The acoustical impedance of holes in the wall of flow ducts

Journal of Sound and Vibration (1972) 24 (1) 133-150

THE ACOUSTICAL IMPEDANCE OF HOLES IN THE WALL OF FLOW DUCTSt D. RONNEBERGER

Drittes Physikalisches lnstitut, Universitiit G6ttingen, 34 G6ttingen, West Germany (Received 12 April 1972, and in revisedform 19 June 1972) The radiation impedance of circular and oblong holes in the wall of a flow duct has been measured as a function of the flow velocity. The boundary layer at the wall of the duct is thin compared to the dimensions of the orifices. At low Strouhal numbers (quasi-static case) and constant boundary layer thickness, the flow resistance of the orifice (real part of the impedance) increases in proportion to the flow velocity. The imaginary part of the impedance corresponds to a constant, negative attached mass above the orifice, i.e. the impedance is spring-like. In the transition range from air at rest to the quasi-static case (high Strouhal numbers) the impedance as a function of the flow velocity describes a spiral in the complex plane. The mechanism causing the flow dependence of the impedance is illustrated by a simple model of the flow above the orifice. As a practical example of the flow-dependent impedance of orifices, the flow-dependent sensitivity of a probe microphone used in flowing media is discussed.

1. INTRODUCTION It is often necessary to d a m p flow ducts acoustically. In most cases the lining material is protected by perforated metal plates from damage by the flow. Sometimes the perforated plates are mounted some distance from the rigid duct wall without any damping material in the interspace; to prevent sound propagation, the interspace is subdivided into separate cavities. Thus one gets Helmholtz resonators, provided the distance between the perforated metal sheet and the wall is smaller than )q/4, A, being the wavelength of the sound.~ Often, the acoustical properties of such duct linings depend strongly on the impedance of the perforated plate. This impedance can be changed considerably by the flow in the duct [1-5]. Thus an optimal design of the duct lining becomes difficult. This is one of the reasons why it is interesting to study the impedance of a single hole in the wall of a flow duct as a function of the flow; in a way, this is a model of the perforated plate. Another example which demonstrates the dependence of the hole impedance on tangential flow is the change of the sensitivity of probe microphones used in flowing media. In section 2 the sensitivity of a probe microphone is discussed in detail. The impedance of a hole in a rigid wall is changed not only by tangential flow but also by flow through the hole or by large particle velocities causing separation of the acoustical flow at the edges of the orifice. Westervelt has explained, in a general way, the dependence of the hole impedance on the flow by the fact that the kinetic energy stored in the oscillating medium t Presented at the symposium, jointly sponsored by the British Acoustical Society and the American Society of Heating, Refrigeration and Air Conditioning Engineers, on "The Acoustics of Flow Ducts", at the Institute of Sound and Vibration Research, University of Southampton, Southampton, England on 10 to 14 January 1972. $ A list of symbols is given in the Appendix. 133

134

D. RONNEBERGER

flowing across the orifice is blown away, so to speak [1 ]. This loss of energy causes an increase of the acoustical flow resistance of the hole and a decrease of the so-called attached mass. This process of "blowing away" has been largely clarified in the case of flow through the hole [6-12]; here the aim is to try to throw some light on that process in the case of tangential flow across the hole. 2. SENSITIVITY OF A PROBE MICROPHONE IN FLOWING MEDIA Figure l(a) shows the sensitivity of a probe microphone which is placed in the axis of a circular duct containing turbulent flow (the dimensions of the probe tube can be read from the sketch above the diagram); the Mach number in the duct axis is the abscissa and the frequency is the parameter. These data were obtained from the pattern of the sound pressure 3m

L

~[,

2 5m ~6mm

Electrodynomic microphone

"•i

4 h o l e s of Imm diometer

Io.~

'

'

'

1

Imp-- I

io 1

(b)

X

-I

X\+.

Nil

,.%

"-~ - 2

-3

0

I O. I

I 02

l 0'3

1 0 4

i 0.5

0

0.~

0-2

0.3

M

Figure 1. Sensitivityof the probe microphone sketched above as function of the Mach number. (a) Microphone in the axis of a duct containing turbulent flow. (b) From the model experiment sketched in Figure 2(a). O, 0"3 kHz; ×, 0'6 kHz; +, 0"9 kHz; A, 1.2 kHz; II, 1"5 kHz. in tubes with smooth changes of the cross-section. Powell has calculated this sound pressure pattern for the principal mode assuming isentropic flow with a plane flow profile [13, 14]; from his formulae one gets

F(pl,p2, M)=[p2(1 +

M ) 2 - p 2 ( l - M)2]( 1 + ~ ) - M 2 ) M

e+'/e-' = const,

(1)

where Pi and P2 are the sound pressures of the waves propagating down- and upstream, M is the Mach number, and ), the ratio of the specific heats. According to Blokhintsev this equation can be interpreted as the law of conservation of the acoustic energy in flowing media [15] (the density of the medium, the sound velocity and the cross-section usually involved in equations for the acoustic energy are functions of the Mach number and in the present case thus included in equation (1)). Measurements have shown that Powell's calculation describes the sound pressure pattern in ducts with turbulent flow, too, and that in particular the loss of acoustic energy at the changes of the cross-section is only small [16].

IMPEDANCE OF HOLES

IN F L O W

DUCTS

135

If the sensitivity (r of the microphone depends on the flow Mach number, the measured sound pressures p in front of (index a) and behind (index b) the change of cross-section yield F(poI,J~o2, M°)

F(/Sb,,/Sb2, Mb)

=

~(Mo)

~,2(Mb) ,

F(PH,fibS,Mb)

,

log [a(Mb)] -- log [a(go)] = ~ l o g ~ ~ .

