Effect of orifice shape on acoustic impedance

Abstract

The effect on normal incidence acoustic impedance of a non-circular orifice shape is examined relative to a circular orifice. The impedance of an adjustable porosity perforate, formed from two identical perforates sliding over each other, is measured. As the orifice shape becomes more non-circular, the measured impedance is found to deviate from the predicted results for a circular orifice of the same area. Several isolated orifices of the same open area but different shapes are tested and compared with a circular orifice. Both low incident sound pressure levels using broadband noise and high incident sound pressure levels using sinusoidal tones are used to evaluate the impedance performance of these isolated orifices. One orifice mimics the unique shape produced by the adjustable perforate and results in a smaller attached mass (or mass end correction) compared with a round orifice. This is consistent with the perforate impedance results. The unique orifice shape does not appear to have measurable differences in acoustic normal resistance at high incident sound pressure levels. However, since the attached mass plays a key role in determining the peak absorption frequency of resonant liners, the reduction in attached mass relative to a circular orifice has implications where these types of orifices are used.

Keywords

Acoustic Impedance Orifice Liner EndCorrection

Introduction

In a study sponsored by NASA¹ regarding the active control of acoustic liner impedance, a novel concept for manipulating the porosity of a perforate was examined in the present study. In this study, two identical perforates with circular orifices were allowed to slide over each other, which in the limiting case, resulted in a perforate with zero percent porosity. This results in a perforate that consists of orifices that can have non-circular shapes. This perforate is proposed as an adjustable buried septum of a two degrees of freedom resonant liner. Changing the impedance of this septum changes the overall impedance, and hence the resonance frequency of a honeycomb liner. Gaeta² has shown how the sliding perforate concept can be used to vary the peak absorption frequency in a flow-duct. Figure 1 shows how the unique non-circular orifices are formed by two perforates with circular orifices. As one perforate is translated over the other, the initially circular orifice becomes oval or elliptical in shape. Moreover, using this sliding method creates a unique orifice cross-section. Note that, for a given orifice open area, the non-circular orifice has a recessed “shelf” on either side of the perforate. For the balance of this paper, this unique orifice will be referred to as the “stepped-oval” orifice.

Figure 1.

Unique orifice shapes formed by the variable porosity perforate concept.

An adjustable or variable porosity perforate as shown in Figure 1 was mounted in a normal incidence impedance tube (as in Gaeta²) as a test sample and exposed to low intensity broadband incident sound. The measured acoustic mass reactance (i.e. that part of the total reactance attributed to the perforate) is plotted as a function of porosity in Figure 2 accompanied by the circular orifice prediction for comparison. It is found that the measured reactance of the variable perforate deviates from the predicted reactance when its porosity is less than the maximum possible. When the variable perforate is near its maximum porosity (i.e., when the perforate hole shape is circular), it behaves closely to the predicted circular orifice reactance. At the higher porosities, the orifices of the variable perforate closely resemble circular orifices in shape. Since one can obtain a given porosity for any number of orifice shapes, it is speculated that the measured differences were not a result of changing the perforate porosity, rather a result of the unique stepped cross-section along the depth of the orifice that the variable orifice assumes. The purpose of this study is to investigate the role that this cross-sectional shape has on the impedance of orifices and thus the variable porosity perforate geometry described above. Previous understanding of the interaction between acoustic waves and orifices is drawn upon to accomplish the goals of this investigation.

Figure 2.

Reactance of variable porosity perforate (symbols) compared to predicted mass reactance of an equivalent circular orifice perforate (solid curve).

