Orifice Impedance under Grazing Flow

Measured with a Single Microphone Method

G. Kooijman∗

Department of Applied Physics, Eindhoven University of Technology, The Netherlands

J. Golliard†

TNO-TPD, Delft, The Netherlands

A. Hirschberg‡

Engineering Fluid Dynamics Laboratory, University of Twente, Enschede, The Netherlands

The effect of one-sided grazing mean flow on the acoustical impedance of rectangular

orifices is measured at low Mach number and low Helmholtz number by means of a single

microphone method. The results are fairly consistent with previous experimental results

obtained by means of a two-microphone impedance tube. Furthermore no significant in-

fluence of the aperture aspect ratio and aperture wall thickness on the non-dimensional

scaled impedance is found, at least for the qualitative trend. Comparison with an existing

theoretical model shows reasonable agreement for the resistance, provided that the exper-

imental results are tentatively corrected for boundary layer- and induced flow effects. For

the reactance no agreement is found.

I. Introduction

In order to suppress noise in ducts, such as combustion engine exhausts and in- and outlets of jet engines,acoustic damping material, protected by perforated plates, can be placed at the walls. Often the damping

material itself is omitted, and the space between the perforate plates and the backing wall is filled withhoneycomb structure. The so obtained acoustic liner is then basically an array of Helmholtz resonators,cf. figure 1. Sometimes one or two additional layers of resonators are used to obtain a double DegreeOf Freedom (DOF) respectively triple DOF liner.1,2 The acoustic properties of a Helmholtz resonator arepartly determined by the impedance of its orifice. Due to the specific application in liners, especially theeffect of grazing flow on the orifice impedance is of interest. This has been investigated experimentallyby several authors, e.g. Ronneberger,3 Goldman and Panton,4 Kirby and Cummings,5 and Cummings.6

These studies concern the impedance of circular apertures or perforated plates5 in duct flow3,5, 6 for very lowStrouhal numbers Sr and thin3,5, 6 or thick4 boundary layers. Generally, it is concluded that above a certainvelocity limit the effect of the flow is to increase the resistance (absorption) and to decrease the reactance(added mass). Ronneberger3 proposed a simple model, which predicts the experimental results well, butis argued to be only valid for thin boundary layers and low Strouhal number. Goldman and Panton4 andCummings6 give empirical formulas which solely predict absorption and no sound production by the orifices.Sound production was measured,6 but was assumed to be due to experimental error. Howe7,8 proposed atheoretical model for linear perturbations. For the specific case of a rectangular aperture with large aspectratio an exact analytical expression was given for the orifice impedance. Strouhal number ranges for soundabsorption as well as production are predicted. The theoretical model of Howe will be summarized in sectionII. Golliard11 experimentally investigated the impedance of a rectangular orifice with large aspect ratio. Atwo-microphone impedance tube was used. He compared his results with the theoretical model of Howe,7,8

∗PhD student, email: g.kooijman@tue.nl†Senior researcher‡Professor

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10th AIAA/CEAS Aeroacoustics Conference AIAA 2004-2846

Copyright © 2004 by G. Kooijman, J. Golliard and A. Hirschberg. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Single Helmhotz resonator

U U a b

Figure 1. Schematic of a single DOF (a) and a double DOF liner (b) with grazing mean flow U .

which will be discussed in section III. It was argued that the measured resistance is quite consistent with thetheoretical prediction, whereas the length correction (reactance) showed significant inconsistency. Recentstudies by Jing et al.,12 Auregan et al.,13,14 and Peat et al.15 proposed modification of the theory, which wasargued to give only to some extent better agreement with experiments. This triggered the interest in furtherexperimental research and study of the theoretical model. In particular we will present new experimentalresults obtained by a single microphone method in section IV and V.

II. Theoretical model

Howe7,8 considered a two-sided grazing main flow U± in the x1-direction with infinitesimally small bound-ary layer thickness (large Reynolds number) over an aperture in a infinitesimally thin wall, cf. figure 2.

U -

U +

p - , ϕ -

p + , ϕ +

x 1

x 2

2 s

U -

U +

p - , ϕ -

p + , ϕ +

x 1

x 2

2 s

b

x 3

U -

U +

p - , ϕ -

p + , ϕ +

x 1

x 2

2 s

U -

U +

p - , ϕ -

p + , ϕ +

x 1

x 2

2 s

b

x 3

Figure 2. Two-sided graz-ing flow over an aperture.