(2)

Pol,/5°5, Pb~, Pb2 are calculated from the maxima and minima of the standing wave patterns in front of and behind the change of the cross-section.

Annular nozzle

..............

Microphone

//./.~

4 holes of

I mmdiameter

O'IF

I

I

I

[

I

[

018 0 ~ b ) f-O'3 I 0'7 0-8 I

o

I

0"9

I

l I'0

0"1

E

0"7 iI

I

0"8

I

I

I

I

I

OI0

~ 0

0"9

I

I

I

80

I

I'0 I

f=l.2 J

I

I

I

0.9

0

f=O-6 I

I (el

(c)

65-''-~ 4

0

I

0 I

80

~o

i :o~,.

Ol

1

10 I

8 0 ~ 6 0 (f)

(d)

6O 0 f=l'5

f" 0'9

0.7

I

I

0.8

I

I

0.9

I

I I-0

f @9

I

I I.O

Re [v (%)/v (0)] Figure 2. Model experiment on the flow dependent sensitivity of a probe microphone. (a) Sketch of the experimental set-up. (b)-(f) Complex ratios of the output voltages V of the microphone at ftow velocity Uo (in m/s) and air at rest; the parameter is the frequency (in kHz). This kind of measurement of the sensitivity of a microphone as a function of the flow velocity is not very precise, since it yields only the ratios of the sensitivities for several pairs of Mach numbers from which the whole curve (r(M) is computed by integration. Therefore a model experiment was conducted by which (r(M) was measured directly but in which, however, the flow profile near the microphone orifices did not agree with the flow profile in the tube. Figure 2(a) shows the experimental set-up. With the aid of an annular nozzle a thin layer of flowing air is produced above the microphone orifices. The same probe tube and microphone orifices were used in this experiment and the experiment of Figure 1(a). The flow velocity is measured by a hot-wire anemometer. The figures 2(b) to 2(f) show the

136

D. RONNEBERGER

ratios of the microphone voltages with and without flow plotted in the complex plane as a function of the flow velocity; the parameter is the frequency. These curves are very similar to the plots of the impedance of an orifice with tangential flow (see Figures 4 and 12). For comparison with Figure 1(a), Figure 1(b) shows the sensitivity of the microphone calculated from Figure 2(b). The curves a(M) resulting from the two different measurements agree qualitatively; the differences arise from the different flow profiles: the thinner boundary layer in the model experiment leads to a stronger dependence of the microphone sensitivity on the flow velocity. Of course the change of the microphone sensitivity due to the flow depends also on the input impedance of the microphone orifices in air at rest, since the relative change of the impedance by the flow is the higher the smaller the impedance. Thus the sensitivity of microphones displaying acoustical resonances depends especially strongly on flow. 3. MEASUREMENT OF THE ACOUSTICAL IMPEDANCE OF A HOLE 3.1. MEASURING EQUIPMENT AND PROCEDURE

The experimental equipment for measuring the acoustical impedance of a hole with tangential flow is outlined in Figure 3. The hole, in the wall of a flow duct, connects the interior of the duct with a cavity, the so-called measuring cavity, the radius R and the height H of which are R = 25 mm and H = 18 mm. A loudspeaker in the wall of the duct generates the sound pressures P2 in the duct and p~ in the cavity. The dimensions of the cavity are small compared to the sound wavelength. Apart from a small region around the orifice the sound pressure is therefore constant all over the cavity; it is measured by a condenser microphone. The complex impedance W of the hole is defined as v

v

/~

- 1 ,

(3)

where v is the particle velocity, C the compliance of the measuring cavity, oJ the angular frequency, and i = (-1) t/2. Thus W can be evaluated from the complex ratio of p2 and p~. For measuringpz an additional sound pressure is generated in the cavity, just sufficient to stop the particle velocity through the hole; then the sound pressures on both the sides of the hole must be equal and P2 can be measured by the microphone inside the cavity. To generate the additional sound pressure (Pz - Pt) inside the measuring cavity a pressure chamber loudspeaker is coupled to the cavity by a very high flow resistance. The particle velocity through the hole is measured by a hot wire anemometer. The polarizing flow necessary for it is also fed into the cavity through a high flow resistance. This flow is switched off while measuringp~ to avoid its influence on the measurement of the impedance. The polarizing flow does not affect the measurement ofp2, of course, since it is independent of the hole impedance which may be changed by the flow. This statement was proved experimentally. The output voltage of the microphone is rectified by two lock-in amplifiers. Their outputs are two d.c. voltages proportional to the real and the imaginary parts, respectively, of the complex ratio of the microphone voltage and an adjustable reference voltage. These d.c. voltages are recorded by a X-Y-recorder. The reference voltage is adjusted so that the X-Yrecorder indicates a real value of unity in the complex plane when Pl is measured. Then it records Pz/Pt when P2 is measured. From this record the impedance normalized to 1/~oC can be read immediately according to equation (3) by shifting the origin by 1 and turning the plane by 90 °• The flow duct has a cross-section of 10 × 10 cm 2 behind the inlet (contraction ratio 25: 1). As shown in Figure 3 the height of the duct is reduced further, being 5 cm at the measuring

IMPEDANCE OFHOLESINFLOWDUCTS

137

position. Thus the flow is steadily accelerated and remains laminar. At the measuring position the boundary layer is about 0.1 mm thick. Downstream of the measuring position the crosssection of the duct is widened to 10 × 10 cm2 again. The flow velocity can be adjusted by throttling the suction side of the blower. The velocity is measured at the measuring position by a Pitot tube; it is 40 m/s at most.

