Experimental study of a slit in the presence of a bias flow under medium- and high-level acoustic excitations

Abstract

This work presents an experimental investigation of the acoustic properties of a slit in the presence of a bias flow under moderate- and high-acoustic excitations. Impedance tube experiments are discussed for a geometry inspired by deep punching resulting in a cut in the plate. The acoustic transfer impedance of the plate is discussed for several bias flow velocities, acoustic excitation, and different frequencies. In the range considered for this study, a bias flow appears to have two main effects, globally enhancing the sound absorption of the plate and creating a protective layer downstream of the plate due to the interaction between the slits. A maximum of the enhancement factor is found at a specific ratio between the acoustic velocity and the bias flow velocity. Two simple asymptotic behaviors are found, dominated by the flow or by the acoustic excitation, respectively. The behavior of the inertance is complex. Globally the inertance decreases with decreasing flow Strouhal number while its dependency on the amplitude of the acoustic velocity is less obvious.

Keywords

Slits non-linear bias flow impedance

Introduction

In combustion chambers of gas turbines, multi-perforated liners are used to provide film cooling of the walls and offer an excellent candidate to disrupt thermoacoustic instabilities in emerging technologies, such as hydrogen combustion. A perforated plate offers an excellent sound absorption ability and provides means to manipulate and re-distribute the acoustic energy loss at the chamber’s walls, making the system stable.^1–5 In these conditions, the plate encounters high-acoustic excitation and displays a non-linear response. Several works suggest that the presence of non-linear effects can deteriorate the performance of the absorbers.^6–9 However, the presence of a bias flow through the orifices impacts the absorption characteristics of the liner. Extensive research has been performed on the so-called bias flow liners and a recent review of such publications is presented by Lahiri and Bake.¹⁰ In the literature, most of the works focus on circular perforations in the presence of a mean flow.^11–15 Salikuddin et al.¹¹ find that the effect of acoustic excitation amplitude is similar to that of a bias flow. Jing and Sun¹³ compare a discrete vortex model and a quasi-steady method to study high-intensity sound absorption at an orifice with a bias flow. Eldredge and Dowling¹⁶ prove the effectiveness of a perforated liner system with a mean bias flow in the absorption of planar acoustic waves in a circular duct. The combination of a bias flow and high-acoustic amplitudes is also studied by Luong et al.¹⁷ to extend the linear theory of Howe^18,19 to predict the attenuation of sound by vorticity production in a bias flow aperture. Furthermore, several works focus on experimental results for the interaction between an acoustic field with a bias flow.^15,20–23 Fabre et al.^24,25 focus on the impact of flow through a thin circular aperture. Zhou and Bodèn^14,16 find that a bias flow makes the acoustic properties more complex compared to the no bias flow case, especially when the velocity ratio between acoustic velocity and mean flow velocity is close to unity. In recent works, Guzmàn-Iñigo et al.,²⁶ Hirschberg et al.²⁷ and Burgmayer et al.^15,28 discuss the effect of the geometry of the perforations on the acoustic response in the presence of flow. Interesting results are obtained with so-called zero mass flow liners where a single-degree-of-freedom liner is attached to an acoustic actuator emitting a secondary high-amplitude sound field, inducing a periodic bias flow in the orifices.^29–32

In this work, we focus on a specific slit geometry, introduced in Aulitto et al.³³ obtained by deep punching resulting in a cut in a metallic plate, displayed in Figure 1. This geometry could present a cheaper alternative to slanted perforations, keeping equivalent characteristics. The acoustic behavior of such perforations in the presence of a bias flow in the linear regime is discussed in Aulitto et al.,³⁴ while Aulitto et al.³⁵ focus on the impact of non-linear effects on the acoustic impedance of the plate.

Figure 1.

Schematic representation of the geometry with coalescence of the micro-jets used for film cooling.

Due to the particular manufacturing process, cavities are formed in the plate upstream and downstream of each slit, and in the presence of a bias flow, the individual jets tend to attach to the wall of the cavity downstream of each slit, coalescing and creating the same effect of film cooling as grazing effusion holes. Figure 1 shows that downstream of the plate, the jets are coalescing and the combination of multiple slits creates the same effect of film cooling as grazing effusion holes.