The acoustic impedance of orifices has been examined by many researchers.^3–8 With regard to the non-linear behavior of orifice acoustic impedance, it is generally acknowledged that Sivian³ was the first to observe the phenomenon of increasing acoustic resistance (thus, impedance) with a corresponding increase of the orifice particle velocity. This observed behavior has many far-reaching implications in the design of resonant absorbers that rely on perforates and single orifices, including classical Helmholtz resonators. Over the years, Ingard and Labate,⁴ Bies and Wilson,⁵ Ingard and Ising,⁶ and Melling⁷ have contributed significantly to the understanding of the impedance of orifices and perforates. These researchers documented the flow-acoustic interaction and formulated semi-empirical theories that have guided acoustic liner design. Typically, orifices used in such liners or absorbers have been circular in shape and thus most of the impedance models are based on round orifices. This gave a physical basis for fluidic models for orifice acoustic impedance such as those proposed by Hersh and Rogers⁸ where they used axi-symmetric flow assumption through orifices. They relied on a discharge coefficient to capture the viscous effects.

Interest in understanding the impedance behavior of the variable perforate containing “stepped-oval” orifice shapes was sparked by a tunable resonant liner concept investigated by Gaeta and Ahuja.^9–10 Using the variable porosity perforate as the buried septum of a two degrees of freedom resonant liner allowed for a mechanical shifting of the peak absorption frequency. Figure 3 shows a schematic of this concept. It was found that the range of this peak frequency shift was not as large as predicted by the simple changing of porosity (or perforate open area). The impedance of the buried septum was expected to contribute to this peak frequency shift and was studied in isolation.

Figure 3.

Application of the variable porosity perforate to a 2DOF resonant liner.

Technical approach

A comparison of the impedance measured for a circular orifice against orifices of different shapes was performed in a normal incidence impedance tube, which had an internal diameter of 29 mm and utilized a JBL2446J acoustic driver in conjunction with a Carvin 1500 power amplifier and a HP 33120A digital function generator. Two phase-matched, quarter-inch Bruel and Kjaer microphones were used to sample sound pressure levels inside the impedance tube. The data were analyzed with an HP 3667A Multi-Channel Signal Analyzer. A frequency span of 0–6400 Hz was used with a 4 Hz bandwidth.

For all orifices tested, the open area was made equivalent; thus, an effective radius based on the orifice area was the same for all shapes. Since the variable orifice geometry, in particular its cross-section, could possibly be modified to produce more convoluted orifice shapes other than the stepped-oval shape, selected perforate orifice shapes with equal areas were also tested. In addition to the stepped-oval orifice, a baseline circular orifice, as well as a square, triangular, and a star-shaped orifice were tested. Since the stepped-oval orifice made from identical perforate sheet metal has essentially twice the thickness as the circular orifice, an additional circular orifice with this double thickness was also tested. Figure 4 shows the orifices tested. Table 1 shows the orifices' pertinent dimensions. The open area of all of the orifices was 0.1935 cm² (0.03 inches²). Plates, which were round disks, with orifices in the center were fabricated from aluminum sheets using an electrical discharge machining (EDM) process.

Figure 4.

Different isolated orifice shapes tested in the impedance tube.

Table 1.

Dimensions for tested orifices.

Orifice shape	Perimeter, mm (in.)	Thickness, mm (in.)
Round	15.59 (0.61)	0.81 (0.032)
Round	15.59 (0.61)	1.62 (0.062)
Square	17.56 (0.69)	0.81 (0.032)
Triangular	20.06 (0.79)	0.81 (0.032)
Star-shaped	37.82 (1.49)	0.81 (0.032)
Stepped-oval	16.22 (0.64)	1.63 (0.064)

The orifice plates were placed at the end of a normal incidence impedance tube and were backed by a cylindrical cavity of depth L, forming a cavity resonator. The two-microphone method¹¹ was used to determine the impedance of the orifice-cavity configuration. Figure 5 shows the experimental arrangement with an orifice plate installed in the impedance tube. The backing cavity provided known impedance that could be accounted for when trying to isolate the impedance of the orifice.

Figure 5.

The experimental set-up for impedance testing of isolated orifices.

Both broadband and single tone excitation were used to acquire impedance data. When the impedance measurements were made with single tone excitation, the frequency chosen was the resonance frequency that was determined from broadband excitation, i.e. when the total normalized reactance was zero. Comparison of measured orifice resistance with analytical models required knowledge of the rms velocity at the center of the orifice. This velocity was measured for selected orifices with a single hot-wire sensor.