=

.... Condenser

=me

mcrophone"'--...~1~ ] I .....

P2

I

~ J PL ~ H

o t {'7 Measuringcavify I I

..... wire probe Loudspeaker

......./ ! Re(p~lp,~l (--Yrecorder (complex )lone of the sound pressure)

~ ~ T:!!~i~i:.~::~)~?;~i~iii:i i ~~ FI0wresistance Pressure chamber loudspeaker

Figure3. Sketchofthemeasuringequipment. 3.2. CORRECTION OFTHEEVALUATION OFTHEMEASURED RESULTS In deriving equation (3) it was assumed that the admittance of the measuring cavity is determined only by the compliance of the air volume inside the cavity. This is not quite correct for the cavity used here since there is a high flow resistance coupling the pressure chamber loudspeaker to the cavity. This flow resistance consists of a 2 cm thick layer of sand glued together by water-glass. The admittance of this layer must not be disregarded especially at low frequencies. In calculating the sound propagation inside the layer one may neglect the mass forces. If--as here--the thickness of the layer is greater than 0-3 times the wavelength inside the layer, the admittance of the layer as seen from the interior of the cavity is equal to the characteristic admittance: L = (1 + i

)

~

= (1 + i)X/-~L'.

(4)

C' and R' are the specific compliance and the specific flow resistance of the glued sand; the constant L' must be determined by a calibration measurement. Incidentally, the acoustical losses by heat conduction at the wails of the measuring cavity can be expressed by an equivalent admittance, Lr = (1 + i ) ~ L ' r (according to reference [17]) which, however, is about 20 times smaller than L in our case. For the determination of the constant L' in equation (4) the impedance of the two circular holes to be investigated was measured as a function of the frequency for air at rest. According to references [18-21] this impedance is sufficiently well known (cf., section 4.1) to make possible the calibration of the measuring cavity. These calibration measurements proved with sufficient accuracy that (i) the additional admittance L of the cavity is indeed given by equation (4), (ii) additionally there exists a small resistive admittance independent of frequency 10

138

D. RONNEBERGER

(probably due to the inlet of the polarizing flow), and (iii) in the present case the formulae for the impedance of the hole must be corrected, if it is assumed that the admittance of the measuring cavity is the same for both the holes. This last point is treated in the next section in more detail. For testing the calibration a narrow-meshed gauze screen (0.065 Oc) was stretched across the 3 mm orifice (No. 1) and a real flow resistance independent of frequency resulted as the difference between the impedances of the orifice with and without the gauze screen. This measurement proved, however, that at very low frequencies the accuracy does not suffice to get information on the imaginary part of the impedance (section 4.2). 4. RESULTS OF THE IMPEDANCE MEASUREMENTS Table 1 lists the dimensions of the four holes to be investigated here. TABLE 1

Dimensions of the holes investigated No. of the hole

Description of the hole 3mm

I ! 2 !

circulor

diameter 2a = 6mm

oblong

4

£D |

The length of all the holes (thickness of the duct wall) is 5 ram. 4.1. IMPEDANCE FOR AIR AT REST

The impedance of a hole in a wall is mainly determined by the mass contained in the neck of the hole and the attached masses on both the sides of the hole (end corrections), if the diameter of the hole is small compared to the sound wavelength (here: 2 × 10-3 < ka < 5 × 10-2). Under the present conditions, however, the influence of the viscosity on the impedance must not be disregarded (the impedance was measured at frequencies between 80 and 800 Hz). The influence of the viscosity may be roughly characterized by the fact that the impedance acquires a real part caused by the friction forces and that the effective cross-section of the hole decreases as a result of the acoustical boundary layer at the walls of the hole. Crandall [18] has calculated the influence of the viscosity on the impedance of a circular hole not taking into account the end corrections. His relatively complicated formulae were computed by Nolle [19] and Thurston [20]; they found good agreement between experiments and the calculations. Furthermore, Thurston and Wood [21] have measured the end corrections for the real and the imaginary part of the impedance and found them to be equal and in agreement with Ingard's calculation. Ingard [22, 23] has calculated the end correction for the imaginary part only, but he has assumed that the hole leads concentrically into a circular tube (i.e. into the measuring cavity in the present case (cf. Figure 3)); his results may be approximated by Ix =0-85(1 - 1.25aiR)a; aiR < 0.6. (5) IK is the end correction for one side of the hole and R is the tube radius.

IMPEDANCE OF HOLES IN FLOW DUCTS

139

Taking into account all corrections stated above yields the theoretical value of the hole impedance which was used for the calibration of the measuring cavity (section 3.2). The calibration measurements, however, yielded an admittance of the cavity which was 7 ~ higher for the 3 mm hole (No. 1) than for the 6 mm hole (No. 2). The only possible explanation for this is the fact that the influence of the back wall of the cavity on the end correction was neglected. Since the end correction on the side of the cavity is not very important for the flow dependence of the hole impedance, that end correction is assumed to be 1~ = 0.85(1 - 1.25air - ,a/H)a.