We focus on the effect of a bias flow on the acoustic transfer impedance of the plate and its absorption properties in presence of medium- and high-acoustic excitations when the plate exhibits a non-linear response. However, we restrict our analysis to conditions of moderately high-acoustic excitation amplitudes, for which the response is dominated by a single frequency so that the concept of impedance remains meaningful. The acoustic transfer impedance of the micro-slit plate is experimentally studied, using impedance tube measurements. Special attention is given to the search for relevant dimensionless parameters and asymptotic behavior for low or high values of these parameters. The first section of the results discusses the enhancing effect of a bias flow on the resistance of the plate. Following, a study of the change of resistance due to non-linearities is provided. Further, the study focuses on the behavior of inertance. Finally, the conclusions are summarized.

Experimental setup

Geometry of the samples

In Figure 2, a sketch of the plate geometry is shown together with a picture of the sample, with the relevant parameters summarized in Table 1.

Figure 2.

Sketches of the plate and the cross-section of a single slit with the definition of parameters and picture of the sample.

Table 1.

Relevant parameters in Figure 2.

External diameter	$D_{p} = 70 mm$
Internal diameter	$D_{i} = 50 mm$
Plate thickness	$t_{p} = 5 mm$
Slit width	$b = 0.5 mm$
External width of the cavities	$w_{c, e} = 5 mm$
Internal width of the cavities	$w_{c, i} = 2.25 mm$
Depth of the cavities	$t_{d} = 2.75 mm$
Thickness of the plate at the slit	$t = t_{p} - t_{d} = 2.25 mm$
Slit length	$l_{s} = 35 mm$
Total slit length	$l_{s, t o t} = 3 \times l_{s} \approx 105 mm$
Distance between slits	$d_{s} = 9 mm$
Angle	$45^{\circ}$
Porosity	$Φ = 2.7 %$

The porosity (of the portion of the plate in the impedance tube) is defined as the ratio between the total open area $A_{open} = 3 b l_{s}$ and the total area of the plate $A_{tot} = π (D_{i} / 2)^{2}$ . The edges in contact with the slits are kept as sharp as possible to avoid effects due to the rounding of the edges. For this study, the impedance tube discussed in Aulitto et al.³⁴ and shown in Figure 3 is used. The setup consists of an impedance tube of internal diameter $D_{i} = 50 mm$ , a wall thickness $t_{w} = 10 mm$ and length $l_{t} = 1000 mm$ . Six pre-polarized $1 / 4$ in microphones (type BWSA, sensitivity $50 mV / Pa$ ) are flush mounted in the tube wall with a spacing of $175 mm$ . The micro-slit plates are positioned at the end of the impedance tube through a sample holder of $L_{h} = 50 mm$ . A loudspeaker ( $25 W$ ) is used as a source of harmonic excitation and an airflow is injected at the top wall of the setup (see Figure 3). The volume flux of air is controlled by a Bronkhorst F-202AV mass flow controller.

Figure 3.

Schematic representation of the impedance tube setup.

The microphone closest to the sample (at position $x = x_{r e f} = - 47 mm$ , where $x = 0$ corresponds to the first edge of the slit) is used as a reference for the measurements, and the calibration of the other microphones. In this study, we focus on relatively low flow speeds with a mass flow measured by the mass flow meter $Q < 10 lpm$ . The acoustic excitation amplitudes are chosen between $86 dB$ , corresponding to the linear limit, to $120 dB$ , within the range of linearity of the microphones.

Definitions

The concept of acoustic transfer impedance is introduced in the frequency domain of frequency $f$ in Aulitto et al.,³⁵ where the validity of assuming an amplitude-dependent effective impedance is proven. The transfer impedance of the plate is defined as the ratio between the complex acoustic pressure difference $Δ \hat{p}$ and the amplitude of the acoustical velocity $\hat{u}$ in a cross-section upstream of the plate. The pressure difference is found by the extrapolation of the plane-wave solutions (on both sides) to the sample surface. The dimensionless transfer impedance of the plate is

Z_{plate} = \frac{Δ \hat{p}}{ρ c \hat{u}} = R e [Z_{plate}] + i I m [Z_{plate}]

(1)

with

i^{2} = - 1

R e [Z_{plate}]

the resistive part of the transfer impedance of the plate (or resistance of the plate) and

