Fluid dynamics of a flow excited resonance, part I: Experiment

Journal of Sound and Vibration (1981) 78(1), 15-38

FLUID

DYNAMICS

OF PART

A FLOW

EXCITED

RESONANCE,

I: E X P E R I M E N T

P. A. NELSON,t N. A. HALLIWELL AND P. E. DOAK

Institute of Sound and Vibration Research, University of Southampton, Southampton SO9 5NH, England (Received 6 January 1981) This is the first of two companion papers concerned with the physics and detailed fluid dynamics of a flow excited resonance. The phenomenon has been examined by using a rather different approach from others to date, in which usually stability theory has been applied to small wave-like disturbances in an unstable shear layer with an equivalent source to describe the radiation of sound providing the feedback. The physics of the flow acoustic interaction is explained in terms of the detailed momentum and energy exchanges occurring in thefluid itself. Gross properties of the flow and resonance are described in terms of the parameters necessary to determine the behaviour of the self-oscillatory system. In this first paper a full experimental investigation of a flow excited Helmholtz resonator is described, in which the detailed fluid dynamical and acoustic data necessary to develop a mathematical model for the flow.was obtained, and a new theory of the interaction process is presented in the companion paper (Part II). The investigation described involved the use of a two-component Laser-Doppler Velocimeter (L.D.V.) and probe microphones to specify completely the velocity and pressure fields and a flow visualization to give qualitative information of the vortex shedding process. The overall aim of the work described in the two papers was to increase fundamental understanding of flow/acoustic interactions.

1. INTRODUCTION In the context of aerodynamic noise generation, the p h e n o m e n o n of flow excited acoustic resonance is both of engineering importance and of fundamental interest. T h e selfexcitation of an acoustic resonance occurs in a wide variety of engineering applications !nvolving fluid m o v e m e n t . Large amplitude oscillation of various types of resonant acoustic systems has been found in compressor stages, tube bhnk heat exchangers, transonic wind tunnels and lined ducts. In general, the production of high sound levels is a consequence of the coupling of the acoustic resonance to some form of periodic flow disturbance resulting from an instability of the fluid motion. Energy is by some mechanism extracted from the flow in order to sustain the acoustic oscillation. It is the physics of this mechanism and the fluid dynamics of the interaction between the unsteadyflow and the a c o u s t i c r e s o n a n c e which will be considered in these two companion papers. A considerable understanding of this type of sound source has been gained through previous work on specific instances of flow excited resonance p h e n o m e n a and it is relevant to review briefly the knowledge and interpretations which have been offered to date. Sound production in an organ pipe is a c o m m o n example of a flow excited resonance and it has received considerable attention. C r e m e r and Ising [1] were successful in t Now at Sound Attenuators Limited, Eastgates, Colchester CO1 2TW, England. 15 0022--460X/81/170015+24 $02.00/0 9 1981 Academic Press Inc. (London) Limited

16

p.A. NELSON, N. A. HALLIWELLAND P. ]E. DOAK

modelling an organ pipe and the jet oscillating in its mouth as a feedback control system. The jet issuing into the pipe is not self-stable and hence wave-like disturbances can propagate along it which travel towards the downstream lip of the mouth such that the jet is alternately directed into and out of the pipe. This oscillatory "hydrodynamic" volume flow into the pipe serves to excite the pipe acoustic resonance. The resonance of the pipe then produces an "acoustic" volume flow in the pipe mouth which in turn controls the growth of the waves on the jet and hence closes the feedback loop. By deriving the details of the complex loci of the controlled system (organ pipe) and the controller (jet), Cremer and Ising were successful in predicting the frequency and amplitude of self-excitation using the criterion that the loop gain should be unity. Their work forms the foundation for many subsequent refinements of organ pipe theory [2], although the underlying principles have largely remained unchanged, This type of approach was again applied in the study of edgetones [3, 4] and the most comprehensive explanation of the phenomenon has been offered by Powell [5]. He has again relied on the development of "instability" waves on the jet which impinge on the edge downstream of the jet exit, Oscillations of the jet about the lip of the edge give rise to a fluctuating force on the edge. The near field of this dipole-like sound source provides the feedback which controls the growth of the instability waves on the jet. For organ pipe oscillation this form of feedback is generally reckoned to be negligible [6] as the pipe resonance appears to be the dominant, "controlled", system. Exact relations between "pipetones" and edgetones are still a matter of debate however. The high amplitude, single frequency acoustic radiation produced by flow over cavities and cut=outs has also been described in terms of a feedback mechanism. Rossiter [7] succeeded in deriving a semi=empirical equation relating the frequencies of tonal produc= tion to the mean flow speed over the cavity and the length of the cavity in the streamwise direction. In this case the initially unstable shear layer over the cavity formed into discrete vortices which were convected downstream, interacting with the downstream edge of the cavity to produce "acoustic pulses" which in turn fed back upstream and triggered the formation of subsequent vortices. To date this basic view has changed little although the feedback process has been found to vary depending on the type of cavity and the Mach number of the flow. The feedback controlling the behaviour of the shear layer can be dominated by the various acoustic resonances of the cavity. A further type of flow excited resonance that has been given considerable attention is the excitation of flow duct resonances by wake shedding from plates parallel to the airstream. Extensive studies by Parker [8, 9] have led to a more detailed study of the process by Cumpsty and Whitehead [I0], The now familiar Parker fl resonance was excited in a wind tunnel by the periodic shedding of vortices in the wake of a flat plate in the centre of a wind tunnel cross section. The unsteady "incompressible" pressure field associated with the wake was considered to be the source of acoustic excitation. The study of the feedback loop in this instance was later completed by Archibald [11] who measured the influence of an externally applied sound field on the vortex shedding in the wake of the plate, A criterion of a loop gain of unity was used to illustrate the conditions necessary for self-excitation of the resonance, As this brief review indicates, there is already a wealth of knowledge of certain characteristics of the behaviour of various flow excited resonances in which discrete frequency sound is produced. Despite this, however, the understanding of the fluid dynamics involved in any of these processes is still of a superficial nature. As most of the foregoing examples illustrate, the analysis of these problems is often confined to the use of stability theory to describe the growth of waves on a jet or shear layer and an equivalent source to describe the radiation of sound providing the feedback. The exact