(5')

H is the height of the measuring cavity and the constant E turns out to be E = 2.1. Some doubt about this assumption arises from the fact that according to reference [22] a rigid back wall should increase the attached mass. After the introduction of this correction the acoustically measured volume of the cavity is equal to the geometric volume. 4.2. IMPEDANCE FOR FLOWING AIR One expects that only the radiation impedance of the orifice---corresponding to the end correction--on the duct side is changed by the flow. Therefore only this part of the impedance is regarded in the following, i.e. the impedance of the neck and the attached mass on the cavity side is subtracted from the measured impedance. I

[

I

I

1

I 0

I 05

I I

I 1.5

0:8

t~ % ~3

o.',

,-oX,,,

o

--0.5

! ~05

Re(W r ) / 0 85paa.,

Figure4. Complexradiation impedanceof an orificewith grazingflow.I, Measuredpoints; ©, interpolated points; 2a = 6 mm, w = 2~r665s-L Figure 4 demonstrates the typical dependence of the radiation impedance on the flow velocity. The radiation impedance is normalized to 0.85 paw--that is, the value of the radiation impedance for air at rest when the viscosity is neglected--and it is plotted in the complex plane. The flow velocity is normalized to aw (ao~/Uo is a Strouhal number). For small Uo/ato the impedance forms a spiral in the complex plane; this spiral can be smaller or larger depending on parameters like a and co. With increasing Uo/ato the impedance approaches asymptotically a straight line parallel to the real axis. On this straight line the imaginary part of the impedance is negative and the real part increases linearly with Uo/aw, which can be seen from Figure 5. There the real and imaginary parts of the normalized impedance are plotted as functions of Uo/aw for the 3 mm hole (No. 1). Contrary to the situation for Figure 4 the flow velocity is held constant here (30 m/s) and the frequency is varied. Thus all the flow parameters are constant, especially those of the boundary layer above the hole. The measuring accuracy does not suffice to get more details about the imaginary part of the impedance.

140

D. RONNEBERGER

The linear increase of the real part of the radiation impedance W, with Uo/aw means that the real part normalized to a value independent of frequency, e.g. pc, is constant for high Uo/a~o (quasi-static case),

Re(W,)= A UO + B ' 0.85paw

aoJ

Re(W,______~)=0.8SAM

1+

pc

A Uo]"

(6)

The flow Mach number M in equation (6) does not mean that the compressibility of the medium has to be taken into account; it only arises from the arbitrary normalization to pc. 50

I

,

25

20

o

~"

~0

qUJ~qpqmeoOoo O ° o • Im i

J

I

U~l~

Figure 5. Real and imaginary parts of the radiation impedance of a circular orifice as functions of the reciprocal Strouhal number; 2a = 3 mm, U0 = 30 m/s. A normalization of the impedance to p U0 would be more appropriate to the problem but not very useful for applications: lira a~/Uo~O

Re(W,) = 0.85A. pU0

(6a)

In Figure 6 the quasi-static flow resistance of the two circular holes is plotted as a function of the Mach number. It is compared with the static flow resistance measured by a small steady flow through the hole, the flow velocity being small compared to Uo. The static flow resistance agrees very well with the quasi-static flow resistance measured acoustically. The measuring points for the two circular holes lie on two parallel straight lines. The reason for the difference of the measured results for the two geometrically similar orifices, can only be related to flow parameters which change in the same way as the diameter o f the orifice. Such flow parameters are the ratio 6/a of the boundary layer thickness to the radius of the orifice on the one hand and on the other hand the Reynolds number containing the radius of the orifice. Quite certainly the boundary layer thickness 3 should be independent of the diameter of the orifice and proportional to the square root of the flow velocity. IfA in equation (6a) is a function of ~/a it should be a function of (a2Uo) -1/2, too; if, however, A depends additionally or exclusively on the Reynolds number, it should additionally or exclusively depend on aUo. In Figure 7 A is plotted as a function of (a s Uo)-1 which is suggested by the

141

IMPEDANCE OF HOLES IN FLOW DUCTS

results of Figure 6. Within the measuring errors the points follow a straight line. The intersection point with the ordinate is 1/0.85 times the slope of the straight lines in Figure 6. One can easily prove that the measuring points would not fall on a single curve if A were plotted as a function of aU0. Thus A does not depend on the Reynolds number here, i.e. the viscosity does not play a role and the change of the radiation impedance by the flow is caused exclusively

I

I

t

I

I

i

O'OB

a/

006

I/J 0-04

i o,~/~"

0"02

~

~

J I

I

,l

I

0-04

0'02

I

O06 M

1

0 O8

1

O-IO

0"12

Figure 6. Quasi-static and static flow resistances of the circular orifices as functions of the flow Mach number. Acoustic: @, .?.a= 3 m m ; A , 2~ = 6 mm. Static: o, 2~ = 3 mm; ~, 2a = 6 mm. l

/

t

w

0.8

0"7

0'6

0'5 o

0

i 2

', I

I 3

[

,,

4

(•2 ~jO}-I [CrT~2m/s] -I

Figure 7. CoefficientA as function of (a2 U0)-1; the symbols have the same meaning as in Figure 6. by the geometry of the flow. Unfortunately the boundary layer thickness was not measured here, but an investigation is in progress emphasizing the dependence of the impedance on the boundary layer thickness. Of course the measuring points for higher values of (a 2 Uo) -l, or, respectively, (8/a) z, will not lie any longer on the straight line of Figure 7, because otherwise the real part of the impedance would get negative. If~/a gets large compared to unity the impedance will depend only on the gradient U' of the flow velocity above the orifice; then one gets Re (W,) = 0.85A" pU' a

'

8 >> 1, a

U' < 1,

(7)

142

D. RONNEBERGER

A' being independent of the boundary layer thickness; if Reynolds number U'aZ/v, one obtains

A

=

4'