Analytical considerations

Before discussing experimental techniques and results, it will be helpful to review the semi-empirical methods used to determine the impedance of orifices and perforates. The acoustic impedance of an orifice can be separated into its resistive and reactive components that roughly correspond to viscous (resistance) and inertial (reactance) properties. Friction within the orifice is the dominant contributor to the resistance at low sound amplitudes, while the conversion of sound into vorticity at the orifice edges is the dominant contributor to the resistance at high sound amplitudes. At the higher amplitudes, the incident sound produces an oscillatory jet through the orifice.⁵ Increased losses in this jet can occur if the vorticity is increased. This follows from well-established findings from the study of steady-flow jets and, in particular, the study of jet mixing enhancement.^12,13 The primary component of the reactance of an orifice is the oscillation of air mass within and around the orifice. When incident acoustic energy vibrates the mass of air trapped within the orifice, it also vibrates a certain portion of the air mass surrounding or attached to the orifice. This “attached mass” of air moves in concert with the air within the orifice. The mass end correction, δ_m, is classically defined by the radiation impedance of a circular orifice seen as a circular piston of air in an infinite baffle.¹⁴ For kr_o ≪ 1, the radiation impedance of the circular orifice is expressed in the bracket A in equation (1) below.

δ_{m} = 2 \underset{A}{\underset{︸}{(\frac{8}{3 π} r_{o})}} \underset{B}{\underset{︸}{(1 - \sqrt{\frac{σ}{2}})}}

(1)

The factor of two in equation(1) accounts for both sides of the orifice. The expression in bracket B is an empirical term that Ingard¹⁵ suggested should account for the proximity of orifices in a perforate. For orifices where the effective diameter is larger than the thickness, the “attached mass” becomes a dominant contributor to the inertial properties of the orifice and hence to its reactance.

Sivian³ gives the acoustic impedance of a perforate subject to a normally incident plane wave and after accounting for the porosity and end correction is given by

\frac{Z_{perf}}{ρ c} = \frac{1}{σ} [\frac{Z_{o}}{ρ c}] = \frac{1}{σ} [\frac{R_{o}}{ρ c} + i \frac{χ_{o}}{ρ c}] = \frac{i ω t'}{σ c [1 - \frac{{2 J}_{1} (k_{s} r_{o})}{k_{s} r_{o} J (k_{s} r_{o})}]} where \frac{x_{o}}{ρ c} = \frac{ω}{c} [t + δ_{m}]

(2)

It was Ingard¹⁵ who first addressed the radiation impedance or end correction of a non-circular orifice. Specifically, Ingard examined isolated rectangular orifices in an impedance tube. His key result was a modification to equation (2) that related the effective radius of the orifice to the radius of the cylindrical impedance tube. For porosities that are of most interest to liners, Ingard's modification is not substantially different than equation (2). Thus, equation (2) seems to be the best estimate for the mass end correction for circular as well as rectangular orifices. The orifice formed by the variable porosity perforate in our study is eye-shaped and for convenience can be considered elliptical (see Figure 1). As will be shown shortly, equation (2) does not predict the perforate reactance very well for the unique orifices produced by the variable porosity perforate.

Calculation of the mass end correction from measured data

The normalized impedance for a perforate with a cavity backing of length L is given by

\frac{Z}{ρ c} = Z_{perf} + Z_{cav} = [\frac{R}{ρ c} + i \frac{χ_{m}}{ρ c}] - i σ \cot (kL)

(3)

The acoustic mass, which occupies the space in and around the orifice, can be computed from measured data and its magnitude can be compared as the shape of the orifice is changed. If we replace the effective orifice or plate thickness t' in equation(1) with an expression for the acoustic mass, namely

m = ρ t' A_{0}

(4)

then the normalized mass reactance can be expressed as

\frac{χ_{m}}{σ ρ c} = \frac{ω m}{A_{o} ρ c}

(5)

