Design of a single degree of freedom acoustic liner for a fan noise test rig

Abstract

Acoustic liners are an essential part of noise reduction technologies commonly applied in aircraft turbofan engines. Fan noise suppression can be achieved by selecting an appropriate liner design with optimal acoustic impedance at the blade passing frequency. Great efforts have been made not only to improve experimental characterization and numerical methods for acoustic liners, but also to understand noise generation mechanisms, which ultimately impacts on the liner design itself. To gain confidence in the liner design process, a liner barrel was developed and fabricated for the Fan Noise Test Rig located at the University of São Paulo. To this end, analytical methods were used to determine the optimal acoustic impedance for the Fan Noise Test Rig, and a flat test sample was fabricated for experimental characterization with flow using both in-situ and impedance eduction techniques at the Federal University of Santa Catarina. A liner barrel of same nominal geometry was fabricated and placed at the Fan Noise Test Rig, and a modal decomposition indicated that the Tyler-Sofrin mode has been successfully suppressed at the first blade passing frequency. Numerical predictions of liner transmission loss considering the flat sample impedance showed good agreement with experimental results.

Keywords

Acoustic liners experimental techniques modal decomposition optimal impedance predictive models

Introduction

Aircraft noise reduction is a topic of continuous research which led to the development of several noise mitigation technologies for turbofan engines, such as chevrons for jet noise and acoustic liners for fan noise.¹ The latter is generally composed of a honeycomb layer with rigid backing and perforated facesheet, which provides acoustic dissipation due to physical damping at the facesheet holes and reactive cancellation due to reflections at the backing structure.² This configuration leads to a significant noise reduction over a narrow frequency band which comprises the fundamental blade passing frequency (BPF) at take-off or approach conditions.³ Alternative configurations are also possible, for example wire-mesh covered or two degree of freedom liners, to enhance absorption of broadband noise and BPF harmonics.² The acoustic liners are commonly employed at the nacelle internal walls, more specifically the inlet and bypass ducts, whereas over-the-rotor and external liners are currently under investigation.^4–6 The acoustic effectiveness of liners depends not only on its geometry, but also on the environmental conditions found in turbofan engines, namely high grazing flow velocity and high sound pressure levels. These two combined effects represent a striking challenge for liner modelling, predictive tools and experimental techniques.

Given the typical liner dimensions, these are usually represented by a locally-reacting wall impedance, which is convenient for analytical and numerical duct propagation models, and avoids the explicit representation of each honeycomb cell and perforate facesheet hole. The conversion between liner geometry and acoustic impedance is achieved by semi-empirical models with non-linear effects included.^7,8 Therefore, maximization of noise attenuation consists of finding the optimal impedance for a given duct geometry, noise source and environmental conditions. Details of fan noise generation are not always available, but duct radius, fan rotation speed, and number of fan blades and outlet guide vanes (OGV) are sufficient to anticipate major features of the generated sound field. These include the dominant excited acoustic modes, as determined by the Tyler-Sofrin rule,⁹ which is valid for subsonic blade tip speeds. Although high-fidelity numerical simulations can be employed to minimize the far-field radiated noise,¹⁰ an elegant and quick estimate of the optimal impedance is given by Cremer,¹¹ which exploits mathematical properties of the in-duct sound field. From that, the corresponding liner geometry can be determined, and the impedance of small test samples can be experimentally assessed in dedicated test rigs with grazing flow to ensure the liner behaves as expected.

In order to gain confidence in the liner design methodology and verify many of the technologies involved, the Fan Noise Test Rig located at São Carlos School of Engineering, University of São Paulo (EESC-USP) has been chosen as a case study for specification and fabrication of a liner barrel. It contains a scaled-down replica of a representative fan-OGV section to investigate fan noise generation in parametric studies.¹² The liner is designed to target the first BPF and upstream acoustic propagation, which corresponds to the inlet section of a turbofan engine. To achieve this goal, the dominant modes are first determined by the Tyler-Sofrin rule, and the corresponding optimal impedance is obtained by Cremer’s method. Liner geometry is then selected based on a semi-empirical model, and a small flat test sample is fabricated. Effective parameters such as percentage of open area and cavity height are obtained with a portable impedance meter, and impedance measurements with grazing flow are carried out at Federal University of Santa Catarina (UFSC) using both in-situ and impedance eduction techniques in the Liner Test Rig. A liner barrel of same facesheet geometry and cavity height is fabricated and placed at the EESC-USP Fan Noise Test Rig. Liner performance is evaluated in terms of transmission loss, and compared to numerical predictions considering the experimentally obtained impedances.

Experimental setup

EESC-USP fan noise test rig

The EESC-USP Fan Noise Test Rig is located at the Department of Aeronautical Engineering of the São Carlos School of Engineering, University of São Paulo. This test facility, as shown in Figure 1, has been developed to investigate fan noise generation and noise source location by means of beamforming techniques.¹³ It is composed of modular sections which allow parametric studies in order to better investigate isolated fan noise sources.¹² The fan stage is a scaled-down replica of the Advanced Noise Control Fan (ANCF) located at the University of Notre Dame, Turbomachinery Laboratory. (The ANCF was originally developed by NASA Glenn Research Center and it was located at the Aeroacoustic Propulsion Laboratory until 2016.) It consists of 16 fan blades and 14 stator vanes in a duct of diameter 0.5 m. The remaining sections have a diameter of 0.6 m, and a smooth contraction is responsible for reducing the boundary layer thickness at the fan stage section. An anechoic termination is located downstream to the fan stage following design recommendations of the ISO 5136 standard. A schematic representation of the test rig can be seen in Figure 2.

Figure 1.

Photograph of EESC-USP Fan Noise Test Rig setup showing (1) inlet, (2) microphone array, (3) fan-stator section, and (4) anechoic termination.

Figure 2.

Schematic representation of the EESC-USP Fan Noise Test Rig with a liner barrel. Dimensions in meters.