A' is also independent of the (8)

a/f,

providing a suitable definition of the boundary layer thickness. Baiter [24] has measured the flow resistance of a hole of 1 mm diameter in the wall of a tube containing turbulent flow. From his measured results one calculates A' = 0.62, independently of the Reynolds number (70 ~< U'a2/v <~220). At the highest Reynolds number the viscous sublayer is ~ mm thick. In Figure 8 the quasi-static flow resistance of the two oblong orifices is plotted as a function of the Mach number. For comparison the two straight lines of Figure 6 for the circular orifices are introduced. If one intreprets the oblong orifice with the flow in longitudinal direction (No. 3) as a "slim 6 mm orifice" and the oblong orifice with the flow in lateral direction (No. 4) as a "wide 3 mm orifice", the results plotted in Figure 8 can be interpreted easily, namely, widening of the orifice perpendicularly to the flow direction causes an increase of the flow resistance. l

t

I

I

~

I

-/../

0-08

006

oJ 0 0 4 O:7

0"02

I 002

[ 0.04

I 006

I 0 08

I 0 I0

I 0"12

0.14

A4

Figure 8. Quasi-static flow resistances of the oblong orificesas functions of the Mach number. The straight lines originate from Figure 6 and stand for the circular orifices (No. 3: 0; No. 4: e). Finally some results shall be reported qualitatively. Rounding of the edges of the orifices results in a smaller dependence of the impedance on the flow velocity. The radius of curvature of the edges, however, has to be rather large to affect the impedance at all. Thus at the measuring conditions quoted above the impedance of a hole of 3 mm diameter and 5 mm length and edges with a radius of curvature of about 1 mm is changed half as much by the flow as the impedance of the same hole with sharp edges. Stretching a narrow-meshed gauze screen (ca. 0.1 pc) across the orifice also produces a smaller dependence of the impedance on the flow velocity. Unexpected and not explained up to now, however, is the result that the impedance depends more strongly on the flow velocity if the gauze screen covers the hole on the side of the measuring cavity or even if the gauze is stretched in a plane which is only a small distance (small compared with the diameter of the hole) beneath the inner surface of the duct [25]. 5. DISCUSSION OF THE RESULTS WITH THE AID OF A SIMPLE MODEL OF THE FLOW ABOVE THE ORIFICE The flow dependent radiation impedance of an orifice can be calculated with the aid of a strongly simplified model of the flow above the orifice. G o o d qualitative agreement with experiment is obtained. Thus the mechanism responsible for the flow induced change of the radiation impedance probably is described correctly by the model.

143

IMPEDANCE OF HOLES IN FLOW DUCTS 5.1. THE MODEL

Above the orifice a free shear layer builds up in which waves are excited by the sound. Figure 9 shows a longitudinal section of the hole; for a fixed time the resulting streamlines are drawn qualitatively. Let us consider the streamline separating at the left edge of the orifice (x --- 0, y = (3); its displacement in the y-direction can be expressed as ~(x, t), it being assumed that ~ has already been calculated as a function of the pressure difference indicated in Figure 9. The particle velocity through the hole then consists of two parts: (i) the area shaded in Figure 9 depends on time, so that 2a

Vl= [l f E(x,t)dx];

(9a)

0 L/o

2

~

2Poet~t

Figure 9. Model of the flow above the orificefor the calculation of the radiation impedance. (ii) at the right edge of the orifice (x = 2a) the acoustic medium is blown into the hole and sucked out by viscous forces depending on the sign of ~(2a, t), yielding I) 2 -~"

1~(2a, t)a: U0;

(9b)

a2 is a constant the value of which is about 0.5. This is the main assumption in the model. To keep the calculational work within reasonable limits some further assumptions for the determination of ~(x, t) are necessary. First, some properties of waves in infinitely spread shear layers are summarized. For waves propagating in the flow direction one obtains the wave numbers k1/2 = k0(1 + i ) ; ko=oJ/Uo, (10) if the wavelength ~ = 2zr/Re (k) is large compared to the shear layer thickness $; Uo is the flow velocity. Thus there is a strongly amplified and a strongly damped wave. The viscosity is neglected in this calculation which is ascribed to Rayleigh. The wave motion decays in the direction normal to the shear layer (y-direction) proportionally to exp(-kly[); thus the thickness of the layer taking part in the movement is ~,/2zr. If one artificially reduces the thickness of this layer to a value d small compared to ~ (by restricting the shear layer, for example, to the space between two parallel pressure release planes (distance 2d) one obtains the same wavenumbers as for the undisturbed shear layer (equation (10)). If h is not large compared to 3, the profile of the shear layer must be taken into account. Michalke has calculated the wave propagation for the tanh-like profile U = (Uo/2) [1 + tanh (2y/3)],

(11)

144

D. RONNEBERGER

which is a good approximation of the actual profile of free shear layers [26]. Figure 10 shows the real and the imaginary parts of the resulting conjugate complex wave numbers as functions of the Strouhal number s = k0 5. For s > 1 there is no solution of Michalke's equations which have been derived without the effect of viscosity. Now, by using the following three assumptions, ~:(x, t) can be calculated as a function of the pressure difference between both the sides of the orifice. The three assumptions are as follows. (i) The orifice is a slit (width 2a) at right angles to the flow direction. l

I

l

I

]

2'0 Re

1.5

1.0

0'5

f

f

l

)

0.2

0,4

0'6

0'8

S

Figure 10. Real and imaginary parts of the wavenumlx~r of a shear layer wave as functions of the Strouhai

number according to reference [26]. (ii) £(x, t) = ,4011 - AIe -tSk'dx - ,42 e -/Ik~dx] elt°t;