The left hand side of equation(5) is obtained from impedance tube measurements and equation(3). Thus, the acoustic mass can be calculated from

m = \frac{A_{o} ρ c}{ω} (\frac{χ_{m}}{σ ρ c})

(6)

Now, the mass end correction, δ_m, can be expressed in terms of the orifice thickness

m = m_{insideorifice} + m_{externaltoorifice} = ρ A_{o} t + ρ A_{o} δ_{m} = ρ A_{o} (t + δ_{m})

(7)

Thus, from equations (6) and (7), the end correction can be calculated using equation(8) below

δ_{m} = [(\frac{χ_{m}}{σ ρ c}) \frac{c}{ω} - t]

(8)

A note on the mass end correction of an ellipse

At this point, it is prudent to ask if the planform shape of the stepped-oval could be considered as an ellipse (see Figure 1) and if the resulting reduction in attached acoustic mass is related to this special shape. The mass end correction for an orifice of area A_o can be expressed as¹⁶

δ_{m} = \frac{A_{o}}{K}

(9)

Here the term K is referred to as the orifice conductivity, for which Rayleigh¹⁷ derived an analytical expression as follows

K = [2 \sqrt{\frac{A_{o}}{π}}] [1 + \frac{ɛ^{4}}{64} + \frac{ɛ^{6}}{64} + \dots \dots]

(10)

where ɛ is the eccentricity, given by

ɛ = \frac{\sqrt{a^{2} - b^{2}}}{a^{2}}

where a and b are the semi-major and semi-minor axes of the ellipse, respectively.

Substituting equation (10) into equation (9) yields

δ_{m} = \frac{\frac{\sqrt{π A_{o}}}{2}}{[1 + \frac{ɛ^{4}}{64} + \frac{ɛ^{6}}{64} + \dots \dots]}

(11)

Thus, for a circular orifice for which the eccentricity = 0, the end correction is consistent with equation (2) (ignoring adjacent orifice effects). Plotting the ratio of the mass end correction of an elliptical orifice normalized with that for a round orifice as a function of eccentricity, as shown in Figure 6, one can see that it is only at eccentricities above 0.95 that there is an appreciable reduction in end correction relative to a circular orifice. Assuming that the stepped-oval orifice is an ellipse, then it has a calculated eccentricity of approximately 0.77. This implies that the elliptical mass end correction should be approximately 1% smaller than the circular orifice. It can, thus, be argued that the elliptical shape of the tested stepped-oval orifice does not have a large impact on the attached mass and it is quite likely that the unique cross-sectional geometry has more to do with the reduced attached mass.

Figure 6.

Effect of the elliptical orifice on the mass end correction.

Results and discussion

A direct comparison of the circular orifice and the stepped-oval orifice will be presented for the purpose of highlighting the different acoustic reactance results. This will be followed by a comparison of other orifice shapes with the circular orifice. Finally, selected impedance results for high incident sound pressure levels will be presented.

Impact of stepped-oval orifice on impedance

The end correction in circular orifices is normally independent of the orifice thickness (see equation (2)). A circular orifice of twice the thickness was tested since the stepped-oval orifice has twice the thickness of the baseline orifice as a check. Under broadband excitation, the impedance of the stepped-oval orifice was measured. For descriptive purposes, the circular or round orifice with thickness 0.813 mm will be referred to as the single round orifice and the round orifice with twice the thickness will be referred to as the double round orifice. Figure 7 shows the total normalized reactance in the vicinity of the first resonance frequency for these three orifices. The single round orifice has the highest resonance frequency, as indicated by the zero reactance value. The next lowest resonance frequency is the stepped-oval shaped orifice followed by the double round orifice. Figures 8 and 9 show the associated normalized resistance and mass reactance of the three orifices, respectively, near the resonance frequencies.

Figure 7.