The fan shaft rotation speed is controlled by an in-house software, which is limited to 4500 rpm. At this condition, the mean axial flow Mach number at the fan stage section is approximately $M = 0.14$ , and fan blade tip Mach number is 0.37. Consequently, noise generation is dominated by rotor-stator interaction tones which, for the current test rig configuration, corresponds to a dominant mode of order (2, 1), as predicted by the Tyler-Sofrin rule.⁹

A total of 77 flush-mounted 1/4” GRAS 40PH and Brüel & Kjær Type 4958 microphones are disposed in three circumferential rings in order to capture sound pressure at the duct walls. Although this arrangement has been conceived for beamforming measurements, it is also possible to perform a modal decomposition in order to evaluate the in-duct modal amplitudes with and without a liner barrel. Each ring is composed of 33, 23 and 21 equally spaced microphones, and they are located 1.43 m, 1.53 m and 1.70 m upstream to the fan plane, respectively. The microphones are connected to a National Instruments PXI-1042Q chassis with NI-4496 and NI-4498 signal acquisition modules at a sampling rate of 51.2 kHz. The signals are acquired for 30 s and post-processed using Welch’s method with 2¹⁵ samples per average, Hanning window and 50% overlap. A Pitot-static tube Dwyer Series 160E was used for axial flow velocity measurements upstream to the acoustic liner.

UFSC liner test rig

The UFSC Liner Test Rig is located at the Laboratory of Vibration and Acoustics, Federal University of Santa Catarina. A schematic of the test rig is shown in Figure 3. It is composed of rectangular ducts with cross section of height 100 mm and width 40 mm. The sample holder section can accommodate liner samples up to 210 mm covering the entire duct height. Quasi-anechoic terminations are located at the test section entrance and exit to minimize reflections. Eight Beyma CP-855Nd compression drivers are placed upstream and downstream of the sample holder section in order to generate sound fields up to 150 dB at the test sample using pure tones. An external flow supply system is able to provide a cross-section averaged flow up to Mach 0.7, as assessed by a 2 mm Pitot tube located at the test rig inlet connected to a KIMO CP-115 differential pressure transmitter. Temperature is measured with a KIMO TM-110 temperature transmitter also located at the test rig inlet.

Figure 3.

Schematic of the UFSC Liner Test Rig.

Two arrays of four flush mounted B&K 4944-A 1/4” microphones are located upstream and downstream of the sample holder section for impedance eduction. For the in-situ measurement, a pair of Kulite MIC-062 microphones is inserted into the test sample. A National Instruments PXIe-4499 module is used for signal acquisition with a sampling rate of 25.6 kHz. For pure tone measurements, the excitation signal is used as a noiseless reference for cross-spectrum estimation using Welch’s method with 30 averages of 25,600 samples and 75% overlap.

Predictive models

In order to design a liner barrel for the EESC-USP Fan Noise Test Rig, predictive models are necessary to determine i) which wall impedance produces a high attenuation at the fan rig given the typical operating conditions, and ii) the corresponding liner geometry.

Optimal impedance

Governing equations

Consider a cylindrical duct of radius R and polar coordinates $(r, θ)$ with uniform flow U in the positive z direction as an approximation of the EESC-USP Fan Noise Test Rig. Speed of sound c₀ and density ρ₀ are assumed constant along the duct. The duct wall can be either rigid or lined with a normalized impedance Z. We seek acoustic modes of the form $\hat{p} (r) \exp (i ω t - i k_{z} z - i m θ)$ , where $\hat{p}$ is the mode shape, $i \equiv \sqrt{- 1}$ is the imaginary unit, $ω = 2 π f$ is the wave frequency in rad s⁻¹, t is time, k_z is the axial (complex) wavenumber and $m = 0, \pm 1, \pm 2, \dots$ is the azimuthal wavenumber. Positive and negative values of m indicate clockwise and counterclockwise propagating modes, respectively. Under these assumptions, acoustic propagation is governed by the Convected Helmholtz Equation,¹⁴

\frac{d^{2} \hat{p}}{d r^{2}} + \frac{1}{r} \frac{d \hat{p}}{d r} + ({(k - M k_{z})}^{2} - k_{z}^{2} - \frac{m^{2}}{r^{2}}) \hat{p} = 0

(1)

where $k = ω / c_{0}$ is the free-field wavenumber and $M \equiv U / c_{0}$ is the mean flow Mach number. Equation (1) is the Bessel differential equation,¹⁵ and its solution for hollow ducts is given by $\hat{p} (r) = A J_{m} (k_{r} r)$ , where A is an arbitrary amplitude, J_m is the Bessel function of the first kind, and k_r is the radial wavenumber, related to k_z for each radial index $n = 1, 2, \dots$ by

k_{r, m n}^{2} = {(k - M k_{z, m n})}^{2} - k_{z, m n}^{2}

(2)

It is possible to rearrange it as

k_{z, m n}^{\pm} = \frac{- k M \pm \sqrt{k^{2} - (1 - M^{2}) k_{r, m n}^{2}}}{1 - M^{2}}

(3)

where + and – denote downstream and upstream propagating modes, respectively.

The wall impedance $Z = \hat{p} / (\hat{v} \cdot n)$ is defined without flow, therefore it must be corrected to account for the slipping velocity. The simplest model is the Ingard-Myers boundary condition,^16,17 which assumes continuity of normal acoustic displacement across an infinitely small boundary layer, such that

\frac{d \hat{p}}{d r} = \frac{{(k - M k_{z})}^{2}}{i k Z} \hat{p} at r = R

(4)

Substituting with the solution to equation (1) gives the eigenvalue equation

i k Z = {(k - M k_{z})}^{2} \frac{J_{m} (k_{r} R)}{k_{r} J_{m}' (k_{r} R)}

(5)

This equation is particularly important because it relates axial wavenumbers k_z to the liner impedance Z. As a result, one may ask which impedance maximizes the attenuation of a particular mode.