(12)

the motion of the shear layer hence consists of a pumping-like and two wave-like motions. The wavenumbers are given by

kl12 = ~2kol - t~XoX, Is >

(13)

where k~' and k~ are the wave numbers computed by Michalke for the tanh-profile (Figure 10), and the term iek]x indicates an additional attenuation preventing the displacement £ from growing too much for small g/a and large koa. Presumably this attenuation is caused by nonlinear interaction between the coherent wave under consideration and low frequency perturbations which for sake of simplicity are assumed to grow in the x-direction proportionally to kx; k is the wave number of the perturbation. Assuming additionally that the interaction is limited to a range of perturbations with wave numbers 0 < k < e'ko and that the decay of the coherent wave is proportional to the amplitude of the perturbations one obtains an attenuation proportional to k2x. o: is a constant to be fitted. A~ and A2 are calculated from the conditions ~:(0,t) = 0 and O£/Ox(O,t) = 0 (Kutta condition):

k'2 Al = k2 - k~'

A2 - kl' - k2"

(14)

which is possible only for k~ ¢ k~. For k~ = k~ = 2k o (s ~> 1) one obtains lim se(xl t) = Ao[1 - (1 + 2ikox)e-lIka~]e~t. k'l-~k'2

(15)

IMPEDANCE OF HOLES IN FLOW DUCTS

145

(iii) In deriving equation (12) for the motion of the shear layer the "source" (v2) at x = 2a was neglected and the shear layer was assumed to be infinitely extended. Thus for determining the excitation coefficient A0 one has to assume a pressure (d'/d)po exp (itot) in the plane y = d' and -(d'/d)poexp (itot) in the plane y = - d ' , which yields P0 to2 pd'

Ao

(16)

independently of d'. Due to the finite extension of the shear layer in our problem the motion of the shear layer is limited to a range of length din the y-direction; dis proportional to a and for obvious reasons is assumed to be equal to the end correction of the orifice (d = 0.85a for the circular orifice). Additional acceleration forces arise due to the additional particle velocity v2, and partly due to v~ ; thus A 0 now depends on d'. Figure 9 illustrates that within a definite layer which is also assumed to have the thickness d, the additional particle velocity does not cause additional pressures at x = 0. Therefore A0 may be calculated from equation (16) provided that d' = d and one obtains the impedance of the layer between the planes y = - d and y = d, according to equation (17). Apart from simplifying the geometry of the orifice here, the results obtained for the infinitely extended shear layer are thus applied to the finite shear layer over the orifice by assuming a finite thickness d of the layer which participates in the motion of the shear layer. That is justified only if the wavelength is small compared to the diameter of the orifice. Then, however, it is possible that the wavelength is comparable with the thickness of the shear layer and it is uncertain whether or not the Kutta conditions is valid for low values of A/8; it has been proved only for high values of A/8 [27]. In our problem we are only interested in the case of A being comparable to or large compared with 2a. Here another omission should be noted: in calculating the motion of the shear layer behind a thin plate Orszag and Crow [28] have shown that equation (12) is correct only for x ~>A/2 (apart from the term i~k2ox in the wavenumbers); thus the plate influences the motion of the shear layer up to half a wavelength downstream from the trailing edge. Finally the assumption is included that the thickness d o f t h e layer taking part in the motion has the same value for small A/a (the impedance has not yet departed from the value for air at rest) and for large A/a (quasi-static case); certainly this is also a considerable simplification. Based on these assumptions one obtains for the impedance of the layer between the planes y = d and y = - d 2po C°'t ~2d---Vl + V2

/

1

= 2itod 1 + 2k0 a(fl - fl*) +

a2

(7 fl*

o

e -l~n-~/z'72 d~ - fl

7 o

)

e -l~°'7-~/~'7~ d~/ +

1 + f l _ fl-------i-

W2a=2itod

/

, for s < l,

(17a)

;

1 o (1 + 2i~) e -2tn-~/vl2 d~ + 1 - 2--~oa

+ ~ [ a o uL 1 -- (1 +4ikoa)e-"k°"-2~"ko ",2

11

, f o r s>~ 1,

(17b)

where fl = k~/ko. On evaluating the experimental results the change of the impedance of the hole was interpreted as a change of the radiation impedance Wr; thus W r =

W2a

--

itopd.

(18)

146

D. RONNEBERGER

5 . 2 . COMPARISON OF THE RESULTS OF THE CALCULATION AND OF THE EXPERIMENT

In Figures 11 and 12 some of the results of this model calculation are summarized: the radiation impedance normalized to poad is plotted in the complex plane as a function of (koa) -l = Uo/aco for various combinations of the parameters ~, 3/a and a2. In Figure 11 3/a is independent of Uo, i.e. these results correspond to measured results taken at constant flow velocity and variable frequency. In Figure 12 8/a is proportional to U -1/2, thus these results correspond to experimental results plotted in Figure 4 where the frequency is constant and the flow velocity is varied. In so doing ~ and az are assumed to be independent of 3/a. There are no essential differences between the corresponding curves plotted in the two figures. i

1

I

1

i

1

j

0"2 0.4 0'4 ~

(Q)

0-8 6 2"0

2"4

2"8

0"2

~OZ'5o.e

(b)

0'4 -0.6

.9 1"2

.... 1"6

c'- = ~ . -

2'0

2

-

x".,:x..~5 0.8

(o)

O'4

0'6 0"2 0.40 0 ~ ' 5

1'2

1"6 2'0

2-4

2"8 3.2

0.8

0"6 ~

-1

1.2 I I

1 0

I I

I 2

1.6 l 3

l 4

2.0 I 5

I

1 6

I

Re(Wr)/..,o~d

Figure 11. Complex radiation impedance of an orifice computed for various combinations of parameters according to equation (17); 8[a is independent of Uo/aoJ.In the graphs (b), (c), (d) the value of one of the parameters of graph (a) is changed in each case. (a) a/a = 0.1, ~ = 0-35, a 2 = 0-5. (b) 3/a = 0"066. (c) ~ = 0"3. (d) a 2 = 0.4. Therefore we can confine our considerations to Figure 12, e.g., here 3 = 30(k0a) 1/2 holds since koa is the only parameter containing Uo in the calculation above. Actually a = ao(Uo ID/O -'/2

(19)

is valid; UolD/v is the Reynolds number containing a characteristic length lD of the flow duet and 3o, too, depends only on the dimensions of the duct. Equation (19) yields

3 = 30(oJalD/v)-x/2 (ko a) 1/2.

