are coefficients.

In the pipe the acoustic behavior can be described as the lossless medium moving with a constant velocity linear wave equation as,

(55)

where, the source term from the equation (54) is,

(56)

where ps is the pressure variation caused by the source cavities and G is the ratio between opening of the cavity and the length of the pitch. The non linear equations (53) and (55) are can be solved using high order schemes such as Optimized Prefactored Compact finite volume (OPCfv) scheme for discretization in space, and Runge-Kutta for time stepping 95,96. This proposed one dimensional coupled non linear model effectively simulates the acoustics corrugated pies. This coupled model is capable of predicting the lock-in frequency as well as the onset fluid velocity. The application of this model for acoustics of corrugated pipe is successfully

demonstrated by Debut et al88 and Popescu et al96.

To summarize, we could see that there are number of theories are available to predict the “whistling” of acoustics of corrugated pipes. The Cummings Acoustics Model CAM given in equation 24, is simple to use and predict the singing frequency reasonably accurately if one knows the geometry details of the corrugations, and one could also use even much simpler form of Binnie’s7 model (ref. Eq. 29). Alternative form of Binni’s7 equation which explains about the role of cavity depth on the resonant frequency given by Lapin84 can be used as well. The drawback of Cummings35, Binnie’s7 and Lapin84 models is they cannot predict the sound pressure level in the corrugated pipe. Singing of corrugated tube due to rotation can be effectively modeled using Bernoulli’s principle as explained by Silverman et al9 and use of Doppler shift effect could be used for virtual music application as explained by Serafin et al15. To calculate the lock-in frequency range using Strouhal number in a particular resonant mode Elliot’s vortex sheet model35 can be used. To predict the acoustic pressure level on the corrugated tube, EBM model22 can be used. The formulation of EBM model is complex, hence the accuracy of the model depends on the assumptions of the pipe geometry and flow losses. Plane Wave Acoustic Model (PWAM) can also be used to predict the resonant frequency, if we know the boundary conditions of the corrugated pipe.

3. Acoustics of Corrugated pipes: The Experiment

From the first paper of Burstyn1 to the very recent study of Nakiboğlu et al73 almost all the researchers carried out the experiments to understand the tone generation in corrugated pipes. The theories described in section 2, are all formulated using the extensive experimental data gathered in the during the experiments. This section introduces some of the experiments carried out on the acoustics of corrugated tubes by researchers.

For the experiment, Binnie7 used the electrically driven fan fitted on the inlet with a short 3 in. pipe and on the outlet with smoothing devices and an orifice-plate followed by a straight 1.25 in. pipe and 23 ft long. He controlled the airflow by changing the fan speed and the opening of the delivery valve. The frequency spectrum and the form of the sound waves emanating from eight different types of corrugated tubes were measured by means of a double beam oscilloscope connected to a calibrated oscillator and to a microphone near the tube under test. He tested the corrugated tubes with internal and external flow. For external flow measurement, Binnie introduced the smooth tube outside of the corrugated tube. The pipe geometries and flow condition is given in reference7. The some of the observation noted are; a) the substantial contraction is necessary at the inlet for short corrugated tubes to sing and contraction is not required for longer tubes, b) lower notes are observed for longer tubes when the flow changed from internal to external, c) some short corrugated tube during internal flow condition, “sing” only when the introduction axial rod, d) presence of resonant frequency is directly proportional to the tube lengths, e) hysteresis of frequency with flow velocity is observed, f) at the exit of the pipe plane waves are observed, and g) sound radiated with equal intensity in all directions.

Crawford8 used the wrist watch for measuring the rotational velocity of the “Hummer” and with the use of his ear, he correlated the rotational velocity to the note generated. By holding a tube outside the car window, Crawford8 measured the relationship between speed of the car and singing frequency of the tube. With his experiment, he found that, it is possible to make the fundamental of the Hummer to sing, by using the relation between the length of the tube and the minimum frequency value. He also studied the role of turbulence in causing Hummer to sing and explained the minimum Reynolds number require to cause the tube to sing using, tube induced turbulence and corrugation induced turbulence. He noted that that minimum Re>2000 required for the tube to singing, but later Cadwell10 repeated the Crawford’s8 experiment for different length of the tubes and proposed Re >500 is enough to make the tube to sing. Crawford8 also noted that the note generated in Hummer is discreet value but the velocity is not quantized. This observation was latter confirmed by other researchers such as, Nakumura et al11,12, Petrie et al16, Elliot35, Kristiansen et al62,68 and Tonon et al22.

3.1. Wind Tunnel Experiments for Corrugated Tube

Nakamura et al12 and Hammache et al20 used the wind tunnel to measure the sound frequency of corrugated pipes. The Experimental set up used by Nakamura et al12 is given in Fig. 10 (a), shows the positions of the microphone and the two hot-wire probes used, along with relevant dimensions of the corrugated tube studied. Hammache et al20 also introduced two hot wire probes near the corrugation to observe the vortex shedding frequency which is shown in Fig. 10 (b). Nakamura et al12 observed the frequency lock in mechanism and noted that during transitional region, two or more sounds with neighboring frequencies, the same was also confirmed by Elliot35. Nakamura et al12 and Hammache et al20 were able to capture the un-locked shedding frequency and lock-in whistling phenomenon when the shedding frequency coincides with the tube resonance. Based on the wind tunnel experiments, Nakamura et al12 proposed a flow-acoustic coupling feedback model based on impinging-shear-layer-instability which was described in detail in the section 2.4.