I m [Z_{plate}]

the reactive part of the transfer impedance of the plate (or inertance of the plate), where

ρ

is the density of air and

c

is the speed of sound in air and

Z_{plate} = z_{plate} / ρ c

In Aulitto et al.³⁵, it is shown that in the same range of frequencies and amplitudes as in the present study, one finds at least an order of magnitude difference in amplitude between the fundamental frequency and higher harmonics. The study is limited to moderately high amplitudes, and in the present work, the concept of impedance is extended in the presence of a bias flow. The study is limited to frequencies $f < 220 Hz$ . To measure the transfer impedance of the plate the procedure explained in Aulitto et al.³³ is followed. The transfer impedance of the sample is obtained by measurements of the reflection coefficient with and without the sample, where the open pipe reflection coefficient $R_{o}$ is measured before each measurement.

In this open pipe radiation impedance measurement, the sample is replaced by a ring in the sample holder so that the geometry (pipe length and position in the room) is the same as when measuring with a sample. The dimensionless radiation impedance is obtained as

Z_{rad} = \frac{1 + R_{o}}{1 - R_{o}}

(2)

Further, the impedance of the sample-loaded termination is obtained by measuring the sample-loaded reflection coefficient

R_{s}

Z_{s} = \frac{1 + R_{s}}{1 - R_{s}}

(3)

Consequently, the dimensionless transfer impedance of the plate is

Z_{plate} = Z_{s} - Z_{rad}

(4)

Since the samples have relatively low porosity, the radiation impedance is much lower than the impedance of the plate. The radiation impedance used to define the transfer impedance of the sample is obtained in the no-flow case. Tests in the presence of flow show a deviation smaller than

1 %

on the impedance of the sample, compared to no-flow tests.

The dimensionless numbers used in this work are presented here.

The Shear number is the ratio between the slit width $b$ and the thickness of the viscous boundary layer $δ_{v}$

S h = \frac{b}{δ_{v}}

(5)

with

δ_{v} = \sqrt{2 μ / ω ρ}

, where

ω = 2 π f

is the angular frequency,

ρ

is the air density (

ρ = 1.18 {kg / m}^{3}

25^{\circ}

C and atmospheric pressure) and

μ

is the dynamic viscosity of air (

μ = 1.85 \times 10^{- 5} kg / ms

25^{\circ}

C). The Shear number is used as a dimensionless presentation of the frequency. The range of Shear numbers considered for the results shown in this work is

1.8 \leq S h \leq 3.5

. This range is chosen to obtain a good signal-to-noise ratio for the linear case.

The acoustical Strouhal number ( $S t_{ac}$ ) is the ratio between the slit width $b$ and the amplitude of the oscillating particle displacement at the slits

S t_{ac} = \frac{ω b}{| {\hat{u}}_{ac} |}

(6)

where

{\hat{u}}_{ac}

is the cross-sectional surface averaged acoustic velocity amplitude defined as

{\hat{u}}_{ac} = \hat{u} / Φ

, with

Φ

the plate porosity and

\hat{u} = \frac{| \hat{p} |}{Φ | z_{plate} |}

(7)

the approaching acoustic-flow velocity,

| {\hat{p}}_{ref} |

the acoustic pressure at the reference microphone and

z_{plate}

the acoustic transfer impedance of the plate. The acoustic pressure is obtained assuming a sensitivity of the reference microphone of

50 mV

. There is an uncertainty of the order of

5 %

in the absolute calibration of this microphone. The acoustic Strouhal number contains both the frequency and the acoustic amplitude.

The behavior of the plate can be studied as a function of the Strouhal number of the flow $S t_{flow}$ based on the cavity width $w_{c}$ , expressed as

S t_{flow} = \frac{ω w_{c}}{U_{s}}

(8)

with

U_{s}

the average flow velocity in the slit,

U_{s} = U_{d} / Φ

, with

Φ

the porosity of the plate and

U_{d}

the average velocity in the duct. In the experiments, the velocity

U_{s}

is defined as

U_{s} = \frac{Q}{Φ A_{tot}} = \frac{Q}{A_{open}}

(9)

with

Q

the mass flow measured by the mass flow meter and

A_{open} = 3 b l_{s}

the total open area.