FLOW EXCITED RESONANCE, PART I

17

mechanics of either the generation of sound by the shear layer or the stimulation of shear layer disturbances by the sound is poorly understood. In the work described in what follows, a specific instance of flow excited resonance was chosen for study and the details of the flow measured in one resonant condition. A rather different approach has been chosen in analyzing the phenomenon in that effort is directed to explaining the physical behaviour in terms of momentum and energy exchanges occurring in thefluid itself. Little attempt has previously been made to describe the gross properties of the flow or resonance as a function of the parameters necessary to determine the behaviour of the feedback system model. The phenomenon chosen for study is that of a flow excited Helmholtz resonator. Choice was largely determined by experimental convenience. Previous work on this type of flow excited resonance by Bolton [12] and Elder [13] had again been directed towards describing experimental observations in terms of a feedback model. Neither dealt with the details of the fluid dynamical interactions involved, although Elder has described measurements of the wave-like behaviour of the shear layer. In this first of the two companion papers the measurement of the mean and fluctuating parts of the velocity field and the fluctuating parts of the pressure field in the neck of a Helmholtz resonator excited to peak amplitude by a grazing flow is described. A complete description of the velocity field was accomplished by using a two-component Laser-Doppler Velocimeter (L.D.V.) [14] which was capable of simultaneous point measurement of the mean and fluctuating parts of the velocity vector in two orthogonal component directions. Further to this accurate fluctuating pressure measurements were possible by use of a small probe microphone which did not influence the flow being measured. Results of a flow visualization are presented which produced qualitative information of the vortex shedding process. 2. ACOUSTICAL CHARACTERISTICS OF THE RESONATOR 2.1. RESONATOR DESION AND ACOUSTIC RESPONSE The basic design of the resonator and jet nozzle is shown in Figure 1. The rectangular nozzle delivered a relatively quiet and steady grazing flow across a long slot which was backed by a rectangular cavity to form a Helmholtz resonator. Basic design of the resonator was largely determined by the requirements of the L.D.V. and the need for far field sound pressure measurements. Under aerodynamic excitation the fundamental frequency (f,) and the flow speed (U~) are related to the slot width (L) by [12"1

fnL/U~o ~- 0.25.

(1)

Considerations of the available air supply allowed for initial design calculations of L = 10 ram, fn 2 6 0 0 Hz and U~-~20 m/s. With reference to Figure 1 the neck of the resonator was made by cutting a slot of 100 mm length across the top of the rectangular box which formed the body. The top was manufactured from 0.6 mm thick aluminium sheet. This produced an essentially two dimensional flow regime (this was checked experimentally) in the streamwise plane through the slot centre where velocity and pressure measurements were taken. The body of the resonator (internal dimensions 95 • 80 x 58 mm) was constructed from 12-5 mm thick aluminium in order to provide rigid cavity walls and two "windows" consisting of 0.5 mm thick glass allowed access for the laser beams. The top of the resonator was extended in the flow direction to prevent flow separation until well downstream of the resonator neck.

18

P. A. NELSON. N. A. HALLIWELL AND P. E. DOAK

Vertical component ....9"reference beam

t

~_ ...J" ~ M a i n

beam ...................Streamwise component

h /f..,.~ e ~ [a [ [ffrx\ I

r

Sketch e gnaw ohis of neck cross-seclion which

.C

)-

I0 mm Figure 1. The resonator and laser beam geometry. A n experiment was performed to measure the frequency response of the resonator when excited by an incident sound wave. The arrangement used is illustrated in Figure 2. A constant sound pressure was maintained at the reference microphone as the sound produced by the loudspeaker was swept through the frequency range. Pressure in the cavity of the resonator was monitored by using a 12.5 mm microphone mounted inside the cavity such that the microphone diaphragm was flush with the inner surface of the cavity. Figure 3 shows the frequency response of the resonator with a clearly defined Helmholtz resonance at about 600 Hz. A more accurate determination of the fundamental

Loudspeaker

I jResona,o r 12"5mm reference

n [ ~ . 5 mm U I U cavity microphone m~cropnone JJ (held adjacentto resonatorsurface)I I "

B and K A.E spectrometer Type 2112

I B and K sine random generatorType 1024

I

B and K levelrecorder Type 2305

I

B and K A.E spectrometer Type2113

Racal digital frequency meter type 9522

Figure 2. Resonator frequencyresponse test.

19

FLOW EXCITED RESONANCE, PART I I

i

I

I

I

I

I0 dB[ Cavity microphone~

"

i

Referencemicrophoneoutput I l I I I 0-1 0-2 0-5 I 2 Frequency(kHz)

5

I0

Figure 3. R e s o n a t o r frequency response.

frequency was undertaken by slowly changing the excitation frequency in the region of 6 0 0 H z and monitoring the cavity p r e s s u r e signal with a digital frequency counter (accurate to within 1 Hz). The peak response of the resonator was 6054- 3 Hz. The quality factor Q of the resonator as measured from the bandwidth o f t h e response curve (614-3 Hz) was Q = 1 0 + 0 . 5 . From measured values of resonance frequency and quality factor it is possible to determine the acoustic impedance of the resonator. The differential equation for the volume displacement ~: of the slug of air in the resonator neck can be written [15] as

M d2se/dt2+R d ~ / d t + ~ / C =p,

(2)

where p is the complex driving pressure, M the inertance of the mass of air in the resonator neck, C the compliance of the air in the cavity and R the resistance which provides the damping of the motion. The impedance z defined as the ratio of driving pressure to the volume velocity in the neck is then z = R + i { o J M - lfioC}.

(3)

Further to this, with po the ambient air density, Co the speed of sound, and V the volume of the cavity, the compliance, inertance and resistance can be calculated from [15]

C = V/poc~,

M = 1/to2C,

R = to,M/O.

(4)

Using the measured values and the appropriate physical constants gives the calculated values as C = 3.1 x 10 -9 (m 4 s 2 kg-1), M = 22 (kg m -2) and R = 8-4 • 103 (kg m -4 s-l). If l is the effective length of the mass of air in the resonator neck, the inertance M = I/S where S is the area of the neck and the inertance thus corresponds to an effective length of l = 18 mm. It is interesting to compare the value of the resistance R with the radiation resistance R, of a piston in an infinite baffle radiating at the same frequency. At low frequencies where the wavelength (h) is much greater than the dimensions of the radiating piston, R, is given by

R , = pocok 2/2~r,

(5)

20

P.A.