The + and − subscripts refer to the x2 > 0 -respectively x2 < 0 region. Inthe aperture a shear layer develops, which separates the flows above and be-neath the wall. Potential flow is assumed, except for the infinitesimally thinregion of the shear layer (vortex sheet), where all vorticity is considered to beconcentrated. ϕ±(ω) and p±(ω) are the Fourier transformed velocity potential-respectively pressure for uniform linear perturbations taken at sufficiently largedistance from the wall. An e−iωt-convention is taken for the harmonic distur-bances from here on, with ω the angular frequency. The pressure difference(p+ − p−)(ω) induces a flow through the aperture, corresponding to an ad-ditional velocity potential φ(x, ω), so that the total velocity potential of theperturbation is:

ϕ(x, ω) = ϕ±(ω) + φ(x, ω) (1)

Regarding the flow as locally incompressible: M 2 = (U±

c0)2 � 1 and He2 = (kL)2 � 1, with c0 the (mean)

sound velocity, k = ωc0

the wavenumber and L the characteristic size of the aperture, the velocity potentialsatisfies the Laplace equation:

∇2φ(x, ω) = 0 (2)

With the boundary condition for the flow velocity in the x2-direction, i.e. normal to the wall: v2(x, ω) = 0at x2 = 0 outside the aperture, the velocity potential is solved with use of the Green’s function:

G(x,y, ω) =−1

4π|x − y|+

−1

4π|x − y′|, y

′ = (y1,−y2, y3) (3)

to obtain :

ϕ(x, ω) = ϕ±(ω) −sign(x2)

2π·

∫ ∞

−∞

v2(y1,±0, y3, ω)

|x − y|dy1dy3, y2 = 0 (4)

where the flow velocity in normal direction in the aperture v2(x1,±0, x3, ω) is unknown. By using therelations:

p(x, ω) = −ρ0(−iω + U±∂

∂x1)ϕ(x, ω) (5)

v2(x1,, 0, x3, ω) = (−iω + U±∂

∂x1)ζ(x1, x3, ω) (6)

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with ρ0 the mean fluid density, this equation can be rewritten into a relation between the pressure p(x, ω)and the normal displacement of the shear layer in the aperture ζ(x1, x3, ω):

p(x, ω) = p±(ω) −ρ0sign(x2)

2π(ω + iU±

∂

∂x1)2 ·

∫

S

ζ(x1, x3, ω)

|x − y|dy1dy3, y2 = 0 (7)

where the integration is now restricted to the area S of the aperture. This equation implies that ζ(x1, x3, ω)is continuous over the vortex sheet (same value at x2 = ±0), which is reasonable in the infinitesimally thinwall case. However, recently Jing et al.12 and Peat et al.15 argued continuity of particle velocity rather than-displacement over the vortex sheet, whereas Auregan et al.13,14 concluded that a intermediate boundarycondition between displacement and velocity continuity, depending on the boundary layer conditions, wasmost realistic to obtain a fit of the experimental results.

Finally, by demanding continuity of pressure over the aperture (at x2 = ±0), an equation for the shearlayer displacement is found:

[(ω + iU+∂

∂x1)2 + (ω + iU−

∂

∂x1)2] ·

1

2π

∫

S

ζ(x1, x3, ω)dy1dy3√

(x′1 − y′1)2 + (x′3 − y′3)

2=p+ − p−

ρ0(8)

After integration with respect to the differential operator on the left hand side we find:∫

S

ζ ′(y′1, y′3, ω)dy′1dy

′3

√

(x′1 − y′1)2 + (x′3 − y′3)

2+ λ1(x

′3)e

iσ1x′1 + λ2(x

′3)e

iσ2x′1 = 1 (9)

where the scaling:

ζ ′ =ρ0ω

2Lζ

π(p+ − p−), x′ =

x

L, y′ =

y

L

is used to obtain a dimensionless form. Here L is the characteristic size of the aperture in stream wisedirection. Furthermore:

σ1 =ωL(1 + i)

U+ + iU−= σ+

1 + i

1 + iµ, σ2 =

ωL(1 − i)

U+ − iU−= σ+

1 − i

1 − iµ(10)

are the dimensionless Kelvin-Helmholtz wave numbers of the instability waves of the shear layer, withσ+ = ωL

U+the Strouhal number based on velocity U+ and L, and µ the ratio of flow velocities: µ = U−

U+.