At low frequencies, using Bernoulli’s frictionless quasi-steady-flow equation

p_{flow} \approx \frac{1}{2} ρ U_{s}^{2}

(10)

one retrieves the limit for

S h \to 0

(

f \to 0

)

| {\hat{u}}_{ac} | = | \hat{p} | / ρ U_{s}

(11)

The quantity

S t_{flow} / 2 π

is an estimation of the ratio of travel time over the cavity of fluid particles in the jet formed by flow separation downstream of the perforation and the oscillation period

1 / f

The Reynolds number based on the slit width $R e$ is defined, for a given volume flux $Q$ through the perforated plate, as

R e = \frac{ρ U_{s} b}{μ} = \frac{ρ Q}{μ l_{s, tot}}

(12)

with

l_{s, tot}

the total length of the slits (

l_{s, tot} = 3 \times l_{s}

). This definition is used because in the experiments the steady mass-flow rate is measured. From this, the mass-flow rate

Q

can easily be estimated.

An acoustic Reynolds number

R e_{ac} = \frac{| {\hat{p}}_{ac} | b}{Φ c μ}

(13)

is defined as if the plate behaves as an anechoic termination, where

c

is the speed of sound. The speed of sound

c

is obtained by the measurement of the ambient temperature as

c = c_{ref} \sqrt{\frac{T}{T_{ref}}}

(14)

with

c_{ref} = 343 m / s

and

T_{ref} = 298.15 K

. This representation shows a

R e_{ac} \approx 1

in the linear case (corresponding to a sound pressure level (SPL)

86 dB

, with

SPL = 20 \log_{10} ({\hat{p}}_{, ac, rms} / p_{ref})

with

p_{ref} = 2 \times 10^{- 5} Pa

. The Mach number

M a = U_{s} / c

considered for this study is

M a << 0.1

, ranging from

M a = 0.0015

M a = 0.009

The dimensionless number $\bar{P}$ is introduced to define the ratio between the flow and the acoustic pressure in the experiments as

\bar{P} = \frac{ρ {U_{s}}^{2}}{2 | \hat{p} |} \approx \frac{p_{flow}}{| \hat{p} |}

(15)

This representation of the dimensionless number is chosen (and not the inverse) to include the no-flow case for

U_{s} = 0

with

\bar{P} = 0

. The parameter

\bar{P}

is the ratio between the dynamic pressure of the steady flow and the acoustic pressure.

Enhancement of the resistance

Maa³⁶ suggested that maximum absorption for a micro-slit plate can be obtained when $Z / ρ c$ is approximately unity. Hence, for optimal absorption, the impedance of the plate should approach the characteristic impedance of air. In this section, the potential of adding a bias flow to enhance the absorption properties is investigated. In the presence of a main flow, a free jet is formed which is delimited by so-called shear layers that display convective instabilities. Acoustic perturbations are convected away by the main flow, leading to formation of large coherent structures, dissipated by turbulence. The range of Shear numbers considered is $1.8 \leq S h \leq 3.5$ .

In Figure 4, the real part of the impedance is shown normalized with the characteristic impedance of air for several flow speeds (Reynolds numbers) as a function of the ratio between the amplitude of the acoustic velocity and the flow velocity. The relevant parameter is the mean flow Reynolds number $R e$ (in legend, the line corresponding to the highest acoustic amplitude is used for representation). Globally, in the presence of flow, one achieves a higher resistance for low- and high-acoustic excitations. This shows that even for high-acoustic excitations, the presence of a flow does produce a significant effect on the resistance. In Aulitto et al.,³⁴ the quasi-steady limit $ρ {\hat{u}}_{ac} / 2 α^{2} Φ$ provides an order of magnitude for the resistance in the linear regime. In this work, the contribution of non-linear effects provides an increase of resistance of one order of magnitude and a smaller reduction of the inertance with respect to the linear case, dominated by viscous effects. This translates into an increase in absorption. The same normalized representation is used in Figure 5, where the real part of the acoustic transfer impedance is shown, normalized with the quasi-steady limit as a function of the ratio between the acoustic velocity ${\hat{u}}_{ac}$ and the mean flow velocity $U_{s}$ . The dimensionless number $R e_{ac}$ is used in the legend to discuss the effect of the amplitude.