NELSON, N. A. HALLI~,VELL A N D P. E. D O A K

where k is the wavenumber of the radiated sound. For radiation at 605 Hz then R, = 8.0 • 103 (kg m - 4 s - l ) . The dominant contribution to the resistive impedance of the cavity is thus due to radiation losses. An estimate of the contribution due to viscous losses can be made from the analysis of Morse and Ingard [16, p. 483] of the viscous resistance of a slit of width L in a two dimensional duct. When L is much smaller than the duct width the resistance due to viscous action is given by

R~, = (2potodJ2rtS) In ( L / 2 h ),

(6)

where d r is the viscous boundary layer thickness (21.t/potO) 1/2 and h is the thickness of the slit. In the present case h = 0 . 6 ram. The value of R , computed at 605 Hz is approximately 0.2 • 103 (kg m -4 s-l). This confirms that the resistive part of the impedance z is dominated by the radiation resistance and as such is an important factor in the flow modelling which will be introduced in the second paper. The high value of radiation resistance in the present case is in marked contrast to the resonator considered by Howe [17] in which viscous losses were found to dominate. This can be attributed to the large neck area and small neck length of the resonator under examination. It is important to note that no account has been taken here of any non-linear contributions to the resonator mouth resistance. Particle velocities are typically 0.01 m/s, which is well below the onset value for non-linear behaviour found by Ingard and Ising [18]. Under aerodynamic excitation, however, particle velocities are considerably higher and in the region where non-linear losses may indeed make some contribution. 2.2.

A E R O D Y N A M I C EXCITATION RESPONSE

The free jet nozzle shown in Figure 1 was used to provide a grazing flow across the resonator slot and the mean velocity (Uoo) of equation (1) at the edge of the boundary layer occurred at a distance of approximately half the nozzle height above the resonator surface (nozzle height = 34 ram). In order to test the acoustic response of the resonator to flow excitation, the flow was varied between 15 m/s and 30 m/s whilst the amplitude and frequency of the cavity pressure fluctuations were again monitored. The results are plotted as a function of the velocity Uoo in Figure 4. Precise frequency of peak excitation was difficult to determine because of the large bandwidth; however, it can be seen to be in the region of 603 Hz and occurs at a value of Uoo of 22 m/s. All subsequent experiments on the aerodynamic excitation were performed at this peak excitation 135

i

I

700

650

130

550

125

~o

25 ' Velocity (m/s)

Figure 4. Frequency response for aerodynamic excitation.

30

N -r-

FLOW EXCITED RESONANCE, PART I

21

condition. It was found that a m e a s u r e m e n t of the frequency of cavity pressure oscillation provided an accurate means of monitoring the excitation. The frequency of the cavity pressure signal was kept within the range of 602 to 608 Hz, which corresponded to Uoo = 224- 0.5 m / s . 2.3. SOUND POWER RADIATION An experiment was conducted in order to determine the total sound power radiated to the far field when the resonator was excited to its nominal peak amplitude. The jet nozzle itself was connected via flexible ducting to the air supply and was m o u n t e d in the centre of a semi-anechoic room. The area in which far field pressure m e a s u r e m e n t s were made was surrounded by anechoic wedges in order to give as near as possible truly anechoic conditions. T h e decay of sound pressure level away from the resonator in three orthogonal directions is shown in Figure 5. Values of SPL shown were measured in 1/3

II0

I00

90 x2

SPL

no

80 ve x2

direction

(SPL-K)dB

70

.+ ve x3 direction

(SPL-2OdB;

60 x3

-ve x 3

direction

(SPL -50 dB', 0-I

'

'

'

O'-S

'

'

'

' I'.0

Distance from source centre (rn)

Figure 5. Decay of SPL with distance from resonator.

octave bands centred on 630 Hz. It can be seen that the decay of SilL is at 6 dB/distance doubling out to approximately 0.8 m where small reflections interfere with the sound radiated directly. Accurate m e a s u r e m e n t s of the far field pressure were made at a distance of 0.5 m which represented approximately one wavelength of the frequency of interest. Twenty m e a s u r e m e n t s were taken corresponding to the surface of an icosahedron surrounding the resonator. Each set of 20 m e a s u r e m e n t s was repeated three times before the arithmetic average SPL for each set and the consequent sound power level of the source was calculated. By this method the average rate of acoustic energy generation by the flow could be estimated to be 6 m W to within an accuracy of +1 mW. Radiation from the resonator was approximately omnidirectional although SPL measurements over the upper hemisphere were up to 5 dB higher than those below. This form of directivity is observed for the radiation from a piston on a spherical baffle [16, p. 344] when the wavelength of the radiated sound approaches the circumferential dimension of the sphere. Far field radiation was dominated by the fundamental frequency