The normalized reactance spectra for the stepped-oval orifice and the circular orifice obtained using broadband input and the associated resonance frequencies (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 8.

A comparison of the normalized resistance between the stepped-oval and circular orifice using broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 9.

A comparison of the normalized mass reactance between the stepped-oval and circular orifice using broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 8 indicates that the additional length of the double round orifice or the stepped-oval shape does not affect the resistance significantly. It is clear from Figure 9 that the double round orifice has a larger mass reactance than the single round orifice. The additional acoustic mass in the double round orifice due to its increased thickness explains its larger total mass reactance relative to the single round orifice. However, the stepped-oval orifice appears to have some intermediate amount of mass within its unique oval shaped orifice does not have as much total reactive mass as the double round orifice. It thus appears that the stepped-oval orifice created from two round orifice has some intermediate amount of mass reactance that lies between the two round orifice thicknesses.

To better understand the mass reactance of the stepped-oval orifice, the attached acoustic mass and mass end correction were calculated for each of the round orifices and the stepped-oval orifice. Since the standard end correction for a circular orifice is dependent on the orifice radius only, the single and double round orifice should exhibit the same amount of attached mass and thus the same end correction. This is borne out in Figures 10 and 11. Figure 10 shows the spectra of the attached acoustic mass, whereas Figure 11 shows spectra of the calculated mass end correction. It is found that the single and double thickness round orifices have essentially the same end correction at their respective resonance frequencies and agree quite closely with the predicted value. However, using the same thickness as the double round orifice, the corresponding attached mass for the stepped-oval shaped orifice is measurably smaller.

Figure 10.

Reduced attached mass for the stepped-oval compared to circular orifice; broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 11.

Reduction of the mass end correction associated with the stepped-oval compared with the circular orifice; broadband input (A_o = 19.36 mm²; Δf = 4 Hz_).

Impact of orifice shape on impedance

Orifices of different shapes but without possessing the unique stepped cross-section as the stepped-oval orifice were tested in the normal incidence impedance tube and compared with the baseline circular orifice. The normalized resistance spectra for the square, triangular, and star-shaped orifices shown in Figure 4 are compared with those for the circular orifice in Figure 12. Figure 12 shows that for the highly non-circular star-shaped orifice, the normalized resistance is noticeably higher (about 60% higher than the round orifice) in the frequency range near resonance. This result implies that a variable porosity perforate that is made with very non-circular orifices (with highly convoluted shapes) may provide a higher degree of absorption than equivalent circular orifices. This implication is indeed borne out in the absorption coefficient comparison of these orifices as shown in Figure 13. Note that the peak absorption corresponds to the resonance frequency (i.e. zero reactance value). Increasing orifice perimeter is accompanied by increasing absorption. The star-shaped orifice has an absorption coefficient that is approximately 40% higher than the equivalent-area round orifice.

Figure 12.

Impact of the orifice shape on the normalized resistance for several orifices near the resonance frequency; broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 13.

Impact of orifice shape on the normalized mass reactance for several orifices near the resonance frequency; broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

The corresponding mass reactance spectra are compared in Figure 14. The normalized mass reactance does not appear to be significantly affected by the orifice shape. This implies that the associated mass end correction for the different shapes tested was not significantly different from the baseline circular orifice. There is a slight trend of reduced mass reactance with increasing orifice perimeter. (In the interest of completeness, it is fair to state that the authors did not test the elliptical shape without a step; as such additional tests without a step are required and the comparisons with predictions for elliptical shapes only will provide 100% confirmation that the step may be impacting the results.)

Figure 14.

Impact of the orifice shape on the absorption coefficient for several orifices near the resonance frequency; broadband input (A_o = 19.36 mm²; Δf = 4 Hz).

The literature on steady jets indicate that the mixing of non-circular jets, especially those that have a tab like structure at the nozzle exit similar to the star-shaped orifice is much better than circular jets.^18,19 It is quite likely that the increased mixing in the near field of the orifice could cause an increase in resistance. Further study is needed to address the detailed mixing of the highly non-circular orifice under acoustic excitation. This mixing translates into a loss of energy that originates from the acoustic energy driving the oscillating mass. Hence, less acoustic energy is reflected back to the source and the absorption coefficient is increased.