Figure 4.

Real part of the acoustic transfer impedance of the plate normalized with the characteristic impedance as a function of the velocity ratio ${\hat{u}}_{ac} / U_{s}$ . The different symbols represent different frequencies.

Figure 5.

Real part of the acoustic transfer impedance of the plate normalized with the quasi-steady limit for high-acoustic excitations as a function of the velocity ratio ${\hat{u}}_{ac} / U_{s}$ .

It can be seen that for ${\hat{u}}_{ac} / U_{s} < 2$ , the behavior of the resistance is dominated by the mean flow. The dimensionless resistance follows the linear quasi-steady asymptote $U_{s} / {\hat{u}}_{ac}$ . For $2 < {\hat{u}}_{ac} / U_{s} < 4$ , one observes a clear transition. For ${\hat{u}}_{ac} / U_{s} > 4$ , the resistance is dominated by the non-linear behavior due to high-acoustic excitation amplitude and tends to a horizontal asymptote. This asymptote corresponds to the high-amplitude limit for the no-flow case. One observes that the effect of a bias flow is similar to the effect of acoustic excitation amplitude.

A third dimensionless representation for the real part of the acoustic transfer impedance is used in Figure 6. The linear quasi-steady limit $ρ U_{s}$ allows quantifying the contribution of convection on the resistance. It can be seen that for ${\hat{u}}_{ac} / U_{s} < 4$ , around $80 %$ of the resistance is due to the linear contribution.

Figure 6.

Real part of the acoustic transfer impedance of the plate normalized with linear quasi-steady limit as a function of the ratio ${\hat{u}}_{ac} / U_{s}$ .

At a given acoustic excitation amplitude (defined in decibels, $dB$ ), the ratio between the resistance and the resistance in the no-flow case can be defined as the** enhancement factor (EF). This factor quantifies the increase in the absorption that can be obtained by introducing a bias flow and can be expressed as

rEF = \frac{Re [z_{plate}]}{Re [z_{\,plate,no-flow}]}

(16)

where rEF stands for the resistance EF. An enhancement of the resistance corresponds to an increase in the bandwidth of the absorption. In Figure 7, the EF is shown as a function of the velocity ratio for several acoustic excitation amplitudes.

Figure 7.

Enhancement factor as a function of the ratio ${\hat{u}}_{ac} / U_{s}$ for several acoustic excitation amplitudes.

One sees that, overall, the presence of a bias flow increases the resistance with respect to the no-flow case. Hence, the EF is larger than one.

For low-velocity ratios ${\hat{u}}_{ac} / U_{s} < 0.4$ , which corresponds to the region dominated by the mean flow, one observes an increase of the resistance with the velocity ratio. The curves show a maximum around ${\hat{u}}_{ac} / U_{s} \approx 1$ , where, after the maximum, one finds a transition zone where the EF decreases with increasing velocity ratios. Jing and Sun¹³ and Zhou and Bodèn¹⁴ found a similar transition range for an orifice in the presence of a bias flow ( $0.3 < {\hat{u}}_{ac} / U_{s} < 4$ ). In this range, they observe a dip in the resistance. For ${\hat{u}}_{ac} / U_{s} > 4$ , the resistance tends to the no-flow case, as shown in Figure 6. A horizontal asymptote close to $rEF = 1$ is found. From the engineering point of view, this means that, given an acoustic pressure excitation, a bias flow velocity can be tuned to achieve the largest resistance, that is, the maximum of sound absorption. In Figure 8(a), a zoom of Figure 7 for ${\hat{u}}_{ac} / U_{s} < 3$ is shown.

Figure 8.

(a) A zoom of the enhancement factor for velocity ratio ${\hat{u}}_{ac} / U_{s} < 3$ . (b) Maximum of the enhancement factor ${rEF}_{m a x}$ shown as a function of the acoustic excitation amplitude in decibels.

The maximum EF is shown in Figure 8(b) as a function of the acoustic excitation amplitude in decibels. This shows that the maximum enhancement decreases with increasing acoustic excitation amplitudes. A linear fit of the experimental results with ${rEF}_{max} = 20.26 -- 0.15 {dB}^{- 1} {\hat{p}}_{ac}$ , with ${\hat{p}}_{ac}$ in decibels is proposed. The parameters of this fit are specific to this geometry.