Cremer’s impedance

A quick estimate of the optimal impedance for a given set of parameters $(k, R, M, m)$ can be found by means of Cremer’s impedance.¹¹ Mathematically, it corresponds to the impedance at which the two least-attenuated modes coalesce¹⁸ i.e. have the same wavenumbers and mode shapes. In this case, the imaginary part of the least-attenuated mode is maximized. It should be noted that Cremer’s impedance does not take into account modal amplitudes or scattering at impedance transitions. However, under suitable conditions, Cremer’s impedance can lead to a reasonable approximation of the optimal impedance when compared to results obtained in more detailed numerical models of the turbofan engine duct using optimization routines.¹⁹

In order to find Cremer’s impedance, the derivative of equation (5) with respect to k_r must vanish,²⁰ so

\frac{\partial}{\partial k_{r}} ({(k - M k_{z})}^{2} \frac{J_{m} (k_{r} R)}{k_{r} J_{m}' (k_{r} R)}) = 0

(6)

which leads to¹⁹

\frac{2 M J_{m} (k_{r} R)}{(1 - M^{2}) k_{z} + M k} + \frac{(k - M k_{z}) R}{k_{r} J_{m}' (k_{r} R)} [J_{m}' {(k_{r} R)}^{2} + (1 - \frac{m^{2}}{{(k_{r} R)}^{2}}) J_{m} {(k_{r} R)}^{2}] = 0

(7)

Equation (7) contains multiple roots and can be solved for k_r (or k_z). In general, the solution with lowest $| Im (k_{z}^{\pm}) |$ leads to the optimal radial wavenumber $k_{r, opt}$ or, equivalently, the optimal axial wavenumber $k_{z, opt}$ . Substituting it in equation (5) leads to Cremer’s impedance

Z_{opt} (k, R, M, m) = \frac{{(k - M k_{z, opt})}^{2}}{i k} \frac{J_{m} (k_{r, opt} R)}{k_{r, opt} J_{m}' (k_{r, opt} R)}

(8)

Impedance models

Once the optimal impedance has been found, it is necessary to determine the respective liner geometry by means of semi-empirical predictive models. In the case of single-degree-of-freedom acoustic liners, predictive models take into account the facesheet thickness τ, hole diameter d, percentage of open area σ and cavity height L, as well as environmental conditions such as skin friction velocity $U_{*}$ or mean flow Mach number M, in order to calculate the liner impedance $Z (ω) = R (ω) + i X (ω)$ . In general, viscous and radiation effects at the holes are computed analytically, whereas non-linear effects are best-fitted to experimental data.

Several models have been proposed over the past years, with special attention to non-linear regimes^21,22 i.e. grazing flow and high sound pressure level. An important difference between the models is the measurement procedure to obtain experimental data, which can be in-situ^23,24 or impedance eduction^25,26 techniques. In this work, we consider two commonly employed models and highlight their differences.

Guess model

One of the earliest models for impedance of SDOF liners has been proposed by Guess.⁸ It assumes asymptotic limits for the viscous and radiation effects which are usually valid for the frequencies of interest. The model is given as follows,

R_{Guess} = \frac{\sqrt{8 ν ω}}{σ c_{0}} (1 + \frac{τ}{d}) + \frac{{(k d)}^{2}}{8 σ} + \frac{(1 - σ^{2})}{σ} (M_{0} + 0.1 M)

(9a)

X_{Guess} = \frac{\sqrt{8 ν ω}}{σ c_{0}} (1 + \frac{τ}{d}) + \frac{ω (τ + τ_{0})}{σ c_{0}} - \cot (\frac{ω L}{c_{0}})

(9b)

where

M_{0} \equiv | v_{0} | / c_{0}

is the root mean square acoustic particle velocity at the hole in terms of Mach number and ν is the kinematic viscosity.

The first term in equations (9a) and (9b) is related to viscous and mass effects and assumes $| K d / 2 | > 10$ , with $K \equiv \sqrt{- i ω / ν}$ the Stokes wavenumber, which is generally satisfied in this work. Also, viscous effects near the hole entrance are considered as a correction length through $(1 + τ / d)$ . The second term in equation (9a) is related to radiation effects and assumes $k d < 0.5$ , which is well satisfied in this work. The second term in equation (9b) is a combination of viscous and radiation effects with the same asymptotic limits provided before. Additionally, it includes a hole end correction τ₀ that takes into account interaction between adjacent holes and non-linear effects due to high sound pressure level,

τ_{0} = 0.85 d (1 - 0.7 \sqrt{σ}) (\frac{1 + 5 \times 10^{3} M_{0}^{2}}{1 + 1 \times 10^{4} M_{0}^{2}})

(10)

The last term in equation (9a) is related to high sound pressure levels and grazing flow on the perforate facesheet, respectively. The constant related to grazing flow has been modified in comparison to the original work of Guess to better fit experimental results at the UFSC Liner Test Rig. Since v₀ depends on the liner impedance, computation of equations (9a) and (9b) is an iterative process because

| v_{0} | = \frac{| p_{0} |}{ρ_{0} c_{0} σ | Z |}

(11)

must be satisfied, with

p_{0} \equiv p_{ref} 10^{SPL / 20}

the root mean square acoustic pressure and

p_{ref} = 20

µPa. Finally, the last term in equation (9b) is related to the backing reactance.

Kooi-Sarin model

It has been early recognized that the boundary layer profile could have a significant impact on the acoustic impedance of orifices.²⁷ In this way, Kooi and Sarin²³ measured the liner impedance using the in-situ technique²⁸ up to velocities of $M = 0.45$ and sound pressure levels of 140 dB with a varying boundary layer thickness. In fact, they observed a strong correlation between impedance and the Strouhal number based on the skin friction velocity $U^{*}$ . Consequently, the Kooi-Sarin (KS) model gives

R_{KS} = \frac{\sqrt{8 ν ω}}{σ c_{0}} (\frac{τ}{d}) + \frac{(5 - τ / d)}{4 σ c_{0}} (9.9 U_{*} - 3.2 f d)

(12a)

X_{KS} = \frac{\sqrt{8 ν ω}}{σ c_{0}} (1 + \frac{τ}{d}) + \frac{ω (τ + τ_{KS})}{σ c_{0}} - \cot (\frac{ω L}{c_{0}})

(12b)

where

τ_{KS}

is a modified end correction length,

τ_{KS} = 0.85 d (1 - 0.7 \sqrt{σ}) [0.92 - 0.75 \frac{U_{*}}{f τ} + 0.11 {(\frac{U_{*}}{f τ})}^{2}]

(13)

In the presence of flow, liner resistance is dominated by the second term on the right hand side of equation (12a), and it decreases with frequency. Furthermore, the model assumes that non-linear effects due to high sound pressure levels are negligible compared to grazing flow effects, which is valid for the EESC-USP Fan Noise Test Rig. The skin friction velocity is estimated from the law of the wall,²⁹

\frac{U (y)}{U_{*}} = 2.5 \ln (\frac{y U_{*}}{ν}) + 5.1.