The terms on the left hand side of equation (9) with coefficients λ1 and λ2 represent the instability waves ofthe vortex sheet. For ω real σ1 = σ∗

2 , so that one of the waves will grow exponentially in the x1-direction,which yields the instability. Note that equally strong grazing flow at both sides: U− = U+ or µ = 1, givesσ1 = σ2 = σ+. The two terms in Eq. (9) are then replaced by: (λ1(x

′3) + λ2(x

′3)x

′1)e

iσ+x′1 . Since in this case

σ+ is real, the motion of the vortex sheet is stable.Equation (9) is (in general numerically) solved for the vortex sheet displacement by applying the Kutta

condition, ζ = ∂ζ∂x1

= 0, at the upstream-edge. The shear layer thus leaves the upstream edge tangentially.With ζ found, the volume flux Q through the aperture can be calculated directly by:

Q(ω) =

∫ ∞

−∞

v2(x1, 0, x3, ω)dx1dx3 =

∫

S

−iωζ(x1, x3, ω)dx1dx3 (11)

with use of equation (6). The acoustic behavior of the aperture is then expressed in the Rayleigh conductivityKR defined as:

KR ≡Q

ϕ+ − ϕ−= iωρ0

Q

p+ − p−(12)

For a rectangular aperture with large aspect ratio, b � 2s, cf. figure 2, an exact analytical expression forthe Rayleigh conductivity was found:

KR =πb

2(F (σ1, σ2) + Ψ)(13)

where σ1,2 are as defined in equation (10), and in this case are based on the half-width s of the slot (so L isreplaced by s). The function F is given by:

F (σ1, σ2) =−σ1J0(σ2)G(σ1) + σ2J0(σ1)G(σ2)

σ1W (σ2)G(σ1) − σ2W (σ1)G(σ2),

G(x) = J0(x) − 2W (x), W (x) = ix(J0(x) − iJ1(x)) (14)

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with J0 and J1 Bessel functions. Ψ is related to the local approximations of the Green’s functions on eitherside of the aperture. For instance, for a three-dimensional free space on both sides:

Ψ±3D = ln(4b

s) − 1 (15)

Figure 3 shows a plot of the Rayleigh conductivity of a slot with aspect ratio b2s

= 10 for one-sided grazingflow, µ = 0, according to equation (13). Here the conductivity:

KR = 2Re(ΓR − i∆R) (16)

is scaled to the hydraulic radius of the aperture: Re =√

2sbπ

. ΓR and ∆R correspond to a reactance respec-

tively resistance of the orifice, which according to the present theory thus only depend on the Strouhal num-ber, for given aspect ratio and flow velocity ratio µ. For ∆R > 0, Q is proportional to

0 1 2 3 4 5 6−1

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

ωs/U+

KR

/Re

ΓR

∆R

Figure 3. Scaled Rayleigh conductivity of aslot with aspect ratio b

2s= 10 in a infinites-

imally thin wall for one-sided grazing flow,U− = 0, according to the theory of Howe.7,8

−(p+ − p−), cf. eqs. (12) and (16). The power P performedby the pressure load on the aperture flow equals P = −(p+ −p−) · Q. So for ∆R > 0 the power is positive: P ∝ (p+ −p−)2 > 0. Energy is thus transferred from the pressure fieldto the aperture flow, which implies sound absorption. It canbe seen in figure 3 that sound absorption takes place for lowStrouhal numbers, ωs

U+< 1.6. For higher Strouhal numbers,

1.6 < ωsU+

< 3.5, ∆R is negative, here sound production occurs.

Grace et al.10 computed the Rayleigh conductivity, accord-ing to the present theory of Howe,7,8 for 8 different shapes ofwall apertures, such as a square and a circle. It was found that,especially for one-sided grazing flow, KR is very similar to thatdisplayed in figure 3 for all geometries.

Furthermore Howe7,9 investigated the influence of wallthickness d on the Rayleigh conductivity. For a thin wall ap-proximation, d

L� 1, and for the wavelengths of disturbances

on the vortex sheet large compared to d (σ+ � Ld), the verti-

cal displacement of fluid in the aperture was regarded indepen-dent on x2, and thus the same as the vortex sheet displacementζ(x1, x3, ω) at the upper and lower end of the aperture at x2 = ± 1

2d. Instead of demanding continuity ofpressure over the vortex sheet as in the infinitesimally thin wall case, now the pressure difference between

the upper and lower end of the aperture can be set equal to the inertia −ρ0d∂2ζ∂t2