Impact of high incident sound pressure levels

It has been established that when very large oscillatory velocities exist in the orifice, an unsteady jet issues from both sides of the orifice.^4,7 This is produced by very high incident sound amplitudes or very small orifices. This phenomenon has been treated as a quasi-steady jet in the literature (see, for instance, Ingard and Ising⁶). As the variable orifice perforate is transitioned to smaller porosities and more non-circular orifice, it was observed that the standard resistance model based on circular orifices did a reasonable job of predicting the normalized resistance even though it over-predicted the mass reactance. These data were obtained for relatively low incident sound pressure levels for a particular frequency via a high-energy broadband signal.

Are there significant differences when the orifices are exposed to higher incident sound pressure levels? To examine this question, impedance tests on the selected orifice shapes described above were conducted using single tones at gradually increasing amplitudes. The frequency of the tone was chosen to be the resonance frequency determined from the broadband tests. After the impedance data were acquired, the same acoustic conditions were reproduced and a single hot-wire was placed in the center of the orifices to measure the root-mean-square orifice velocity to correlate with the normalized impedance. Only the axial (orifice axis) component of velocity was measured. The hot-wire data were sampled at a rate of 55 kHz and taking 1024 samples.

Figure 15 shows the normalized resistance of the tested orifices under single tone excitation as a function of increasing tone amplitude and hence the orifice velocity, which was measured as described above. This a log–log plot, and on this basis the curve is almost linear at the higher values of the particle velocities at the orifice, indicating that the acoustic resistance varies non-linearly with particle velocity at higher orifice velocities and thus at the higher levels of incidence amplitude. If the data were plotted on linear scale, the resistance at the lower velocities and thus incidence amplitudes will vary linearly with the orifice velocity and thus with the incidence acoustic amplitude. Thus, in general, the acoustic resistance is found to increase in a linear fashion with incidence sound pressure at low sound pressure levels and non-linearly at higher sound pressure levels. At the lower levels, the star-shaped orifice resistance is distinctly higher than that of the other orifice shapes. These lower incident sound pressure level results are consistent with the broadband results shown in Figure 12. At the lowest measured orifice velocity condition (approximately 0.04 m/s), a trend is discernable with increased orifice perimeter in that the larger the orifice perimeter, the larger the resistance. However, as the sound pressure is increased and the orifice velocity is increased above 0.5 m/s, the resistance curves for all of the shapes tend to collapse on each other. Figure 16 shows an increase in absorption coefficient with increasing sound amplitude for the tested orifice shapes. As with the resistance data, the shape of the orifice seems to have an effect on the absorption at the lower orifice velocities while the data tend to merge together at the higher velocities.

Figure 15.

Result of increasing SPL, and hence the orifice velocities on the normalized resistance for different orifice shapes; Single tone inputs (A_o = 19.36 mm²; Δf = 4 Hz).

Figure 16.

Result of high orifice velocities on the absorption coefficient for different orifice shapes; single tone inputs (A_o = 19.36 mm²; Δf = 4 Hz).

The literature suggests that a superposition of the low orifice velocity (linear) impedance model and the quasi-steady jet model (non-linear) should be a reasonable predictor of orifice resistance for a range of incident sound pressure levels. Melling,⁷ who separates the resistance into a linear part and a non-linear part, gives an expression for the specific acoustic resistance, R, as follows

R = \frac{R_{lin}}{ρ c} + \frac{0.6}{C_{D}^{2}} (\frac{1 - σ^{2}}{σ}) \frac{u_{o - rms}}{c}

(12)

where R_lin is the linear regime resistance.