(14)

Since the flow profiles at EESC-USP Fan Noise Test Rig and UFSC Liner Test Rig are different, liner impedance should also be affected, even if the cross section averaged mean flows are the same.

Optimal liner

To illustrate the liner design process, we target the first BPF at rotational speed of 3800 rpm, which gives $f = 1013$ Hz. At this condition, the mean axial flow velocity is $M = 0.114$ . According to Cremer’s impedance, for Tyler-Sofrin mode (2, 1) and $R = 0.25$ m, the optimal normalized impedance is $Z_{opt} = 0.8226 - 0.4351 i$ . For the liner geometry, the facesheet geometry is fixed to $d = 1.0$ mm and $τ = 1.0$ mm, which leaves the percentage of open area and cavity height as free parameters to match the optimal impedance. Using Guess model with 120 dB SPL leads to $σ = 2.78 %$ and $L = 30.3 mm$ , whereas KS model gives $σ = 4.84 %$ and $L = 44.5$ mm. In fact, these models have very distinct predictions of liner resistance, as it will be shown in the comparison with experimental results. As a consequence, the percentage of open areas are also very different in order to achieve the same optimal resistance. In this case, radiation effects on liner reactance can be more or less pronounced, which impacts on the necessary cavity height to match the optimal reactance.

This procedure is represented graphically in Figure 4. The optimal resistance increases with frequency, whereas the optimal reactance decreases with frequency. Once a target frequency is selected, the liner design geometry can be determined to match the optimal impedance at this condition. Axial flow velocity increases with fan rotation speed, and therefore it is taken into account in both optimal impedance and model prediction. Furthermore, boundary layer properties also vary with flow velocity, and have been curve-fitted to experimental data at different rotation speeds.

Figure 4.

Optimal impedance variation with BPF. Liner design is selected such that the corresponding impedance curve matches the target impedance.

Although Cremer’s impedance leads to a prediction at low frequencies in Figure 4, it should be noted that mode (2, 1) becomes cut-on only around 650 Hz. As previously stated, equation (7) contains multiple roots, and care must be taken to ensure that the obtained optimal impedance is meaningful, especially in the low frequency range.³⁰

Flat liner test samples

Before fabrication of a liner barrel, flat liner test samples were considered not only to check the fabrication procedure, but also to assess its impedance experimentally and compare it with predictive models. The previously presented optimal liner geometry is taken as a reference, with minor modifications. First, the percentage of open area of acoustic liners is generally above 5%. In this way, a perforation pattern has been selected to ensure a nominal $σ = 5.67$ %. Since the holes are partially blocked due to the honeycomb walls, the effective percentage of open area should be slightly lower. Secondly, a nominal cavity height of 40 mm has been selected, which should be also slightly reduced due to adhesive layers at the backplate. Facesheet thickness and hole diameter are both kept with 1 mm.

Fabrication and assembly

In order to simplify the liner barrel fabrication process, a squared honeycomb has been considered, which is also applied in the flat liner test sample. The honeycomb walls are 1 mm thick and each cell is 9.20 mm wide, as shown in Figure 5(a). Although the cell dimensions remain similar to traditional honeycombs, the thicker walls should considerably reduce the effective percentage of open area, and consequently the effective liner geometry should be closer to the optimal one. Care has been taken to avoid additional blockage of facesheet holes due to the presence of adhesive layers. A 3 mm thick backplate was bonded to the squared honeycomb to avoid leakage between cells and ensure a fully reflective condition. The final assembly of a flat liner test sample can be seen in Figure 5(b).

Figure 5.

Photographs of the flat liner test sample. (a) Details of the squared honeycomb assembly. (b) Final assembly including the perforate facesheet.

Normal incidence impedance measurements

Preliminary tests on the flat liner test sample have been performed using a Brüel & Kjær Type 9737 portable impedance meter system, as shown in Figure 6. It is composed of a 29 mm diameter impedance tube with two microphones separated by 20 mm. Normal incidence impedance measurements were performed directly on the assembled liner sample using the flanged termination. Although very convenient from the practical point of view, this termination introduces uncertainties in the low frequency range due to the apparent mismatch between impedance tube area and exposed honeycomb area.³¹ A broadband source has been used to measure the liner impedance between 500 Hz and 6400 Hz at overall sound pressure levels (OASPLs) up to 150 dB. The non-linearity factor (NLF) and effective percentage of open area were obtained with pure tones at increasing levels close to the liner resonance frequency.

Figure 6.

Normal incidence impedance measurement of a commercial liner test sample using the Brüel & Kjær Type 9737 portable impedance meter system. It is composed of (1) a personal computer, (2) LAN-XI Front End and Power Amplifier, (3) impedance tube and (4) test sample.

Effective percentage of open area and non-linearity factor

It is known that facesheet percentage of open area is inversely proportional to liner resistance, as indicated by the impedance model.²³ Consequently, it is essential to correctly estimate the liner effective percentage of open area, especially at such low values, in order to better predict its non-linear behavior i.e. when exposed to grazing flow or high sound pressure levels. To this end, the Brüel & Kjær portable impedance meter is used with increasing levels of pure tones at a suitable frequency e.g. the liner resonance frequency,²⁴ such that the resistance curve is obtained as a function of the acoustic particle velocity. From that, the curve slope a can be related to the effective percentage of open area $σ_{eff}$ by²⁴

a = \frac{ρ_{0}}{2 {(C_{D} σ_{eff})}^{2}}

(15)

where

C_{D} \approx 0.76

is the discharge coefficient. Since the liner resonance frequency is close to the frequencies of high uncertainty in the impedance tube, a slightly higher frequency has been chosen for the pure tone measurements with increasing OASPL. From the curve slope, it is also possible to estimate the liner non-linearity factor

NLR = R_{200} / R_{20}

, which is defined as the ratio between the resistance at acoustic velocities of 200 cm s⁻¹ and 20 cm s⁻¹. This parameter is a measure of the resistance sensitivity to non-linear effects, and should be useful to anticipate the effect of grazing flow on the liner impedance. In practice, measurements are performed at predefined sound pressure levels, such that R₂₀₀ and R₂₀ are usually interpolated and/or extrapolated from the resistance curve.