= ρ0dω2ζ. From equation

(7) the equivalent of equation (8) is then deduced for a wall of finite thickness:

dω2ζ(x1, x3, ω) + [(ω + iU+∂

∂x1)2 + (ω + iU−

∂

∂x1)2] ·

1

2π

∫

S

ζ(x1, x3, ω)dy1dy3√

(x′1 − y′1)2 + (x′3 − y′3)

2=p+ − p−

ρ0(17)

The Rayleigh conductivity of rectangular orifices was calculated for several thicknesses and for one-sidedand even two-sided grazing flow.9 It was argued that the introduction of a finite wall thickness in all casesmodifies the stability of the shear layer motion (in the two-sided grazing flow case the motion also becomesunstable). For one-sided grazing flow the Rayleigh conductivity was found to be very similar to the infinitelythin wall case. Especially the low Strouhal number region of sound absorption remains nearly unaffected.For walls with d

L> 0.05 the region of sound production vanishes. It was however questioned, whether the

thin wall approximation still holds in this case.

III. Previous experimental results

Golliard11 investigated the effect of one-sided grazing flow on the impedance of large aspect ratio rectangu-lar orifices (0.7 cm × 10 cm and 1.4 cm × 10 cm) in a relatively thin wall (1 mm). He used a two-microphoneimpedance tube. Experiments were conducted for several turbulent boundary layer displacement thicknessesδ1 between 1.4 and 5.5 mm. The results were presented as a length correction rather than a conductivity.The complex fluid motion through the aperture can namely be regarded as the motion of a volume of fluid

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with effective length leff and area equal to the aperture area S0. Integration of the linearized Euler equationfor momentum:

∂v

∂t= −

∇p

ρ0(18)

over this fluid volume, which is regarded as incompressible gives:

leff =S0(p+ − p−)

iωρ0Q=

S0

KR

(19)

with use of equation (12), where the area averaged velocity v = QS0

. Furthermore leff can be decomposed in

a real part (reactance) and a resistance, scaled to the characteristic impedance ρ0c0: r = 1ρ0c0

<(p−−p+

v)) =

k · =(leff ), using eq. (19). The effect of flow can now be written as:

rflow = k · (=(leff ) −=(leff,0)) =SrM

L· (=(leff ) −=(leff,0))

δflow = <(leff ) −<(leff,0) (20)

with Sr now based on the full width of the aperture, L = 2s. leff,0 is the effective length in case of no meanflow. rflow is denoted as the resistance due to the flow, and δflow as the length correction due to the flow.Golliard11 introduced scaling of rflow to M , and δflow to L, to obtain the dimensionless quantities:

rflow =Sr

L· (=(leff ) −=(leff,0))

δflow =1

L(<(leff ) −<(leff,0)) (21)

With use of equation (19) and the theoretical expression forKR, eq. (13), we obtain the theoretical prediction:

rflow =2Sr

π· =(F )

δflow =2

π(<(F ) −<(F(U=0))) (22)

Where the fact is used that =(F(U=0)) = 0 (so that leff,0 is real) in the model. This implies that the radiationresistance is not accounted for, which is justified for He� 1. Since F , eq. (14), is a function of the Strouhalnumber and the flow ratio µ only, the predicted rflow and δflow are clearly also a function of Sr and µ only.The exact geometry of the space surrounding the aperture, which is reflected by ψ, is therefore not relevant.This shows the advantage of using the parameters rflow and δflow to represent orifice impedance.

Figure 4 sketches the experimental results of Golliard11 obtained for the 1.4 cm × 10 cm slot for δ1 = 1.4mm (1) and δ1 = 5.5 mm (2), compared to the theory of Howe,7,8 equation (22). Qualitatively, the samebehavior of rflow and δflow is found for all boundary layers as shown in the figure. However, the horizontalscaling as well as the amplitudes of the ’oscillations’ of the experimental curves were found to depend on theboundary layer thickness. The amplitudes of the ’oscillations’ were larger for smaller δ1

Lratios. The results

for rflow and δflow were subsequently plotted against the Strouhal number based on an effective convectionvelocity Uc of vorticity in the aperture, where Uc

Udepends on the ratio of boundary layer thickness to L. This

gave a better agreement for the horizontal scaling between the results obtained for different boundary layer.Also the measured first region of sound absorption (rflow > 0) and subsequently production (rflow < 0)coincided better with the theoretical prediction after this rescaling. The measured second region of soundproduction, which is very clearly visible in figure (4), and the further oscillations in rflow were not predictedby the theory for one-sided grazing flow. However some improvement was reached by comparing the measuredrflow with the theory for two-sided grazing flow, with µ = U−

U+≈ 0.2. This would imply a correction for the

flow induced by viscous entrainment beneath the orifice. We do however not present these corrections here,because they are not yet well established.