(19')

For Figure 12(a) 3o/a = 0.05 was used which is a value characteristic in the experiments; moreover a2 = 0-5 and a = 0.35. In Figures 12(b), (c) and (d) one of the three parameters of Figure 12(a) under consideration is changed. Thus the influence of each parameter on the shape of the impedance curve becomes clear. Evidently the shape of the curve is rather insensitive to changes of 30/a; otherwise Figure 12 would show greater differences from Figure 11. On the other hand, the shape of the impedance curve is strongly changed by changes of

IMPEDANCEOF HOLESIN FLOW DUCTS

147

and a2; ~ must not be too small, since otherwise the resulting curve is in no way similar to the curves plotted here. The impedance curves shown in Figure 12 agree qualitatively well with the experimental curves (c.f. Figure 4). By choosing the parameters skilfully one can even bring the calculated curves to coincide with the exl~rimental curve. The calculated curves as a function of Uo/aoJ, however, change two to three times as fast as the experimental curves. Because of the considerable simplifications in the calculation this does not surprise one very much. 1

Io

l

i

[

[

I

i

0'2

0.~

(a)_

0"6

l'2

0'2 ~0"5 0"4

2"0

2'4

28

O:8

0~.2

o ~

l'6

(b)

~

~'~ ~'2

l

1'2

1'6

2°0

2"4

2-8

--=U°/a~

^ O' 5

(c) 07

8 "~

~

,',o

20

2'4

2'8

0"2

o.G ~ -I

"~ I -I

I 0

f I

t6

I t t 2 3 4 Re (Wr )/pe.~d

/

2 ' ~ f 5

I 6

I

Figure12. Complex radiation impedance of an orifice plotted in ana]ogy to Eigure l l ; 8/a = 8o/a( Uo/ao~)-LI2. (a) aola = 0.05, ~ = 0.35, a2 = 0,5. (b) aola = 0.033. (c) = = 0.3. (d) a2 = 0"4. I t is a p p r o p r i a t e to e x p a n d e q u a t i o n (lTa) f o r small koa to compare the calculated r a d i a t i o n impedance w i t h the experimental one at h i g h Uo/aco: Wr l Uo 28/a ( 4a2-2+• p~od ~ a2(1 - 0t/2) aco a2(1 - ~/2) 2 + i 3a2--~: ~

) Uo 1 , --aoJ>> 1.

(20)

The calculated real part of the impedance increases linearly with Uo/aoJ as does the measured real part (Figure 5), the constant part being negative for the calculated real part as well as for the measured one; likewise the calculated and the measured imaginary parts are independent of Uo/ao~. In view of the considerable simplifications the calculation is based on it is not possible to derive quantitative results from these qualitative agreements, e.g. to determine ~ and a2 as functions of ~/a for a comparison between the results shown in Figure 7 and the results of the calculation. According to the good qualitative agreement between the calculated model and the experimental results one can be rather sure that the basic ideas of the model are correct. 6. SUMMARY The radiation impedance of several orifices in the wall of a flow duct was measured as a function of the flow velocity and of the frequency. The thickness of the wall boundary layer is small compared to the diameter of the orifices.

148

D. RONNEBERGER

The attached mass of the orifice is "blown away" even at low flow velocities (Uo/ao ~ 1); characteristically for this process the impedance as a function of the flow velocity follows a spiral in the complex plane. At large Uo/aCo(quasi-static case) the mass corresponding to the imaginary part of the impedance is constant and negative, i.e. the radiation impedance becomes spring-like. The flow resistance of the orifice at large Uo/aco is independent of the frequency and increases linearly with the flow velocity. If the boundary layer thickness is held constant independently of the flow velocity, one even obtains proportionality between the quasi-static flow resistance and the flow velocity. Widening of the orifice perpendicularly to the flow direction increases the flow induced flow resistance of the orifice. The most essential experimental results are well described qualitatively by a simple model of the flow above the orifice. This model is based on the assumption that a thin flow shear layer builds up above the orifice. In this shear layer waves are excited by the sound pressure difference between both the sides of the orifice. The particle velocity through the orifice is determined on the one hand by the temporal change of the shear layer displacement upon the orifice; on the other hand the flow is controlled by the shear layer motion at the downstream edge of the orifice, i.e. corresponding to the displacement of the shear layer at the downstream edge the medium either flows into the orifice or is sucked out. The validity of this model, of course, is in doubt, if the boundary layer thickness is no longer small compared to the diameter of the orifice. ACKNOWLEDGMENT The author is indebted to Dr D. Bechert for a fruitful discussion on the flow model. The investigation has been supported by the Deutsche Forschungsgemeinschaft. REFERENCES 1. P. J. WESTERVELT1951 Journal of the Acoustical Society of America 23, 347-348. Acoustical impedance in terms of energy functions. 2. E. MEYER,F. MECI-IELand G. KURTZE1958 Journal of the AcousticalSociety of America 30, 165174. Experiments on the influence of flow on sound attenuation in absorbing ducts. 3. H. H. HELLERand S. E. WIDNALL1968 Journal of the Acoustical Society of America 44, 885-896. Dynamics of an acoustic probe for measuring pressure fluctuations on a hypersonic re-entry vehicle. 4. E. FEDERand L. W. DEAN 1969 NASA CR-1373. Analytical and experimental studies for predicting noise attenuation in acoustically treated ducts for turbofan engines. 5. U. Kt~zE and C. ALLEN 1971 Journal of the Acoustical Society of America 49, 1643-1654. Influence of flow and high sound level on the attenuation in a lined duct. 6. L. J. SrWAN1935 Journal of the Acoustical Society of America 7, 94-101. Acoustic impedance of small orifices. 7. U. INOARDand S. LABA~ 1950Journal of the AcousticalSociety of America 22, 211-218. Acoustic circulation effects and non-linear impedance of orifices. 8. P. J. WES~VELT and P. W. StuCK 1950 Journal of the Acoustical Society of America 22, 680 (Abstr.). The correlation of non-linear resistance, flow resistance and differential resistance for sharp-edged circular orifices. 9. G. B. TmmSTON, L. E. HARGROWand B. D. COOK 1957 Journal of the Acoustical Society of America 29, 992-1001. Nonlinear properties of circular orifices. 10. F. ME¢I~L, W. SCmLZ and J. DIETZ 1965 Acustica 15, 199-206. Akustische Impedanz einer luftdurchstr6mten (~ffnung. 11. U. I N G ~ and H. ISING 1967 Journal of the Acoustical Society of America 42, 6-17. Acoustic nonlinearity of an orifice. 12. D. RO~mnEgGER 1968 Acustica 19, 222-235. Experimentelle Untersuchang zum akustischen Reflexiomfaktor von unstetigen Querschnitts~inderungen in einem luftdurchstr~mten Rohr. 13. A. POWELL1959 Journal of the Acoustical Society of America 31, 1527-1535. Propagation of a pressure pulse in a compressible flow.