22

P. A,

Ns

N. A.

HALLIWELL

AND

6

P. E. D O A K

O

m O

._= O ~~a "O O

k O

6

6

FLOW EXCITED RESONANCE, PART I

23

of the resonator. Although the second harmonic was detectable it was some 25 dB lower than the fundamental. 3. FLOW VISUALIZATION In order to obtain a qualitative assessment of the nature of the flow in the resonator neck, a second resonator was used which had a Perspex side allowing photographs of the flow to be taken in the plane of the resonator top surface from one side. Oil mist was used to visualize the flow and this was inserted through a smoke probe which entered the rear of the resonator and released smoke just underneath the upper surface. When the resonator was "singing" a stroboscope light was used to illuminate the neck region and this was triggered from the oscillating cavity pressure signal. A phase shifting circuit was introduced which allowed strict control of the point in the vortex shedding cycle at which the flow was frozen. A series of 12 photographs were taken for successive phase shifts at increments of 30 degrees. The results are shown in Figure 6. Vortex shedding which occurred was shown to be correlated along the entire span of the resonator neck confirming that at the mid-point the flow could essentially be considered as two dimensional. The results of any flow visualization study must be interpreted with caution. The photographs taken indicate the periodic formation of discrete vortices near the upstream lip of the resonator neck. These vortices appear to grow in size as they are convected towards the downstream lip of the neck. The vortices appear to be "pushed out" of the resonator before being convected downstream over the upper surface of the resonator. It should be remembered that photographs of streaklines produced by particles injected into the flow will not give a good representation of streamlines. Hama [19] has illustrated that an apparent "growth" in streakline oscillations can be produced by a neutrally stable shear layer disturbance. More quantitative information is thus required before any firm conclusions can be reached regarding the formation and growth of the discrete vortices observed. Nevertheless, useful information can be deduced from the photographs. Figure 7 shows a series of sketches of the flow in which the positions of the observed vortices have been identified by tracing the photographs taken. A discrete vortex is formed during the first half of the cycle after the cavity pressure reaches a minimum. During this period the cavity pressure is increasing and the air below the neck is being displaced into the cavity. The vortex reaches the mid-point of the resonator neck on completion of the first half cycle. At this stage the cavity is at its maximum downward displacement. During the next half cycle, as the cavity pressure falls and the air below the neck is displaced upwards, the vortex convects towards the downstream lip. On completion of the cycle, the vortex has impinged on the downstream lip and has almost been ejected as the air below the neck reaches its maximum upwards displacement. During the second half cycle between the photographs taken at 0.47 T and 0.97 T the vortex has travelled a distance of 5 mm which implies a convection speed of 6 m/s. After ejection from the resonator neck, during the third half cycle shown in Figure 7, the vortex appears to travel a distance of the order of 10 mm although the "stretched" shape makes location difficult. This does, however, imply an increased convection speed of 12 m/s. Confirmation of this is illustrated in the fourth half cycle, where the trailing end of the vortex again appears to travel a distance of about 10 mm between the photographs corresponding to 1.47 T and 1.9 T after the start of the first cycle. The increase in convection speed is important evidence as to the effect of Coriolis accelerations in the fluid, as will be discussed in the companion paper.

24

P.

6

k

~

~n 6

A.

NELSON,

N.

A.

HALLIWELL

o

-

k

k

~n

--

6

k

k

6

0

AND

P.

E.

DOAK

--

u~ ::k t~

~n

q

~n

",D II b~ o

o e~ e~

o

cO k oJ

r-:

@ k

6

0

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t~

~n

k Lo

913

g. t.0

~n

FLOW EXCITED RESONANCE; PART I

25

A final feature of note from the flow visualization is the behaviour of the fluid at the upstream lip of the resonator neck. The shear layer of the upstream lip of the neck always appears to be leaving in a direction tangential to the upper surface of the resonator. This implies that the Kutta condition is satisfied at the upstream lip of the neck. The consequences of this and the other results revealed by the flow visualization will be discussed in more detail in the companion paper. 4, THE VELOCITY FIELD IN THE RESONATOR NECK 4.1. GENERAL The use of Laser-Doppler Velocimetry (L.D.V.) is now well established and has been applied to a wide variety of flow problems since its use was first demonstrated by Yeh and Cummins [20]. A comprehensive survey of the underlying principles and alternative techniques can be found in the text by Durst, MeIling and Whitelaw [21]. The instrument used in the present work was that devised by Rizzo and Halliwell [14] which measures two orthogonal components of velocity simultaneously in a flow. In this case the geometry was arranged so that the streamwise and vertical components of velocity were measured (see Figure i). Intersection of the light beams with the sides of the resonator box restricted the measurement plane to that shown in the figure. 4.2.

EXPERIMENTAL P R O C E D U R E AND RESULTS

The 30 mW H e - N e gas laser used in the L.D.V. had a beam of 1.5 mm in diameter which when focused down with the reference beams produced a measurement volume giving a streamwise resolution of 0.1 mm. It was more convenient experimentally to move the flow with respect to the L.D.V. and the jet nozzle and resonator were mounted on traverse gear which could be stepped in the streamwise and vertical flow directions. Overall accuracy in positioning the measurement volume relative to the resonator was less than 0.25 mm. Results were taken over a measurement grid of points spaced 0.5 mm apart. The provision of adequate seeding for the flow was one of the most important aspects of the use of the L.D.V. The Doppler signal processing used was frequency tracking demodulation [22] which requires a continuous signal for correct operation. Seeding was provided in the form of oil mist particles introduced into the flow both inside the resonator and upstream at the jet nozzle exit. Full details of the experimental arrangement can be found in reference [23]. All the measurements made were with the resonator at peak aerodynamic excitation. A probe tube was used to monitor the pressure inside the resonat6r and it was ensured that the frequency of the pressure fluctuations remained in the range 602-608 Hz when taking measurements. The voltage outputs from the frequency trackers were analogues of the flow velocities and subsequent time averaging with use of a low pass filter gave values of the mean flow at a point in the flow directly on a digital voltmeter. Mean flow velocities in the two component directions were recorded. The values measured were repeatable to within +0.2 m/s. The accuracy of the measurements was limited by the positional accuracy o f the traverse gear and the drift in the air supply) although the latter was kept within the limits defined by the allowable change in the frequency of the cavity pressure oscillation. A field representation of the measured mean How vectors in the resonator neck is shown in Figure 8. The fluctuating velocity field in the resonator neck was first examined by measuring the power spectra of the velocity fluctuations in the two component directions. The

26

P. A. NELSON, N. A. HALLIWELL AND P. E, DOAK -

"

.

.

.

10 m/s

Contour corresponding to slreamwise meon flow values of I m/s

.

Figure 8. Mean flow vectors in the resonator neck.

0

I

(a)

I

-I0 -20 -30 -40 -riO -60

0

(b) -tO -20 -30 -40 -50 -60 i

i

IOO

IOOO Frequency (Hz)

Figure 9. Power spectra of the resonator neck flow. (a) Streamwise velocity fluctuations; (b) vertical velocity fluctuations.