For the measured non-dimensional scaled length correction δflow, cf. figure 4, no reasonable agreementwith theory was found, even not after the corrections described above. It should be noted here that the

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E x p e r i m e n t s G o l l i a r d T h e o r y H o w e

r f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 01 . 52 . 0

0

1

2

0

d f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 0

0

0

12

E x p e r i m e n t s G o l l i a r d T h e o r y H o w eE x p e r i m e n t s G o l l i a r dE x p e r i m e n t s G o l l i a r d T h e o r y H o w eT h e o r y H o w e

r f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 01 . 52 . 0

0

1

2

0

r f l o w~ r f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 01 . 52 . 0

0

1

2

0

d f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 0

0

0

12

d f l o w~ d f l o w~

w L / U2 4 6 8 1 0- 2 . 0- 1 . 5- 1 . 0- 0 . 5

0 . 51 . 0

0

0

12

Figure 4. Experimental results of Golliard11 compared to the theory of Howe,7,8 for the 1.4 cm × 10 cm slotfor one sided grazing flow, U+ = U , µ = 0; (1) δ1 = 1.4 mm, (2) δ1 = 5.5 mm.

theoretical prediction for δflow shows a double limit behavior. While the limit U+ → 0 implies Sr → ∞, thelimit values of F , eq. (14) are different in both cases:

limU+→0

F = 0, limSr→∞

F = −2 (23)

From equation (22 this gives for the non-dimensional scaled reactance:

limU+→0

δflow = 0, limSr→∞

δflow = −4

π(24)

This double limit behavior is in principle not surprising, because the theory cannot hold for Sr → ∞, dueto inviscous assumption. On the other hand, for lower Sr the theory should still hold, which is however notconfirmed by the experimental results.

Recent study by Peat et al.15 also showed that experimental results for the resistance of circular orificeswere fairly consistent with the predication of Howe’s theory,7,8 when Sr was based on an effective vorticityconvection velocity Uc. Agreement for the reactance was however poor. The theoretical predicted reactancewas much improved when considering the velocity continuity boundary condition,12 as discussed in sectionII. On the other hand the results for the resistance became worse, especially sound production in criticalStrouhal ranges was now not predicted anymore. Furthermore, surprisingly the Strouhal number did notneed to be based on the convection velocity to fit experiment with theory, when velocity continuity was used.One might therefore question the physical meaning of the correction for convection velocity. Actually thetheory of Howe does already include the fact that vorticity is not convected at the velocity U+.

Finally it can be noted that both experiments and theory give a decreasing reactance and a increasingresistance below a certain Sr value. This is also consistent with the general result of the studies mentionedin the introduction.3,4, 5, 6

IV. Experiment

In the present experiments a single microphone method is used to measure the influence of one-sidedgrazing flow on the impedance of an orifice. The experimental set-up, which is placed in a semi-anechoicroom, is shown in figure 5a. An open wind tunnel generates a flow over a Helmholtz resonator with adjustableorifice dimensions. Damping material is placed in the resonator to suppress flow-induced cavity tones. Aharmonic signal is sent to a loudspeaker to excite the shear layer developing in the orifice. The harmonicpressure in the resonator is measured by a single microphone (PCB 116A). The transfer of loudspeaker signalto microphone signal is measured by means of a digital lock-in. The signal transfer measured without flowand measured with a flow could be translated to a impedance change due to the flow with the analyticalmodel described below.

The Helmholtz resonator used in the experimental set-up is shown schematically in figure 5b. The innerdimensions are Wr‖×Wr⊥×Lr = 25 cm × 16 cm × 70 cm, with Wr‖ and Wr⊥ the width parallel respectivelyperpendicular to the flow direction, and Lr the depth. The pressure field in the Helmholtz resonator can