Results

When measuring liners with low percentage of open area, impedance results are very sensitive to the flanged termination position due not only to the area mismatch between exposed honeycomb cells and impedance tube, but also to small changes in the average number of facesheet holes per honeycomb cell. Additionally, imperfections in the fabrication process could affect the liner homogeneity, and therefore three different locations were selected for the normal incidence impedance measurements. Results compared favorably, and for brevity we show only the results obtained at a single location.

The flat liner impedance curve is shown in Figure 7 at three different OASPL. Two important features can be observed: the increase in resistance with OASPL, as expected for non-linear liners, and the anti-resonance near 4500 Hz, which corresponds to a cavity height of approximately 38.5 mm. At high OASPL, liner acoustic dissipation is dominated by vortex shedding at the facesheet holes,³² and for that reason the effect of sound scattering to neighbor honeycomb cells at lower frequencies is less pronounced. Overall, these results resemble a typical acoustic liner impedance curve, which gives confidence to the fabrication of a liner barrel for the EESC-USP Fan Noise Test Rig.

Figure 7.

Acoustic impedance of the flat liner test sample using a broadband source and increasing OASPL.

In order to obtain the liner effective percentage of open area and non-linearity factor, measurements were performed using pure tones at 2000 Hz from 130 dB to 150 dB in steps of 5 dB. Results are shown in Figure 8, together with interpolated and extrapolated values of resistance. Based on the curve slope a and equation (15), the effective percentage of open area of the flat liner test sample is $σ_{eff} = 4.6$ %. Compared to the nominal percentage of open area, it represents a cell blockage of approximately 20% due to the honeycomb walls and adhesive layers at the facesheet. Finally, the adjusted values of R₂₀₀ and R₂₀ allow to compute the non-linearity factor, which gives a value of 4.8, as also typically found in low porosity SDOF liners.

Figure 8.

Slope of the flat liner test sample resistance with increasing acoustic particle velocity using pure tones.

Grazing incidence impedance measurements

Results obtained with a normal incidence impedance tube give a glimpse of the liner non-linear behavior, but results from dedicated test rigs with flow are essential to predict liner performance in its final application. To this end, the flat liner test sample was tested at the UFSC Liner Test Rig using both in-situ and impedance eduction methods.

In-situ technique

The in-situ technique, as proposed by Dean,²⁸ considers two microphones located at the facesheet and backplate of a single honeycomb cell. Assuming a fully reflective backplate and plane wave propagation inside the honeycomb cavity, it is possible to show that the in-situ impedance is given by

Z_{meas} = \frac{- i H_{fb}}{\sin (k L)}

(16)

where

H_{fb}

is the complex ratio between facesheet and backplate acoustic pressure. The in-situ impedance is local in the sense that each cavity has its own effective percentage of open area, and consequently its own impedance. Moreover, the in-situ impedance does not consider the honeycomb wall thickness which also modifies the average effective percentage of open area, as seen for example by the normal incidence impedance tube or impedance eduction techniques. From the predictive impedance models, the facesheet impedance

Z_{f}

can be calculated by removing the backing reactance from the measured impedance

Z_{meas}

, such that

Z_{f} = Z_{meas} + i \cot (k L)

(17)

which can be corrected by

ϵ = σ / σ_{eff}

, defined as the ratio between local and effective percentage of open area. Accordingly, the corrected liner impedance is given by

Z = ϵ Z_{f} - i \cot (k L)

(18)

In order to achieve a constant SPL e.g. 130 dB at the facesheet Kulite microphone, the instrumented cell is selected close to the liner leading edge. Otherwise, the high attenuation at the liner resonance frequency would be extremely demanding for the acoustic drivers. When using a downstream acoustic source, the liner test sample is rotated, such that the instrumented cell is closer to the acoustic source. Although this procedure may lead to small differences in the in-situ impedance due to the boundary layer development along the sample holder section, instrumentation of another cell would also introduce new uncertainties because of the local effective geometry.

Impedance eduction

The key idea of impedance eduction methods is to compare the measured acoustic field to the numerical one with a given initial impedance guess, for instance an impedance prediction given the liner geometry, and from that iterate the liner impedance until both fields converge. The mode-matching method has been selected over the Finite Element Method³³ and other similar numerical techniques due to its reduced computational cost and high accuracy,³⁴ which are essential characteristics for inverse methods.

The mode-matching method has been used for impedance eduction by several authors.^35–38 Although similar in nature, each of these works adopt different assumptions and solution strategies. The procedure here described follows the same approach from Spillere et al.,³⁴ and for that reason details are omitted for brevity.

The governing equations are similar to equation (1) – cylindrical coordinates are replaced by cartesian coordinates. Accordingly, the Convected Helmholtz Equation is given by

\frac{d^{2} \hat{p}}{d y^{2}} + ({(k - M k_{z})}^{2} - k_{z}^{2}) \hat{p} = 0

(19)

where modes of the form

\hat{p} (y) \exp (i ω t - i k_{z} z)

were assumed. Boundary conditions are given as

\frac{d \hat{p}}{d y} = {\begin{array}{l} 0 at rigid walls, \\ \frac{i}{k Z} {(k - M k_{z})}^{2} \hat{p}, at lined walls, \end{array}

(20)

where the Ingard-Myers boundary condition^16,17 was assumed for lined walls.