Change of resistance due to non-linearities

In this section, the change of resistance due to the presence of non-linearities at moderate- and high-acoustic amplitudes is discussed. In the linear case, for the full range of frequencies, the resistance is constant. In absence of main flow, viscous effects are dominated by dissipation in the laminar Stokes boundary layer. In the absence of main flow, at high-acoustic amplitudes, the response of the plate is dominated by formation of vortices due to flow separation that dissipates kinetic energy by turbulence. In the presence of the main flow, the behavior is more complex, with potential movements of the flow separation point. Globally, the effect of viscosity is the same as in the linear case, leading to dissipation in the Stokes layer, flow separation, and dissipation by turbulence. Figure 9 shows the change of resistance $Δ Re [z_{plate}] = Re [z_{plate}] - Re [z_{plate, linear}]$ normalized with the resistance in the linear case ( $Re [z_{plate, linear}]$ ) and presented as a function of the Shear number.

Figure 9.

Change of the real part of the acoustic transfer impedance of the plate normalized with the linear case as a function of the Shear number $S h = b / δ_{v}$ for several Reynolds numbers.

The linear contribution $Re [z_{plate, linear}]$ obtained at the lowest experimental amplitude ( $86 dB$ ).

Each of the figures corresponds to a fixed flow speed (Reynolds number). Each curve corresponds to a different value of the parameter $\bar{P} = p_{flow} / | {\hat{p}}_{ac} | = ρ U_{s}^{2} / (2 | {\hat{p}}_{ac} |)$ . It can be seen that, globally, one observes an amplitude-dependent resistance that increases with increasing amplitude (and decreasing $\bar{P}$ ) for all mean flow speeds. For $R e < 67$ , the resistance decreases with the Shear number.

Increasing the flow speed, the effect of the non-linearities decreases and for $R e = 100$ one observes a non-monotonous behavior of the resistance.

For low values of $\bar{P}$ ( $\bar{P} < 10$ ), the behavior of the resistance is heavily affected by the presence of a bias flow.

In Figure 10, the change of resistance is shown as a function of the velocity ratio ${\hat{u}}_{ac} / U_{s}$ for several acoustic excitations amplitudes. The change in resistance increases with increasing velocity ratios. In the first region, ${\hat{u}}_{ac} / U_{s} < 4$ , the dependence on the velocity ratio is quadratic, transitioning to a linear dependence and approaching the quasi-steady behavior for higher velocity ratios.

Figure 10.

Change of the real part of the acoustic transfer impedance of the plate normalized with the linear case as a function of the velocity ratio ${\hat{u}}_{ac} / U_{s}$ .

The change in resistance is seen to be proportional to the ratio ${\hat{u}}_{ac} / U_{s}$ for high-acoustic amplitudes and to the ratio, $({\hat{u}}_{ac} / U_{s})^{2}$ for low and moderate amplitudes.

Considering the parameter $\bar{P} = p_{flow} / | {\hat{p}}_{ac} | = ρ U_{s}^{2} / (2 | {\hat{p}}_{ac} |)$ , in the quasi-steady limit, when $\bar{P} << 1$ , the acoustic velocity is

{\hat{u}}_{ac} ≃ \sqrt{\frac{2 {\hat{p}}_{ac}}{ρ}}

(17)

One obtains

\frac{{\hat{u}}_{ac}}{U_{s}} = \sqrt{\frac{1}{\bar{P}}}

(18)

Therefore, the change in resistance is expected to be proportional to

\sqrt{\bar{P}}

for high-acoustic amplitudes and to

\bar{P}

for low and moderate amplitudes.

In Figure 11, the change of resistance due to non-linear effects is shown for similar $\bar{P}$ as a function of the inverse of the acoustic Strouhal number.

Figure 11.

Change of the real part of the acoustic transfer impedance of the plate normalized with the linear case as a function of the inverse of the acoustic Strouhal number ( $S t_{ac} = ω b / u_{ac}$ ) for similar pressure ratios $\bar{P}$ : (a) scaling with $\sqrt{\bar{P}}$ and (b) scaling with $\bar{P}$ .