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Loudspeaker Amplifier

Signal generator

Digital data - acquisition / lock - in Microphone

Damping material

Open wind tunnel

Loudspeaker Amplifier

Signal generator

Digital data - acquisition / lock - in Microphone

Damping material

Open wind tunnel

(a) Schematic of the complete set-up

x = 0

x = L

x = x m

Sd

p +p -

S r

x = 0

x = L r

x = x m

S 0

p +p -

S r

x = 0

x = L

x = x m

Sd

p +p -

S r

x = 0

x = L r

x = x m

S 0

p +p -

S r

x = 0

x = L

x = x m

Sd

p +p -

S r

x = 0

x = L r

x = x m

S 0

p +p -

S r

p +p -

S r

p +p -

S r

(b) Schematic of the Helmholtz resonator used in theset-up

Figure 5. Experimental set-up for measuring orifice impedance with the single microphone method.

be decomposed in two plane waves with complex amplitudes p+ and p− travelling in positive- respectivelynegative x-direction. The Fourier transform of the pressure in the resonator is thus:

p(x) = p+(x) + p−(x) = p+eikx + p−e

−ikx (25)

The Fourier transform of the velocity in the resonator is deduced from the pressure p(x) and the linearizedEuler equation for momentum, eq.(18):

v(x) =p+e

ikx − p−e−ikx

ρ0c0(26)

The flow in the aperture is considered to be incompressible with area averaged velocity vn. Conservation ofmass at the interface between aperture and resonator volume (at x = 0) then implies:

vnS0 = v(0)Sr =p+ − p−

ρ0c0Sr (27)

with S0 and Sr the area of the aperture respectively resonator, cf. figure 5b. Integration of the momentumequation, eq (18), over the aperture gives:

iωρ0vnleff = p(0) − pex = (p+ + p−) − pex (28)

with pex the applied pressure above the aperture. Combination of equations (27) and (28) gives:

p+ =pex

(R0 + 1) + ik Sr

S0(R0 − 1)leff

(29)

where R0 = p−

p+the value of the reflection coefficient R(x) at x = 0:

R(x) =p−(x)

p+(x)= R0e

−2ikx (30)

The pressure at the microphone position xm, cf. figure 5b, is then given by:

p(xm) =pex(eikxm +R0e

−ikxm)

(R0 + 1) + ik Sr

S0(R0 − 1)leff

(31)

Writing p(xm)0 and p(xm)f for the pressure at the microphone in case of no flow respectively in case of acertain flow present, the change of the effective length of the aperture due to the flow, ∆leff = leff − leff,0,can be expressed as:

∆leff =((R0 + 1) + ik Sr

S0(R0 − 1)leff,0) · (

p(xm)0p(xm)f

− 1)

ik Sr

S0(R0 − 1)

(32)

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with use of equation (31), provided that the excitation pressure pex is the same in both cases. ∆leff canstraightforwardly be translated to the quantities defined in equation (21).

Besides p(xm)0 and p(xm)f , leff,0 and R0 are needed to calculate ∆leff from equation (32). Thesequantities are determined by measuring the frequency response of the resonator (without flow) without- andwith damping material, using a frequency analyzer (HP35652A). First the pressure p(xm) is measured in asufficiently wide frequency band around resonance without damping material in the resonator. In this casethe reflection coefficient at the bottom of the resonator, x = Lr, equals R(Lr) = 1 , so that R0 = e2ikLr , cf.equation (30). The measured frequency response p(xm) is then fitted with that predicted by equation (31)to obtain leff,0. Subsequently the frequency response with damping material is measured, and again fittedwith equation (31), in which the now known leff,0 is substituted, to obtain the R0 with damping material.Note that R0 here has the form R0 = |R0|e

iφe2ikLr . The damping material thus introduces an |R0| < 1,which (primarily) results in a lower quality factor (broadening of the resonance peak), and an additionalphase φ, which (primarily) alters the resonance frequency.

V. Results

The influence of one-sided grazing flow on impedance is measured for four rectangular apertures, cf.figure 6: two apertures with L = 2 cm respectively L = 9 cm in a 3 cm thick wall, denoted as ’2 cm NE’ and’9 cm NE’, and two apertures with L = 2 cm respectively L = 8 cm in a 3 cm thick wall but with a sharp1 mm thin upstream edge of 1 cm length, denoted as ’2 cm SE’ and ’8 cm SE’. All apertures have a widthof 16 cm perpendicular to the flow. The distance from the windtunnel outlet to the apertures’ upstreamedge is 27 cm. The boundary layer is turbulent in most cases. The boundary layer displacement thickness istypically δ1 = 0.9 mm. Measurements are conducted at several frequencies, f = ω