Pressure and axial velocity fields can be approximated by the sum of $N$ upstream and downstream propagating modes at each duct section,

p (y, z) = \sum_{n = 1}^{N} (A_{n}^{+} Ψ_{n}^{+} (y) \exp (- i k_{z, n}^{+} z) + A_{n}^{-} Ψ_{n}^{-} (y) \exp (- i k_{z, n}^{-} z))

(21a)

u (y, z) = \frac{1}{ρ_{0} c_{0}} \sum_{n = 1}^{N} (A_{n}^{+} Ξ_{n}^{+} (y) \exp (- i k_{z, n}^{+} z) + A_{n}^{-} Ξ_{n}^{-} (y) \exp (- i k_{z, n}^{-} z))

(21b)

with

Ξ_{n}^{\pm} (y) = \frac{k_{z, n}^{\pm} Ψ_{n}^{\pm} (y)}{k - M k_{z, n}^{\pm}}

(22)

where

A_{n}^{\pm}

are the modal amplitudes and

ψ_{n}^{\pm}

are the mode shapes. The latter is found together with the axial wavenumbers

k_{z, n}

by solving equation (19) as a generalized eigenvalue problem with appropriate boundary conditions. The modal amplitudes are then found by assuming continuity of mass and momentum at each interface,³⁹ which results into

(1 - M^{2}) \int_{S} \bar{f} (p^{(q + 1)} - p^{(q)}) d S - \frac{M^{2}}{i k} \oint_{Γ} \bar{f} (\frac{p^{(q + 1)}}{Z^{(q + 1)}} - \frac{p^{(q)}}{Z^{(q)}}) d Γ = 0

(23a)

(1 - M^{2}) \int_{S} \bar{f} (u^{(q + 1)} - u^{(q)}) d S + \frac{M}{i ρ_{0} c_{0} k} \oint_{Γ} \bar{f} (\frac{p^{(q + 1)}}{Z^{(q + 1)}} - \frac{p^{(q)}}{Z^{(q)}}) d Γ = 0

(23b)

where q = 1, 2 is the interface number, S is the cross-section,

\bar{f}

is a suitable complex conjugate test function, for example the mode shapes in a rigid-walled duct, and Γ is the interface contour. The main difference to the traditionally assumed continuity of pressure and axial acoustic velocity is the presence of contour integrals in equation (23), where the Ingard-Myers boundary condition has been assumed. In general, the choice of matching condition is of little importance in the context of impedance eduction, provided some guidelines for the cost function are followed.³⁴

Substitution of equations (21a) and (21b) into equations (23a) and (23b) lead to a system of $2 N$ equations at each interface. Inputs are the plane wave amplitudes $A_{1}^{\pm}$ propagating towards the liner sample, which can be determined from an over-determined plane wave decomposition upstream and downstream to the lined section. Once the remaining modal amplitudes are determined, pressure can be computed at the i-th microphone $p_{i}^{num}$ . These are compared to the experimental pressure $p_{i}^{exp}$ by the cost function

F (Z) = \sum_{i = 1}^{8} | \frac{p_{i}^{exp} - p_{i}^{num} (Z)}{p_{i}^{exp}} |

(24)

The liner impedance Z is updated until the cost function is minimized.

Results

Both in-situ and impedance eduction methods are compared to the normal incidence impedance measurement in the absence of flow. For the in-situ impedance, both effective percentage of open area and height obtained by the Brüel & Kjær impedance meter were considered in the post-processing. Results obtained with upstream and downstream acoustic sources in the test rig are essentially the same in the absence of flow; therefore, only results with downstream source are shown.

As it can be seen in Figure 9, a good agreement is found at frequencies above 1200 Hz. Results obtained using the Brüel & Kjær impedance meter with the flanged termination at lower frequencies do not allow for a straight comparison with other methods because of sound scattering to neighbor honeycomb cells. In this frequency range, small differences are also observed between the in-situ and impedance eduction techniques. A comparison with the Guess model indicates that the in-situ impedance is most likely to be correct, whereas the educed impedance suffers from two combined effects: small lined length compared to the wavelength, and small liner attenuation, which ultimately leads to a high uncertainty. Close to the liner resonance frequency, the Guess model predicts a larger resistance with a shifted peak frequency. Impedance models that take into account the hole discharge coefficient²⁴ could potentially improve the resistance prediction. At higher frequencies, e.g. above 1500 Hz, the in-situ reactance is systematically below the educed reactance, normal incidence measurement and model prediction. This trend could be related to the small distance between the facesheet microphone and adjacent holes, which may reduce the mass reactance end correction at the facesheet holes. Fortunately, this should be minimized in the presence of flow.

Figure 9.

Comparison between liner impedance obtained with BK impedance meter, UFSC Liner Test Rig without flow and prediction from Guess model.

In order to predict the liner behavior in the presence of flow at the EESC-USP Fan Noise Test Rig, measurements were performed with a cross-section averaged velocity of Mach 0.151, using only upstream acoustic sources, and repeating the measurement with downstream acoustic sources. Since the liner is assumed as locally reacting, and consequently independent of the angle of incidence, any difference between in-situ and educed impedances using upstream and downstream acoustic sources could indicate potential flaws in the acoustic theory.

As shown in Figure 10, the group of in-situ and educed impedances exhibit a large difference at low frequencies e.g. below 1500 Hz, similar to the no-flow case. Whereas the in-situ reactance follows the predicted values, the educed reactance is even below the $- i \cot (k L)$ term, which may suggest that the educed impedance is incorrect. At higher frequencies, the differences are significantly reduced i.e. approximately $0.2 ρ_{0} c_{0}$ in resistance, and negligible in reactance. In this case, it is possible that hole blockage due to honeycomb walls, which leads to a higher resistance, has not been correctly accounted for in the in-situ correction.

Figure 10.

Comparison between in-situ and educed impedances with cross-section averaged flow of $M = 0.151$ using upstream (US) and downstream (DS) acoustic sources. Also shown is the impedance prediction given by Guess and KS models.

The difference in impedance when using upstream or downstream acoustic sources with both in-situ and impedance eduction techniques may arise from different reasons. In the case of impedance eduction, the flow is assumed as uniform, which may introduce a significant error especially for upstream acoustic propagation⁴⁰ i.e. when using a downstream acoustic source. As a result, the uniform flow duct propagation model will include any shear flow effect directly into the liner impedance. The use of alternative boundary conditions that take into account boundary layer effects are also not sufficient to remove this discrepancy in two-dimensional models.³⁴ Given the typical duct dimensions of liner test rigs, duct propagation models for impedance eduction should include the three-dimensional flow profile. Regarding the in-situ impedance, it is difficult to determine the local boundary layer characteristics at the liner leading and trailing edges, and whether this change would be sufficient to cause the observed differences in resistance. Another hypothesis is related to boundary layer refraction effects, which point the downstream waves towards the duct walls, and the upstream waves towards the duct center.⁴¹ Such phenomenon could be implicitly captured by the facesheet microphone, leading to an apparent mismatch between impedances as seen from the acoustic field propagating upstream or downstream.