The second term in equation (12) represents the non-linear portion of the total specific resistance and dominates when the orifice velocity is large; and the first term dominates when the velocity is small. This semi-empirical expression uses the concept of discharge coefficient as a crude measure of loss incurred by the unsteady oscillatory jet present at high orifice velocities. For the Reynolds number typically encountered (Re_D > 1000) in these orifices, a discharge coefficient of 0.6 is nominal. The origin of this value of discharge coefficient is from steady orifice flow metering data from Ingard and Ising⁶ and Hersh et al.²⁰ who fitted experimental data with their theory and deduced a value of C_D. They reported that at amplitudes approaching 170 dB, the discharge coefficients approached a value of 0.4, while at lower amplitudes, near 100 dB, the value was near 0.7. As the jetting in the orifice diminishes with decreasing sound amplitude, the C_D should approach unity. Figure 17 compares the measured normalized resistance of the star-shaped and circular orifices with the predicted resistance from equation (12) for two values of C_D, namely, 0.5 and 0.6. It appears that a C_D of 0.5 provides a better fit for the high velocity regime, but clearly the lower velocity portion of equation (12) does not capture the higher resistance of the star-shaped orifice. Note that the star-shaped orifice is substantially different from the other orifices tested only in effective orifice perimeter (see Table 1). Further study is needed to explore this phenomenon relative to the trend observed in Figure 12 . Further progress in exploring the physics of high-amplitude acoustic waves interacting with orifices has recently been made in the literature^21–24. Using direct numerical simulation (DNS), the behavior of a slit-type orifice under high amplitude incident sound has been studied.²² Good agreement with experimental results suggests that DNS will be a valuable tool for understanding the absorption mechanisms involved.

Figure 17.

Comparison of the star-shaped and the circular orifices with the normalized resistance prediction, highlighting the differences between the low and high orifice velocities; Single tone inputs (A_o = 19.36 mm²; Δf = 4 Hz).

The effect of large orifice velocities on the mass end correction of orifices can thus be summarized as follows: The attached mass oscillating in concert with the mass inside the orifice remains attached until a critical velocity is reached. Above this velocity, the amount of attached mass is reduced. This critical velocity is typically around 1 m/s and the mass end correction is reduced to roughly one-half of its low velocity value at velocities above 10 m/s. Earlier it was shown that the stepped-oval orifice had a reduced mass end correction at lower orifice velocities (see Figure 11). Figure 18 shows the mass end corrections for the tested orifice shapes as a function of orifice velocity. Note that the stepped-oval orifice has the lowest correction at the lower velocities, as expected. The square, triangular, and star-shaped orifices lie somewhere between the circular and stepped-oval correction at the lower velocities. Except for the stepped-oval, this trend is inversely proportional to orifice perimeter.

Figure 18.

Comparison of different orifice shapes with the mass end correction prediction; Single tone inputs (A_o = 19.36 mm²; Δf = 4 Hz).

It is clear that near the orifice rms velocity of 1 m/s, the mass end corrections for all of the orifices start diminishing in a linear fashion, as expected. The data show that the rate of decreasing end correction is similar for all of the orifices except the star-shaped orifice that decays at a lower rate. Coupled with the differences in acoustic resistance at lower orifice velocities and the differences in the mass end correction at higher orifice velocities, the star-shaped orifice presents motivation for future work involving more convoluted orifice shapes that could possibly be a result of a unique variable porosity perforate. Furthermore, the concept of bias flow²⁵ relies on orifice non-linearity for its effective performance. Bias flow acoustic liners have small amounts of air pumped through them so that a steady jet is superimposed on the unsteady jet produced by incident sound. The changes in acoustic impedance from this controlled input air allow for the liner absorption to be tuned. Thus, the role of orifice shape on impedance is the key in the effectiveness of the concept of bias flow for sound absorption.

Selected flow visualization of orifice jetting

While a detailed study of the energy structure of the jet issuing from different shaped orifices is beyond the scope of this study, some flow visualization was performed on some of the orifices tested in the impedance tube but without the backing cavity. Images of the jetting structure from the orifices were obtained via a particle image velocimetry (PIV) system. An acoustic driver was attached to one end of a tube and an orifice plate was attached to the other end. The tube was approximately 20 inches long. The tube was filled with incense smoke and the driver was excited with a single tone input.