2π, and mean flow velocities

U .The results for rflow and δflow as defined in equation (21), calculated with use of eq. (32), are shown in

figure 7 respectively figure 8 as function of the Strouhal number ωLU

. It can be seen that the general trendfor the scaled resistance and reactance is quite similar for all apertures and frequencies, although at low Sr

more scatter is observed. For the 2 cm slots similar features for rflow and δflow occur at smaller Sr than forthe 8 and 9 cm slots. This can be due to the larger δ1

Lratio for the 2 cm slots, as argued by Golliard.11

Reasonable consistency with the previous experimental results presented in section 3, figure 4, is found. Thetwo regions of sound absorption, rflow > 0, at Sr < 1 and Sr = 2.5 − 3 can also be observed in the presentexperiments for Sr < 2 and around Sr = 3.5 for the 2 cm slots resp. for Sr < 2 and around Sr = 4.5 forthe 8 and 9 cm slots. The regions of sound production, rflow < 0, around Sr ≈ 2 and Sr = 4 found byGolliard can be seen here, figure 7, around Sr = 2.5 and Sr = 4.5 (less pronounced) for the 2 cm slots, andaround Sr = 3 and Sr = 6.5 for the 8 and 9 cm slots. Furthermore the 8 and 9 cm slots clearly show anadditional region of sound absorption around Sr = 8, whereas for Sr < 2.5 two regions of sound absorptionseparated by a small region of production may be present (for most frequencies). Also the Sr < 2 regionwhere δflow is negative and decreases with decreasing Sr, and the region where δflow is positive aroundSr = 2 − 2.5, found in the previous experiments, figure 4, are seen here for Sr < 2.5 respectively aroundSr = 3 for the 2 cm slots, figure 8a, and for Sr < 3 respectively around Sr = 3.5 for the 8 and 9 cm slots,figure 8b. Although the low Sr region seems to display some additional features in case of the 8 and 9 cmslots. In both two experimental results (for different boundary layers), shown in figure 4, it can be seen thatthe maximum in δflow approximately occurs at a Strouhal number where the second region of absorptionbegins. This can also be seen for all present experimental results, indicating that the scaling of rflow against

Sr with respect to the scaling of δflow against the Sr is the same for both previous and present experiments.

a

3 c m

2 o r 9 c mb

3 c m

2 o r 8 c m1 m m

1 c m

a

3 c m

2 o r 9 c ma

3 c m

2 o r 9 c mb

3 c m

2 o r 8 c m1 m m

1 c m

b

3 c m

2 o r 8 c m1 m m

1 c m

Figure 6. Configuration and dimensions of the used apertures, a) normal edge aperture (NE), b) sharp-edgeaperture (SE).

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r f l o w

r f l o w

2 c m N E

- 2 . 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 52 . 02 . 5

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

2 c m S E

- 2 . 5- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 52 . 02 . 5

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 9 5 H z

(a)

r f l o w

r f l o w

9 c m N E

- 5- 4- 3- 2- 1012345

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

8 c m S R

- 5- 4- 3- 2- 1012345

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

(b)

Figure 7. rflow for: a) 2 cm normal edge slot and 2 cm sharp edge slot, b) 9 cm normal edge slot and 8 cmsharp edge slot

d f l o w

2 c m N E

- 1 . 2- 1 . 0- 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 00 . 20 . 40 . 6

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

2 c m S E

- 1 . 2- 1 . 0- 0 . 8- 0 . 6- 0 . 4- 0 . 20 . 00 . 20 . 40 . 6

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 9 5 H zd f l o w

(a)

9 c m N E

- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 5

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

8 c m S E

- 2 . 0- 1 . 5- 1 . 0- 0 . 50 . 00 . 51 . 01 . 5

0 1 2 3 4 5 6 7 8 9 1 0w L / U

7 4 H z 1 0 5 H z 1 4 5 H z 2 0 4 H z 2 9 5 H z

d f l o w

d f l o w

(b)

Figure 8. δflow for: a) 2 cm normal edge slot and 2 cm sharp edge slot, b) 9 cm normal edge slot and 8 cmsharp edge slot

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The absolute difference in the horizontal scaling of the results may again be argued to be due to the smallerboundary layer thickness to aperture width ratio in the present experiments. The observation in the previousexperimental results, section III, that the amplitudes of the ’oscillations’ in rflow and δflow are larger forsmaller boundary layer thickness to aperture width ratios is confirmed by the present results. Furthermore,the order of the absolute values of the oscillations in rflow and δflow are comparable for the previous andpresent experiments.