As a final remark, the reasonable agreement between predictive models and experimental results should be analyzed with care. For instance, the KS model appears to follow the educed resistance, although it considers the test rig centreline skin friction velocity. Due to the three dimensional nature of the flow profile, it is difficult to determine the average skin friction velocity to correlate with the educed impedance. Moreover, the resistance prediction given by the Guess model is close to experimental results because the constant multiplying the Mach number has been reduced from 0.3 in the original publication⁸ to 0.1. Since it only takes into account the mean flow Mach number, this constant should be different when the liner is applied to the EESC-USP Fan Noise Test Rig because of the different boundary layer characteristics. Nevertheless, only experimentally obtained impedances will be considered for the fan noise test rig numerical model to determine which impedance better predicts the liner barrel transmission loss.

Liner barrel

The reasonable agreement between predictive models and experimental results provides confidence for the liner barrel fabrication. For that reason, the same liner geometry and fabrication procedure has been applied to the liner barrel. In order to facilitate the squared honeycomb assembly, as shown in Figure 11(a), the liner barrel was divided into three sections (hereinafter labeled as sections 12, 23 and 13) with splices of 12.5 mm. Moreover, the facesheet is first perforated and then bent into its final form, which can alter the hole shape, and consequently the effective percentage of open area and discharge coefficient. A photograph of the final facesheet bonded to the squared honeycomb is shown in Figure 11(b).

Figure 11.

Photographs of a liner barrel section. (a) Squared honeycomb assembly for the liner barrel. (b) Details of the perforated facesheet.

Quality control

In numerical models, the liner is assumed as a locally-reacting homogeneous impedance. As a result, it is important to check liner homogeneity i.e. facesheet properties and cavity height at different locations. This procedure would also highlight any difference between the flat sample and liner barrel impedances. To this end, measurements were made at five positions in each section, as schematically shown in Figure 12, using the Brüel & Kjær portable impedance meter with a broadband noise source for impedance measurement, and pure tones for extraction of non-linear parameters.

Figure 12.

Measurement points using the Brüel & Kjaer portable impedance meter equipped with a curved flange termination.

Impedance, NLF and $σ_{eff}$ can be expressed as a mean value and its expanded uncertainty,

MR = \bar{I} \pm t_{s} s

(25)

where

MR

is the measurement result,

\bar{I}

is the mean indication and s is the standard uncertainty. The t-Student coefficient t_s for 95.45% confidence and fifteen measurement points is 2.1953. The mean impedance and confidence interval of the liner barrel is presented in Figure 13, together with the flat liner test sample impedance. Since both liners are based on the same facesheet geometry and cavity height, results should be as close as possible. In reality, the high uncertainty at lower frequencies e.g. below 1000 Hz makes any comparison unreliable. At higher frequencies, the flat sample resistance is within the confidence interval, whereas the liner barrel reactance is well below the flat sample reactance, which is a hint that the effective parameters are different. The mean values and respective expanded uncertainties of NLF,

σ_{eff}

and L are presented in Table 1. Since the liner barrel cavity height remains very similar to the flat sample height, the low frequency reactance should be close to results obtained at the UFSC Liner Test Rig. On the other hand, the effective percentage of open area appears to be higher, and consequently NLF is lower. In this case, the liner barrel is less sensitive to flow effects, and therefore the resistance should be lower than the flat sample resistance at the same flow velocity.

Figure 13.

Liner barrel mean impedance and 95.45% confidence interval. Also shown is the flat liner test sample impedance.

Table 1.

Mean values and expanded uncertainties for non-linearity factor, effective percentage of open area and cavity height.

	Non-linearity factor	$σ_{eff}$ (%)	L (mm)
Mean value, $\bar{I}$	3.67	6.90	37.59
Standard uncertainty, s	0.69	0.80	0.87
Expanded uncertainty, $t_{s} s$	1.51	1.75	1.91

Results at EESC-USP fan noise test rig

Modal decomposition

In order to evaluate liner performance at the EESC-USP Fan Noise Test Rig, we consider a modal decomposition for the following reasons. First, fan noise can be estimated in terms of modal amplitudes by considering a rigid-walled configuration, which in turn can be used as an input for analytical and numerical models. Secondly, it is possible to compute the attenuation of each individual mode by comparing modal amplitudes in rigid and lined configurations. In this case, it is possible to determine whether the dominant Tyler-Sofrin mode is efficiently suppressed by the liner barrel.

Since the microphone array has been chosen for a beamforming technique,¹³ the microphone positions are not ideal for a modal decomposition. In particular, the small number of axial positions limit the maximum number of upstream and downstream modes for a given m that could be identified by the modal decomposition. Consequently, we assume the upstream section (inlet) is perfectly anechoic, such that up to three radial upstream propagating modes can be found. This condition is sufficient to identify all possible cut-on modes of the first BPF in the whole range of fan rotation speeds.

Acoustic propagation in a circular duct of the form $\hat{p} (r) \exp (i ω t - i k_{z} z - i m θ)$ with uniform flow is governed by the Convected Helmholtz Equation, as given by equation (1). In the case of rigid walls, the normal derivative of acoustic pressure vanishes, so that $J_{m}' (k_{r, m n} R) = 0$ . By using the dispersion relation (2), it is possible to notice that only modes with $k_{r, m n} \leq k^{2} / (1 - M^{2})$ are cut-on, and therefore carry acoustic energy. Consequently, the acoustic field is approximated by the sum of upstream cut-on modes,

p (r, θ, z) = \sum_{m = - \infty}^{\infty} \sum_{n = 1}^{N_{c, m}} A_{m n}^{-} U_{m n} (r) \exp (- i m θ - i k_{z, m n}^{-} z)