Figures 19 to 21 show evidence of the orifice jetting for the round, star-shaped, and stepped-oval orifices, respectively, at a moderate sound pressure level. A train of convecting vortical structures is evident in all cases. While no significant differences in the jetting are visible between the circular and star-shaped orifices, there is a noticeable difference in the stepped-oval jetting. It is immediately evident from Figure 21 that the oscillatory jet issuing from the stepped-oval orifice is deflected at some angle from the orifice centerline axis. The recessed space on one side of the orifice, which is uniquely created by the overlapping circular orifices (see Figure 1), acts as a likely local pressure sink that deflects the jet. Well-defined vortical structures still persist several diameters downstream. The asymmetrical nature of the jet near the orifice itself may indeed be the cause for the reduced mass end correction. This qualitative evidence of the nature of the stepped-oval orifice acoustic flow field coupled with the quantitative data from the impedance tube suggest that the shape of an orifice has an impact on the acoustic impedance and that in some circumstances, using circular orifice analytical results is inadequate.

Figure 19.

Oscillatory jet issuing from a circular orifice under excitation from a plane acoustic wave at 120 Hz; note the convecting vortical structures.

Figure 20.

Oscillatory jet issuing from a star-shaped orifice under excitation from a plane acoustic wave at 120 Hz.

Figure 21.

Deflected oscillatory jet issuing from a stepped-oval orifice under excitation from a plane acoustic wave at 120 Hz.

Conclusion

The variable porosity perforate, which motivated this study, is formed from two identical perforates with circular orifices. When one perforate slides over the other, a unique stepped-oval shaped orifice results. This type of perforate results in a lower mass reactance than that achieved with two coaxial circular orifices and impacts the peak absorption frequency of resonant liners that may utilize such a perforate. This paper quantifies the acoustic impedance of this unique orifice at low and high incident sound pressure levels. Furthermore, since the variable porosity perforates obtained by sliding perforates with orifices that may not be round, may produce even more convoluted shapes, a study of several isolated orifices of equivalent area but different shapes were examined. In addition to the baseline circular orifice and the stepped-oval, square, triangular, and star-shaped orifices were tested. The main findings of this study are summarized as follows:

The stepped-oval orifice has a smaller mass end correction than an equivalent round orifice. It has a “vectored” oscillatory jet that exists as a result of the unique overlapping orifice cross-section. This impacts the acoustic liner control concept that may utilize the variable porosity perforate described here. It tends to limit the amount of peak absorption frequency control since the mass end correction plays a key role in determining the resonant frequency of the liner. The vectored jetting was observed with flow visualization and the resulting asymmetrical unsteady flow field is quite likely the cause for the reduced mass end correction. Further study is needed to confirm this.

At relatively low incident sound amplitudes, and thus at low orifice velocities, the star-shaped orifice demonstrated higher acoustic resistance, which led to a higher absorption coefficient. This can impact the variable orifice perforate concept if more convoluted orifice shapes are produced by another mechanical method or set of perforates.

At higher incident sound amplitudes, and thus at higher orifice velocities, the acoustic resistance appears to be independent of orifice shape. This result implies that existing fluid mechanical models that describe high amplitude orifice impedance are adequate for the description of the variable orifice perforate when exposed to high amplitude sound. However, this result is somewhat surprising given the lower amplitude results. A more detailed fluid dynamic study is required to understand this phenomenon.

Footnotes

Acknowledgements

The authors are grateful to Mr. Mike Jones and Tony Parrott of NASA Langley Research Center for their support and invaluable advice throughout the course of this work. Portions of this work also appeared in AIAA Paper AIAA-2001-0662.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Portions of this study were funded by NASA Langley Research Center under Grant NAG1-1734.

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