FLOW EXCITED RESONANCE, PART I

27

I.S.V.R. PDP 11/45 digital computer was used to acquire both the streamwise and vertical tracker outputs directly and evaluate the power spectral density of the velocity signals. Power spectra were produced having a resolution of 10 Hz which were accurate to within • dB with 95% confidence. Figure 9 shows typical power spectra of the velocity fluctuations in the flow in the two component directions. The measuring point was at a position xt = 5 mm and x2 = - 3 mm, where the xl and x2 co-ordinates are in the streamwise and vertical directions, respectively, and the origin is at the corner of the upstream lip of the neck on the upper surface of the resonator. In both directions, the velocity spectra are dominated by the fundamental frequency of the vortex shedding (605 Hz). Higher harmonics of the shedding are clearly visible on the spectra and up to four higher harmonics were detected on the spectra measured at various points in the flow. The fundamental frequency was always at least 5 dB higher in amplitude than the second harmonic on all the spectra sampled. The variation of the amplitude and relative phase of the fundamental frequency of the two velocity components was investigated thoroughly throughout the measurement plane. The amplitude of the velocity fluctuations in each component direction at the (a)

(b)

I

Figure 10. (a) Streamwise and (b) vertical r.m.s, velocity fluctuations. Contour increment 0.1 m/s; -, 1.0; =, 1.5;-, 2.0;_=-,2.5 m/s.

28

P. A. NELSON, N. A. HALLIWELL AND P. E. DOAK

fundamental frequency was measured by filtering the tracker output signals with a 1/3 octave band filter centred on 630 H z and measuring the levels on a true r.m.s, voltmeter. Results were repeatable to within + 0 . 2 m / s , although in some cases, particularly for the vertical c o m p o n e n t near the centre of the neck, the results were repeatable only to within • m / s . Errors were largely determined by positional accuracy which was m o r e important in regions of large velocity gradient. Results for the r.m.s, values of the streamwise and vertical velocity c o m p o n e n t s are shown in the contour plots of Figures 10(a) and (b) respectively. T h e phase of the velocity fluctuations at the fundamental frequency was measured relative to the phase of the cavity pressure fluctuations. R e s o n a t o r pressure was monitored with a condenser microphone and a 1 m m p r o b e tube and care was taken to correct for phase shifts introduced through this and the 1/3 octave filter sets used on both velocity and pressure m e a s u r e m e n t channels. Relative phase difference was measured by using a H e w l e t t - P a c k a r d analogue correlator. Conventional phase meters were found unsuitable since any unsteadiness in the input velocity signal produced rapid fluctuations in

(a)

(b)

Cf

Figure 11. Phase of (a) streamwise and (b) vertical velocity fluctuations. Contour increment 20~ -, -240~ =, -180"; =, -120~ ~ -60*; ~, 0~ -, +60 ~

FLOW

EXCITED

E E

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30

P. A. N E L S O N , N. A. H A L L I W E L L A N D P. E. D O A K

phase meter readings. Results for the phase of the streamwise and vertical components of velocity are shown in the contour plots of Figures ll(a) and (b) respectively. Results were repeatable to within • ~ this again being limited by positional accuracy. This error was greater in the region of the flow just below the resonator neck where - 0 . 5 mm > x 2 > - 1 - 5 mm for the streamwise component measurements. In this region, the phase of the streamwise component was very difficult to measure accurately. Finally, the time histories of the frequency tracker outputs were recorded at a variety of positions over the measurement grid in the resonator neck. The unfiltered tracker outputs giving the voltage analogues of the streamwise and vertical velocity components were recorded by using a storage oscilloscope. The oscilloscope traces were photographed by using a Polaroid camera. The results are shown in Figure 12. 4.3. DISCUSSION The discussion presented in this section is confined to an initial analysis of the principal featues of the results obtained. A detailed analysis of the implications of the velocity measurements is given in the companion paper. The dominance of the streamwise component of the mean velocity is clearly shown in Figure 8 where significant vertical mean flow components only become evident at the lower levels of the shear layer and towards the upstream end of the neck. The results show the entrainment of air from within the resonator cavity. This is the result of a large-scale slowly recirculating mean flow inside the cavity. This was visible during the flow visualization and is driven by mean flow entering thecavity below the downstream lip. This travels around the inside of the cavity and is entrained into the underside of the shear layer towards the upstream lip. The spreading of the shear layer is also indicated in Figure 8 where the contour corresponding approximately to a streamwise mean flow of 1 m/s is shown. The spread of the layer appears roughly linear with distance from the upstream edge. This is confirmed by the collapse of the streamwise mean flow data shown in Figure 13. A reasonable collapse of data has been produced by normalizing the streamwise mean flow values by the velocity fflmax at the upper edge of the shear layer and plotting the results against i

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0]7

n 0"8

Figure 13. Collapse of mean flow data. Value of xt(mm): O, 2; [3, 3; A, 4; ~7, 5; <>, 6; 6~, 7; EB, 8; A, 9.

F L O W EXCITED RESONANCE, P A R T I

31

-(xl/x2). The value of filn~a, was taken to be 12 m/s and the origin is again the point at the corner formed by the upstream lip and the upper surface of the resonator. The results of the measurement of the fluctuating streamwise velocity component provide some of the most important information regarding the vortex shedding process. The presence of discrete vortices is clearly shown by the difference in relative phase between the streamwise velocity fluctuations in the upper and lower halves of the shear layer (see Figure ll(a)). The phase change occurring when the shear layer is traversed in the vertical (x2) direction is of the order of 180 ~ This can be understood by using the simple model illustrated in Figure 14. The streamwise velocity fluctuations induced above the path taken by a vortex convecting in the xl direction will experience a positive peak velocity whilst a point below the path experiences a negative peak velocity. The time histories recorded also illustrate this point well. Figure 12 shows the change in form of the streamwise component time histories between the upper and lower halves of the shear layer. The form of time histories shown thus is in good agreement with those predicted qualitatively from the simple model of convecting vortices illustrated in Figure !4.

Vortex core path

13-

Time histories of induced velocity of point A (above the path of the vortex core)

Time histories of induced velocity at point B (below the path of the vortex core)

Figure 14. Simple vortex model.

The position of the 180 ~phase change in the fluctuating streamwise velocity component defines the position of the path taken by the centres of the vortex cores, This path is given to within +0.5 mm by the experimental results. The amplitude of the fluctuating streamwise component is also at its least value in this region. The results are thus consistent with streamwise convected vortices having a solid body rotation in the core. The vortices will induce only small streamwise velocity fluctuations close to the path taken by the cores. This is further illustrated by the time histories in Figure 12, where fluctuations measured in the region of the path of the cores becomes small and irregular. The form of these fluctuations explains why the phase of the streamwise fluctuations was difficult to measure repeatably in the region of the path of the cores. The value of

32

P.A.