I M P E D A N C E OF HOLES I N F L O W D U C T S

149

14. A. POWELL 1960 Journal of the Acoustical Society of America 32, 1640-1646. Theory of sound propagation through ducts carrying high-speed flow. 15. D. I. BLOKHINTSEV1946 Akustika Neodnorodnoi Dvizhushcheisya Sredi Leningrad. Translation: Acoustics of a non_homogeneous moving medium N A C A TM 1399. 16. D. RONNEBERGER and W. SCmLZ 1966 Acustica 17, 168-175. SchaUausbreitung in luftdurchstrSmten Rohren mit Querschnittsver~derungen und Str~Smungsverlusten. 17. L. CREMER 1948 Arch. El. Obtr. 2, 136-139. Clber die akustische Grenzschicht vor starren W~inden. 18. I. B. CRANDALL1927 Theory of Vibration Systems andSound. New York: D. Van Nostrand. 19. A . W . NOLLE 1953 Journal of the Acoustical Society of America 25, 32-39. Small-signal impedance of short tubes. 20. G.B. THURSTON 1952 Journalofthe Acoustical Society of America 24, 653-656. Periodic fluid flow through circular pipes. 21. G. B. THURSTONand J. K. WOOD 1953 Journal of the Acoustical Society of America 25, 861-863. End corrections for a concentric circular orifice in a circular pipe. 22. U. INGARD 1948 Journal of the Acoustical Society of America 20, 665-682. On the radiation of sound into a circular tube, with an application to resonators. 23. U. INGARD 1953 Journal of the Acoustical Society of America 25, 1037-1061. On the theory and design of acoustic resonators. 24. H. J. BAITER 1959 A VA-Bericht 59/A/34. EinfluB einer StrSmung durch die Druckentnahmebohrung auf die Bestimmung des statischen Drucks strOmender Flfissigkeiten. 25. D. RONNEBERGER and B. MICKELEIT Acoustica to be published. Akustiche Irnpedanz von tiberstrSmten Offnungen. 26. A. MICnALrd~ 1965 Journal of Fluid Mechanics 23, 521-544. On spacially growing disturbances in an inviscid shear layer. 27. D. BECHERT and E. PFIZENMAIER1971 Deutsche Luft- und Raumfahrt Forschungsbericht 71-09. IDber die AbfluBbedingung an der Dfisenaustrittskante bei einer schwach instation~iren D~isenstriSmung. 28. S. A. ORSZAG and S. C. CROW 1970 Boeing Scientific Research Laboratories D1-82-0953. Instability of a vortex sheet leaving a semi-infinite plate. APPENDIX LIST OF SYMBOLS

radius of the orifice coefficient for calculating the particle velocity A O[Re (Wr)/O'85pato]/a[Uo/ato] for Uo/ato ~. 1 Ao, A~, A2 excitation coefficient of boundary layer waves C sound velocity C compliance of the measuring cavity C ' specific compliance d thickness of the layer above the orifice taking part in the motion H height of the measuring cavity a

a2

i

Im(z) k ko

~/-1 imaginary part of a complex number z wave number

o,/Uo ID characteristic length of the flow duct

L, L r

L', L'r M P R R"

Re(z) S

u0 U' /)

end correction of the orifice corrections of the admittance of the measuring cavity coefficient of calibration of L and Lr Mach number sound pressure radius of the measuring cavity specific flow resistance real part o f a complex number z ko8 flow velocity

aU/ay particle velocity

150

D. RONNEBERGER

W impedance Wr radiation impedance coefficient of the attenuation of the boundary layer wave y ratio of the specific heats thickness of the flow boundary layer coefficient used for calculating the end correction A wavelength of the boundary layer wave A~ sound wavelength v kinematic viscosity displacement of the shear layer in y-direction p density of the medium cr sensitivity of the microphone to angular frequency