VI. Conclusion

The effect of one-sided grazing flow on the acoustic impedance of rectangular orifices is measured by meansof a single microphone method. Different aperture geometries (small or large aspect-ratio and thick wall orthick wall with thin upstream edge) give the same qualitative trend. Furthermore the results reasonablyconfirm earlier experiments of Golliard,11 who used a different measuring method (two microphone impedancetube) and large aspect ratio slots in a thinner wall. No significant influence of wall thickness and aperturegeometry, in any case not for the qualitative behavior of the impedance is thus found. Boundary layerthickness is found to influence the amplitude of oscillations in the scaled resistance, rflow, -and reactance,

δflow, versus the Strouhal number (however not the qualitative trend). The amplitudes are larger for smallerboundary layer thickness to aperture width ratio. The theoretical model of Howe7,8 predicts the resistancereasonably well, provided that the experimental results are corrected for an effective convection velocity inthe aperture and for induced flow velocity below the orifice, as proposed by Golliard.11 Further research onthe physical significance of the convection velocity correction should be undertaken. The reactance is poorlypredicted by the theory. Modification of the theory by applying velocity continuity instead of displacementcontinuity on the shear layer improves consistency with experiment to some extent, but does not seem tosolve the problem completely.12,13,14,15

Acknowledgments

The authors would like to thank Freek van Uittert and Igor Denissen. This work is supported by theDutch Foundation for Technical Sciences (STW) under project no. esf. 5645.

References

1http://www.nlr.nl/public/facilities/aeroacoustic/mmt2R.E. Motsinger, R.E. Kraft; Design and performance of duct acoustic treatment. in: H.H. Hubbard; Aeroacoustics of

flight vehicles, theory and practice, vol. 2: noise control; Acoustical Society of America, 1995; ISBN 1-56396-406-6.3D. Ronneberger; The acoustical impedance of holes in the wall of flow ducts; J. Sound and Vibration, vol. 24(1), pp.

133-150, 1972.4A.L. Goldman, R.L. Panton; Measurement of the acoustic impedance of an orifice under a turbulent boundary layer; J.

Acoust. Soc. Am., Vol. 60, No. 6, pp. 1397-1404, 1976.5R. Kirby, A. Cummings; The impedance of perforated plates subjected to grazing gas flow and backed by porous media;

J. Sound and Vibration vol. 217(4), pp. 619-636, 1998.6A. Cummings; The effect of grazing turbulent pipe-flow on the impedance of an orifice; Acustica, Vol. 61, pp. 233-242,

1986.7M.S. Howe; Acoustics of fluid-structure interactions; Cambridge University Press, 1998; ISBN 0-521-63320-6.8M.S. Howe; The influence of mean shear on unsteady aperture flow, with application to acoustical diffraction and self-

sustained cavity oscillations; J.Fluid Mech., vol. 109, pp. 125-146, 1981.9M.S. Howe; Influence of wall thickness on Rayleigh conductivity and flow-induced aperture tones; J. Fluids and Structures,

vol. 11, pp. 351-366, 199710S.M. Grace, K.P. Horan, M.S. Howe; The influence of shape on the Rayleigh conductivity of a wall aperture in the

presence of grazing flow; J. Fluids and Structures, vol. 12, pp. 335-351, 1998.11Joachim Golliard; Noise of Helmholtz-resonator like cavities excited by a low Mach-number turbulent flow; PhD thesis,

l’Universite de Poitiers, Fr, 2002; ISBN 90-6743-964-9.12Xiaodong Jing, Xiaofeng Sun, Jingshu Wu, and Kun Meng ; Effect of grazing flow on the acoustic impedance of an orifice;

AIAA Journal, vol.39, no. 8, pp. 1478-1484, August 2001.13Y. Auregan, R. Starobinski, and V. Pagneux; Influence of grazing flow and dissipation effects on the acoustic boundary

conditions at a lined wall; J. Acoust. Soc. Am., 109(1), pp. 59-64, Jan. 2001.14Y. Auregan, M. Leroux; Failures in the discrete models for flow duct with perforations: an experimental investigation; J.

Sound and Vibration 265, pp. 109-121, 2003.

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15Keith S. Peat, Jeong-Guon Ih and Seong-Hyun Lee; The acoustic impedance of a circular orifice in grazing mean flow:Comparison with theory; J. Acoust. Soc. Am., 114(6), pp. 3076-3086, Dec. 2003.

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