N E L S O N , N. A. H A L L I W E L L A N D P. E. D O A K

the streamwise mean flow velocity along the path of the cores is 6 m/s (4-I m/s). This is the convection speed of the vortices for the length of the path travelled before reaching the region of the downstream lip. The results in this area are not well enough defined to enable an accurate deduction to be made of the path taken by the vortex cores. The amplitude of the streamwise velocity fluctuations is greatest in the upper half of the shear layer near the upstream lip of the neck. As shown in Figure 10, the peak fluctuating streamwise velocities induced by the passage of the vortices become smaller in the downstream direction. This is again confirmed by the time histories in Figure 12, At the positions (xl = 2 ram, xa = 0 ram) and (xl = 3 mm, x2 = 0 ram), for example, the peak positive velocity fluctuations are large and strongly defined. The fluctuations become weaker and more irregular in the downstream direction. The contour plot in Figure I0 also indicates a spreading of the vortices as they travel downstream. The area influenced by the streamwise velocity fluctuations induced appears to increase roughly in accordance with the spreading of the mean shear layer. Velocity fluctuations in the vertical direction are likely to contain any fluctuations resulting from the reciprocating flow driven through the resonator neck by the cavity pressure fluctuations. Any fluctuating flow of this kind will be combined with the vertical velocity fluctuations induced directly by the vortices. The order of magnitude of the velocity fluctuation in the neck produced by the cavity pressure fluctuation can be calculated from the compliance of the cavity deduced in section 2,1. The SPL in the cavity at peak aerodynamic excitation was 134 dB re 2 • 105 N/m 2. This corresponds to an r.m.s, value of I00 N/m 2 for the cavity pressure fluctuations. This implies a volume flow into the cavity corresponding to an r.m.s, neck velocity of i m/s. The maximum values of the vertical velocity fluctuations were found to occur in the region of the flow between the centre of the shear layer and the downstream lip of the neck (see Figure 10(b)). This is again well illustrated by the time histories of the velocity fluctuations. Note also that the form of the vertica! component time histories is predicted by the convecting vortex model illustrated in Figure 14. The measured values of the phase of the vertical component fluctuations show that in the centre of the neck the vertical velocity fluctuations phase lag the cavity pressure by 90 ~ The reciprocating flow inside the cavity also lags the pressure by 90 ~. The relative phase of the reciprocating flow in the neck will depend on the acoustic impedance presented to the cavity pressure fluctuations. However, it is feasible that the maximum vertical velocity fluctuation found in the centre of the neck is due to a constructive interference between the vortex induced velocity and the reciprocating flow driven through the neck by the cavity pressure fluctuations. In particular, the substantial and well defined phase differences between the vertical and streamwise velocity components are important factors in the Reynolds shear stress. With u~ and u~ defined as the fluctuations in the streamwise and vertical directions~ the Reynolds shear stress pou'~u~ (the overbar denoting time average) is given by go Ut/~ ~ 2t = pOU~rm~U2rm~COS(~p~--~2), where go is the ambient density, Ulrm~and l/2rms are the r.m.s. values of the measured velocity components, and ~ and ~2 are the relative phases of the streamwise and vertical components, respectively. The measured values of the r.m.s. amplitudes and phases have been used to plot the contour map of the Reynolds shear stress in the shear layer shown in Figure 15. There are clearly regions of negative and positive Reynolds shear stress above and below the path of the vortex cores. These stresses are concentrated at the upstream end of the shear layer. The precise nature of the shear stress pattern near the downstream end is difficult to determine within experimental error. The implications of these and the other deductions made will be discussed more fully in the companion paper.

FLOW EXCITED RESONANCE, PART I

33

Figure 15. Time averaged Reynolds shear stress. Contour increment 0-25 N/m2; -, - 3 . 0 , --, - 1 . 5 ; ; , 1.5; _=-,3.0 N/m 2.

5. THE MEASUREMENT OF THE FLUCTUATING PRESSURE FIELD IN THE RESONATOR NECK 5.1. THE MEASUREMENT OF FLUCTUATING STATIC PRESSURE

The measurement of static pressure fluctuations in an airflow necessarily involves the positioning of some form of probe in the airstream. This introduces two specific problems. Firstly, it is necessary to assess the effect on the airflow caused by insertion of the probe and secondly the pressure sensed by the probe will not necessarily be the same as the pressure in the flow before the probe was inserted. Fluid brought to rest at the sensing point of the probe will result in the corresponding stagnation pressures being registered in the pressure measurement. The static pressure fluctuations in the neck of the resonator were measured by using a 0.5 mm diameter microphone probe tube made by Briiel and Kjaer for use with a 12.5 mm condenser microphone. The probe was held such that the tube pointed vertically downwards into the resonator neck. The open end of the probe tube was thus in the plane parallel to the upper surface of the resonator. The details of the experimental procedure will be discussed fully in the next section. However, the performance of the probe will first be considered in the light of the two questions posed above. The nature of the flow being measured is such that it is very unlikely that the introduction of the probe tube in any way disturbed the fluctuating flow field at the frequency of interest. The vortex shedding at peak excitation is so well "locked in" that even the introduction of a 4 mm probe tube held vertically into the resonator produced no change in amplitude or frequency of the cavity pressure fluctuations. The wake produced by the probe is obviously approximately determined by the size of the probe. For the small (0.5 mm) probe used the wake flow will be considerably smaller than the scale of the vortices being measured and further to this for a Strouhal number of 0.2, based on diameter, fluctuations of pressure in the wake will be =4 kHz in frequency, which is well above the frequency of interest. It is thus evident that the relevant structure of the fluctuating flow field is undisturbed by the insertion of the 0.5 mm tube held in the vertical direction. There remains to consider the magnitude of the error introduced by the measurement of stagnation pressures at the end of the probe where flow in the vertical direction will be brought to rest. Detailed analysis of the errors involved is quite complex [16, p. 70] but an order of magnitude figure can be arrived at quite simply. Siddon [24] has undertaken careful experiments to assess the error in measuring the fluctuating static pressure when using a conventional pitot tube arrangement. He concluded that the error is of the order of the stagnation pressure of the velocity component normal to the sensing holes This conclusion was also reached by Strasberg [25]. For the magnitude of the

34

P.A.