3.2. Rotting Corrugated Tube Experimental Setup

The apparatus used by Silverman et al9 to measure the Voice of the Dragon’s resonant tones in systematic way is shown in Fig. 11. The corrugated tube was mounted on a thin wooden slab which was attached to a spoked Bicycle tire wheel free to rotate in a vertical plane; a counterweight was fixed at the opposite end of the slab so that the centre of mass of the system lay on the axis of the wheel. A motor, whose speed was controlled by a rheostat, was placed in direct contact with the outer rim of the wheel.

A microphone, aligned along the axis of rotation, recorded the tones at different rates of rotation; the sound was amplified and then Fourier spectrum was analyzed by computer. The rotational frequency of the tube was determined by means of a stroboscope and counter/ timer. The test set up used to measure the axial pressure difference across the tube, axial air speed at the fixed end and tangential velocity at the rotating end is shown in Fig. 11(c). With the help of experimental data, the authors9 successfully developed a theory based on Bernoulli’s principle, which is outlined in section 2.3, to explain the singing of Voice of the Dragon. Later, Serafin et al15 used this model and with the inclusion of Doppler shift effects, they developed a virtual acoustic corrugated singing tube for audio analysis. Hence, this experimental set up can be used to understand the whistling mechanism while whirling or rotating a corrugated tube.

3.3. Experimental Setup Used in Vacuum Cleaner Type Application

To study the noise problem in vacuum cleaners which contains the corrugated tubes, Petrie et al16 used a simple set up which is shown in Fig. 12. The set up was constructed in which the velocity could be varied from 8-60 m/s through a range of ducts from 15-40 mm nominal internal diameter. They varied the pitch and depth of the corrugation in the ranges 2.5-8.0 mm and 2.5-7.5 mm respectively. The number of corrugations was varied between 1 and 340, and several lengths of smooth duct were used to increase the total length. The settling chamber, which was lined with foam and had internal baffles, was used to smooth the flow from the fan and reduce the fan noise transmitted to the duct. Sound measurements were taken by using a 1 inch microphone and Brüel and Kjær 2107 analyzer. A constant temperature hot wire anemometer was used to measure the flow velocity. Initially the air was blown through the duct and in later experiments, the air was sucked through the corrugated duct, after it had been found that turbulence generated within the smooth inlet length L1 had the effect of destroying the whistling. They carried out the experiments with smooth and wrinkled wall tubes (ref. Fig. 12) and found that the wrinkled wall reduces the whistling amplitude considerably. They also noted that with the introduction of smooth wall upstream of the corrugated tube (L3 in Fig. 12), sound pressure level reduced. The level of reduction varied depends upon the L3 value. Based on the experiments they conclude that the turbulence inside the corrugation is sound production and introduction of wrangling or smooth pipe upstream altering the turbulence which gives the less sound pressure level at the pipe exit.

Kristiansen et al68 measured the sound pressure level (SPL), sound intensity and phase of the sound waves inside of corrugated tube along the entire length. The test set up is shown in Fig. 13. The arrangements is similar to the Petrie et al16, but they used the two microphone and one they placed on a guide rail which measures axial acoustic properties. By moving the guide rail, the required acoustic data can be measured for the entire length of the tube. The more details of tubes and instrumentation used for the measurement can be found in ref. 68. The typical acoustical data obtained using this set up for 615 mm corrugated tube is shown in Fig.14. From the figure we could observe the presence of standing waves and flow-acoustic lock in mechanism. They also measured the input impedance of smooth wall and corrugated wall suing well known two-microphone technique97-99. The experimental set up used by Taylor et al27 for measuring the impedance of corrugated tube is shown in Fig. 15, and the similar set up also used by Kristiansen et al68. The measured impedance curves of corrugated pipe and smooth pipe by Kristiansen et al68 is shown in Fig. 16. The data shows that that for corrugated tube the impedance curve is shifted 8 to 9 % left side of the smooth tube. The Kristiansen et al68 expressed that, the shift in impedance due to presence of corrugation is the reason for reduced resonant frequency which is the same order of 8 to 9 % of smooth pipe. Taylor et al27 also measured the Insertion Loss (IL)of the corrugated tube and observed that the IL values are lesser than the smooth tube. The test setup and the typical IL curve measured is shown in Fig.17.

Kristiansen et al68 noted that net acoustic energy stream in corrugated tube is always in the same direction for nearly the total length of the pipe, which is against the direction of the flow. The same observation also has been reported by Binnie6 in his experiment with water in an open horizontal channel with vertically corrugated sides. Kristiansen et al68 experiment also reveals that the sound source is present at the pressure node which was agreed with the observation made by Hammache at al20 and the constant value of Strouhal number of 0.3 is observed during the pipe resonances.

3.4. Experimental Set up of Corrugated Tube as Sidebranch Resonator

Tonon et al22 and Nakiboğlu et al23 modeled the corrugated tube as multiple side branch resonators. The test set up used by them is shown in Fig. 18. Based on the experiments and the use of CAM, PWAM and EBM model, they22,23 successfully demonstrated that the multiple side branch system can indeed be used to explain the whistling phenomenon in corrugated tube.

To summarize, the literature review reveals that different reaches used different experimental set up to study the acoustics of corrugated pipe. Some experiments are simple to set up8 and some are complex and involve lot of instrumentation22.