For each Reynolds number considered in Figure 9, the lines corresponding to $\bar{P} < 10$ are considered. This representation offers a better collapse of the results than a representation with the flow Strouhal number. The change of resistance is scaled (multiplied) with the parameter $\sqrt{\bar{P}}$ (Figure 11(a)) and with the parameter $\bar{P}$ (Figure 11(b)).

One sees a good collapse of the results for both representations with the dimensionless change of resistance approaching order unity for increasing acoustic Strouhal numbers. Although, a better global collapse is observed for a change in resistance scaled with $\bar{P}$ (Figure 11(b)), some discrepancies are found with the highest $\bar{P}$ , corresponding to the flow-dominated results.

Change of inertance due to non-linearities

In this section, the behavior of the imaginary part of the acoustic transfer impedance (inertance) is discussed. The inertance is responsible for a shift in the frequency of the peak of absorption. In Figure 12, the change of inertance due to the presence of a bias flow is shown as a function of the velocity ratio. One sees that, for low and moderate acoustic amplitudes, the inertance is lower in the presence of flow, whereas for high amplitudes, it is slightly higher. A deep in the inertance can be seen around the same ${\hat{u}}_{ac} / U_{s}$ as the peak of resistance. Overall, the change in inertance is not as high as the change of resistance observed in Figure 7. There appears to be a dependence on the flow Strouhal number, with the mean flow dominating the inertance.

Figure 12.

Imaginary part of the acoustic transfer impedance of the plate normalized with the no-flow case as a function of ${\hat{u}}_{ac} / U_{s}$ .

In Figure 13, the change of inertance with respect to the linear case is shown as a function of the inverse of the flow Strouhal number. One observes a fair collapse of the results for $1 / S t_{flow} < 0.4$ with a fairly constant correction. Above $1 / S t_{flow} \approx 0.6$ , the change of inertance increases linearly with increasing $1 / S t_{flow}$ .

Figure 13.

Change of the imaginary part of the acoustic transfer impedance of the plate normalized with the linear case as a function of the inverse of the flow Strouhal number ( $S t_{flow} = ω w_{c, e} / U_{s}$ ) for similar pressure ratio $\bar{P}$ .

Conclusions

Experimental results for the non-linear acoustic transfer impedance in the presence of a bias flow are discussed. Maa³⁶ suggested that optimal absorption can be obtained with a micro-slit plate backed by a cavity when the resistance of the plate matches the characteristic impedance of air. Introducing a bias flow produces a significant change in the resistance of the plate, which can potentially enhance the absorption properties of the plate. The presence of the flow shows a double effect. On the one side, it increases the resistance, increasing the bandwidth of absorption. On the other side, the inertance decreases, shifting the peak of absorption toward higher frequencies. For low acoustic excitations, the behavior is dominated by the flow, and a simple linear quasi-steady model $ρ U_{s}$ captures around $80 %$ of the resistance of the plate. For high-acoustic excitation, the behavior is fully non-linear, with the resistance approaching the resistance in the no-flow case. A transition range is found for a ratio of the amplitude acoustic velocity and the mean flow velocity between 0.3 and 4. Such a transition range is also observed in the literature for circular perforations, where a minimum of the resistance is seen in this transition zone.^13,14 For this particular geometry, a maximum of the resistance is seen, when the mean flow velocity is in the same order as the amplitude of the acoustic velocity. It is found that the resistance depends on the square of the velocity ratio for low and moderate amplitudes, while for high amplitudes, a linear dependency is found. The same behavior is found when considering the change of resistance due to non-linearities, which appear to be a function of the ratio between the dynamic pressure and the acoustic pressure. The inertance shows a more complex behavior with a strong dependence on the flow Strouhal number.

Footnotes

Acknowledgements

The authors wish to thank H. Faghanpourganji for their support with the measurement setup for the discharge coefficient. We acknowledge the precious help of L. Hirschberg during the final stages of our measurements. This work is part of the Marie Skłodowska-Curie Initial Training Network Pollution Know-How and Abatement (POLKA). We gratefully acknowledge the financial support from the European Commission under call H2020-MSCA-ITN-2018 (project number: 813367).

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Alessia Aulitto

Vertika Saxena

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