N E L S O N , N. A. H A L L I W E L L A N D P. E. D O A K

vertical velocities measured in the resonator neck Nelson [23] was able to conclude that the fluctuating static pressure could be successfully measured to within an error of less than 5% at most measuring points. The agreement between theory and experiment obtained, which will be discussed in the companion paper, would seem to validate this assumption. 5.2.

EXPERIMENTAL

PROCEDURE

AND RESULTS

When making the pressure measurements the measuring probe was fixed and the nozzle and resonator were traversed relative to this in the same way as when the velocity measurements were taken. The nominal position of each measurement point was assumed to be the centre of the circular end of the probe tip. Amplitude and phase measurements were taken at each point in the measurement grid. Phase was again measured relative to the cavity pressure by using the analogue correlator, Care was taken to correct for insertion losses and phase shifts through the measurement channels. All measurements were taken at peak aerodynamic excitation. Results for the fluctuating static pressure in the resonator neck in terms of amplitude and phase are shown in the contour plots of Figures 16(a) and (b) respectively. It was found that the repeatability of the pressure measurements taken was extremely good. The amplitude at any given point could be measured to within 4-1 N / m 2 and the phase could be measured to within 4-5~ Measured values became uncertain only in one particular region of the flow near the downstream edge of the slot. In this region, the pressure amplitude measured was relatively low and the phase changed very rapidly with position. This made the pressure results in this region sensitive to the accuracy of positionai location of the probe tip. Nevertheless, the amplitude and phase could still be measured repeatably to within 4-2 N / m 2 and +15 ~ respectively. 5,3,

DISCUSSION

The maximum pressure amplitude occurs in a region near the upstream lip of the resonator neck and approaches an r.m.s, value of 70 N / m 2 (see Figure 16(a)). Velocity measurements and flow visualization both indicate that the discrete vortices shed are formed in this region. It thus seems feasible that the maximum pressure fluctuation is associated with the periodic formation of vortices having a static pressure defect associated with the vortex cores. It is interesting to note, however, that the region of minimum static pressure fluctuations (=15 N / m 2) lies towards the downstream edge of the resonator neck and is still on the path taken by the vortex cores. The relative phase of the pressure fluctuations shown in Figure 16(b) shows a very rapid change in the region of minimum pressure amplitude. Phase changes through approximately 140 ~ occur in a distance of typically I mm traverse. The precise value of the relative phase in this region was thus difficult to determine. However, this region of uncertainty was confined to an area measuring 1.5 mm in the streamwise direction and 2.0 mm in the vertical direction centred on the point having co-ordinates xl = 7 ram, x2 = -0"5 mm. The reason for this unusual behaviour of the measured amplitude and phase in this area is not easy to determine upon initial examination. It certainly appears to be related to the passage of vortices through the region. It is possible that the pressure fluctuations produced by the periodic convection of the pressure field associated with the vortices interferes with pressure fluctuations related to the reciprocating neck flow driven by the cavity pressure fluctuations. More detailed consideration of the fluid dynamics involved is required before any firm conclusions can be reached as to the cause of this interference pattern and this will be undertaken in the second paper.

FLOW EXCITED RESONANCE, PART I

35

in)

I

I

(b)

.

.

Figure 16. Pressure fluctuations (a) r.m.s, amplitude, contour increment 2.5 N/m2; - , 20; =, 30; -, 40; - , 50; - 60 N / m 2 and (b) phase, contour increment 10~ -, - 6 0 ~ , - 3 0 ~ -=, 0~ - , 30~ - , 60 o.

Finally, it is worth emphasizing that the pressure measurements made provide a very good estimate of the true fluctuating static pressure in the resonator neck. Even in the region of lowest pressure amplitude, the measured r.m.s, fluctuation of 15 N/m 2 was well above the fluctuating stagnation pressure. The velocity measurements show that the r.m.s, value of the stagnation pressure in this region is only 2.5 N/m 2 [23]. It is thus

36

P.A. NELSON, N. A. HALLIWELLAND P. IE. DOAK

impossible that the measured interference pattern is a result of the measurement of anything other than static pressure fluctuations. 6. CONCLUSIONS A series of careful experiments has been undertaken in order to gain an understanding of the fluid dynamics of the interaction between unsteady fluid motion and an acoustic resonance. The flow excited resonance chosen for study was a Helmholtz resonator excited by a grazing boundary layer flow. The flow produced is such that detailed measurements can relatively easily be made of the mean and fluctuating parts of the velocity field and the fluctuating pressure field in the resonator neck. The resonator used in the study had a natural frequency of 605 Hz. At low levels of excitation the resistive part of the acoustic impedance of the resonator was dom"inated by radiation resistance. When excited with the grazing flow, the resonator reached its peak amplitude of excitation at 605 Hz. At this frequency, the sound pressure level in the cavity reached 134 dB and 6 mW of acoustic power was radiated to the far field. A flow visualization study indicated that a mean feature of the flow in the resonator neck was of a series of periodically shed vortices. These vortices were formed during the first half cycle of the motion as the pressure in the cavity changed from its peak negative to its peak positive value. Discrete vortices appear to reach the mid-point of the resonator neck in the streamwise direction when the pressure in the cavity reaches its peak positive value. By the end of the cycle, when the cavitypressure returns to its peak negative value, the vortices straddle the downstream lip of the resonator neck. The flow visualization also indicated that the mean convection speed of the vortices was 6 m/s when the vortices traversed the width of the neck. This convection speed appeared to increase to 12 m/s once the vortices had been ejected from the resonator neck at the downstream lip. Another important feature of the results was that the shear layer leaving the upstream lip always appeared tangential to the upper surface of the resonator. Measurements of the mean flow velocity in the resonator neck showed a shear layer thickness growth which was approximately linear in the downstreamdirection. The mean flow speed at the upper edge of the shear layer was 12 m/s, whilst the velocity ohanged across the layer to a flow speed near 1 m/s, associated with a large scale, relatively slow, recirculating mean flow inside the cavity. Behaviour of the shed vortices can be deduced from the measurements of the fluctuating streamwise velocity component. The presence of discrete vortices is confirmed by the 180 ~ change in the phase of the streamwise component between the upper and lower halves of the shear layer. The position of this phase change enabled the location of the path taken by the vortex cores to be made, which could be defined to within 1 mm, and also confirmed that the vortices travel at a mean convection speed of 6 m/s across most of the neck width. In the region of the downstream lip the path taken by the vortices is difficult to establish from the measured results. Streamwise velocity fluctuations are consistent with the passage of vortices having a solid body rotation in their core. This is also confirmed by the recorded time histories of the velocity fluctuations. The amplitude of the streamwise velocity fluctuations is highest towards the upstream edge of the slot and decays in the downstream direction, which implies that the growth of the unstable shear layer into discrete vortices takes place rapidly at the start of the cycle and is followed by a spreading of the vortices as they are convected downstream. This spreading of influence of the vortices is roughly in accordance with the growth of the mean shear layer.

FLOW EXCITED RESONANCE, PART I

37

Measurements of the phase difference between the vertical and streamwise velocity components show well defined regions of negative and positive Reynolds shear stress in the upper and lower halves of the shear layer, respectively. These stresses are concentrated at the upstream end of the shear layer whilst the stress distribution becomes less well defined towards the downstream end of the layer. The fluctuating static pressure in the resonator neck has been successfully measured by using a small microphone probe tube, and the results produced show a complicated interference pattern. This pattern appears to be related to the pressure fluctuations created by the periodic formation and convection of vortices. It is possible that these pressure fluctuations interfere with pressure fluctuations associated with the recriprocating flow in the resonator neck. The experiments undertaken have thus produced a comprehensive and detailed knowledge of the behaviour of the flow in the resonator neck at peak self-excitation. However, many features of the experimental results are not easy to explain at first sight. Certainly, there is some form of interaction between the vortically induced flow field and the reciprocating flow driven by the cavity pressure fluctuations. Before any further conclusions can be reached as to the nature of this interaction, careful consideration is required of the precise definition of these interacting flows. The companion paper will be devoted to a thorough analysis of the experimental results in an attempt to establish more precise quantitative details of the vortex shedding process. These details will then be used to examine the m o m e n t u m and energy balances in the shear layer. Particular attention will be paid to achieving an understanding of how 6 m W of far field acoustic power is extracted from the flow acoustic interaction.

ACKNOWLEDGMENT P. A. Nelson wishes to acknowledge the financial support of Sound Attenuators Limited, Colchester, throughout the period of this research work and the technical assistance of Mr A. G o m m in constructing the experimental apparatus.

REFERENCES 1. L. CREMER and H. ISING 1967 Acustica 19, 143-153. The self-excited vibrations of organ pipes (in German). 2. N. n . FLETCHER 1979 Annual Review of Fluid Mechanics 11, 123. Airflow and sound generation in musical wind instruments. 3. G. B. BROWN 1937 Proceedings of the Physical Society 49, 493-507. The vortex motions causing edgetones. 4. N. CURLE 1953 Proceedhzgs o[ the Royal Society, Series A 216, 412-424. The mechanics of edgetones. 5. A. POWELL 1961 Journal of the Acoustical Society of America 33, 395-409. On the edgetone. 6. S. A. ELDER 1973 Journal of the Acoustical Society of America 54, 1554-1563. On the mechanism of sound production in organ pipes. 7. J. E. ROSSITER 1966 Aeronautical Research Council Reports and Memoranda, R&M No. 3438. Wind-tunnel experiments on the flow over rectangular cavities at subsonic and transonic speeds. 8. R. PARKER 1966 Journal of Sound and Vibration 4, 62-72. Resonance effects in wake shedding from parallel plates; some experimental observations. 9. R. PARKER 1967 Journal of Sound and Vibration 5, 330-343. Resonance effects in wake shedding from parallel plates: calculation of resonant frequencies. 10. N. A. CUMPSTY and D. S. WHITEHEAD 1971 Journal of Sound and Vibration 18, 353-369. The excitation of acoustic resonances by vortex shedding.

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P. A. NELSON, N. A. HALLIWELL AND P. E. DOAK

11. F. S. ARCHIBALD 1975 Journal of Sound and Vibration 38, 81-103. Self-excitation of an acoustic resonance by vortex shedding. 12. J. S. BOLTON 1976 M.Sc. Dissertation, Institute o/Sound and Vibration Research, University o/Southampton. The excitation of an acoustic resonator by pipe flow. 13. S. A. ELDER 1978 Journal of the Acoustical Society of America 64, 877-890. Self-excited depth-mode resonance for a wall mounted cavity in turbulent flow. 14. J. E. RlZZO and N. A. HALLIWELL 1978 Review of Scientific Instruments 49, 1180-1185. Multi-component frequency shifting self-aligning laser velocimeters. 15. L. E. KINSLER and A. R. FREY 1962 Fundamentals o/Acoustics. New York: John Wiley, second edition. See p. 186. 16. P. M. MORSE and K. U. INGARD 1968 TheoreticalAcoustics. New York: McGraw-Hill. 17. M. S. HOWE 1976 Journal of Sound and Vibration 45, 427-440. On the Helmholtz resonator. 18. U. INGARD and H. ISING 1967 Journal of the Acoustical Society of America 42, 6-17. Acoustic nonlinearity of an orifice. 19. F. R. HAMA 1962 Physics of Fhdds 5, 644-650. Streaklines in a perturbed shear flow. 20. Y. YEH and H. Z. CUMMINS 1964 Applied Physics Letters 4, 176-178. Localised flow measurements with an H e - N e laser spectrometer. 21. F. DURST, A. MELLINO and J. H. WHITELAW 1976 Principles and Practice o/ Laser Doppler Anemometry. London: Academic Press. 22. T. H. WILMSHURST and J. E. RIZZO 1974 Journal of Physics E: Scientific Instruments 7, 924-930. A n autodyne frequency tracker for laser doppler anemometry. 23. P. A. NELSON 1981 Ph.D. Thesis, University o/Southampton. Aerodynamic sound production in low speed flow ducts. 24. T. E. SIDDON 1969 Universityo/Toronto UTIASReportNo. 136. On the response of pressure measuring instrumentation in unsteady flow. 25. M. STRASBERG 1963 AGARD Report No. 464. Measurements of fluctuating static and total head pressure in a turbulent wake.