(26)

with

N_{c, m}

the number of radial cut-on modes at azimuthal order m, and

U_{m n} (r) \equiv N_{m n} J_{m} (k_{r, m n} r)

the mode shape normalized by

N_{m n} = \frac{\sqrt{2}}{J_{m} (k_{r, m n} R) \sqrt{1 - \frac{m^{2}}{{(k_{r, m n} R)}^{2}}}}

(27)

except

N_{01} = \sqrt{2}

. Accordingly, acoustic pressure at the i-th microphone is calculated by

p_{i} = \sum_{j = 1}^{N_{c}} A_{j}^{-} N_{j} J_{m_{j}} (k_{r, j} R) \exp (- i m_{j} θ_{i} - i k_{z, j}^{-} z_{i})

(28)

where cut-on modes have been grouped into a single index j, and

N_{c}

is the total number of cut-on modes which is a function of frequency. In matrix form, it can be written as

p = Qa

(29)

where p is the measured pressure vector of size

(77 \times 1)

, a is the modal amplitude vector of size

(N_{c} \times 1)

, and Q is the modal matrix of size

(77 \times N_{c})

. The modal matrix is given by

Q_{i, j} = N_{j} J_{m_{j}} (k_{r, j} R) \exp (- i m_{j} θ_{i}) \exp (- i k_{z, j}^{-} z_{i})

(30)

Assuming that downstream propagating modes are negligible, the upstream propagating modal amplitudes can be determined by

a = Q^{†} p

(31)

where

^{†}

denotes the Moore-Penrose pseudo-inverse. The condition number of Q indicates the quality of microphone spacing for the modal decomposition,⁴² which has been verified to be approximately 1 at 1400 rpm and 5 at 4400 rpm.

Results

In order to obtain results in a rigid-wall configuration, the liner barrel was covered with speed tape, as shown in Figure 14(a). In this case, noise levels were similar to the ones obtained in the original test rig setup. In the lined configuration, the liner edges and splices were also covered to avoid any leakage, as shown in Figure 14(b). Measurements were made in shaft rotation speeds from 1400 rpm to 4400 rpm in steps of 200 rpm. An example of microphone pressure spectrum is shown in Figure 15. It is possible to observe a great reduction in SPL at the first BPF in the lined configuration, as expected. In this case, the second BPF becomes the dominant noise source. The liner barrel also contributes to a small reduction in broadband noise.

Figure 14.

EESC-USP Fan Noise Test Rig setup with liner barrel. (a) Rigid wall configuration. (b) Lined wall configuration.

Figure 15.

Example of microphone pressure spectrum in rigid and lined configurations.

Application of modal decomposition on both rigid and lined configurations can be seen in Figure 16 for shaft rotation speed of 3800 rpm (1013 Hz). As expected, the Tyler-Sofrin mode (2, 1) is dominant at this condition, and at least 15 dB higher than any other mode. In the lined configuration, the higher-order modes are well attenuated because of their lower cut-off ratio, which means that wave propagates mostly in the radial direction i.e. towards the liner.⁴³ However, the high attenuation of the Tyler-Sofrin mode indicates that liner impedance should be close to its optimal, and therefore the liner design methodology can be considered successful. Interestingly, mode (0, 2) has not only gained energy, but has also become the dominant mode after the lined section, which is particularly important for the far-field radiated noise, for example. A comparison with mode (0, 1) may suggest that, at this azimuthal mode order, most of the acoustic energy has been scattered from n = 1 to n = 2.

Figure 16.

Modal amplitudes in rigid and lined configurations.

As a final remark, the modal amplitudes have been obtained in the duct section of $R = 0.3$ m i.e. it does not represent the modal amplitudes at the fan section, where $R = 0.25$ m.

Numerical prediction

Since the flat liner impedance has been obtained with flow, it is possible to use it as a boundary condition in numerical models to predict liner attenuation in the final application and, from that, compare to experimental results. Although the liner barrel effective parameters are slightly different from the flat sample, it should have a minor influence on impedance at low Mach numbers. The numerical procedure is based on the work of Acosta et al.⁴⁴ and briefly described as follows. The governing equations are solved using the Finite Element Method (FEM) over a duct geometry corresponding to the EESC-USP Fan Noise Test Rig. A compressible irrotational background mean flow is first computed as an approximation of the experimental mean flow field. Subsequently, a pressure field governed by the Convected Helmholtz Equation is computed over the potential mean flow. At lined walls, the weak form of Ingard-Myers boundary condition is considered.⁴⁵

Finite element model

The commercial code ACTRAN/TM has been chosen to conduct a frequency-domain simulation of the acoustic field at the EESC-USP Fan Noise Test Rig. The three-dimensional model includes the fan plane, lined section with splices, converging section, and anechoic outlet plane, as indicated in Figure 17(a). Probes are located at the same microphone positions, such that the modal decomposition can be applied. The governing equations are solved using linear elements in a free tetrahedral mesh with local refinements at the splices. The mesh contains 18 elements per wavelength at the maximum frequency of interest, and it is shown in Figure 17(b). Boundary conditions are specified in terms of modal basis at the fan plane and non-reflecting boundary at the outlet.

Figure 17.

Finite element model of the EESC-USP Fan Noise Test Rig. (a) Schematic of the simulation domain. (b) Unstructured mesh used for the simulation. Red dots indicate probes located at microphone positions.

As previously mentioned, the experimental modal amplitudes are known at the duct section of $R = 0.3$ m. However, inputs to the model are the modal amplitudes at the fan plane. Additionally, the number of cut-on modes at the microphone and fan sections are different at certain frequencies. Therefore, preliminary simulations have been conducted with rigid walls. In this case, the modal amplitudes have been best-fitted to match the experimental acoustic field at the microphone section in a rigid-wall configuration. Assuming the liner does not modify noise generation at the source, the same modal amplitudes have been used at the fan plane for simulations with a lined section.

Axial acoustic power

In order to compare liner attenuation at different rotation speeds, we consider the axial acoustic power, defined as the integral of axial energy flux in the cross-section,

P = 2 π \int_{0}^{R} 〈 I \cdot e_{x} 〉 r d r

(32)

where

〈 I \cdot e_{x} 〉

is the time-averaged axial intensity,⁴⁶

〈 I \cdot e_{x} 〉 = \frac{1}{2} Re {(ρ_{0} u + \frac{p M}{c_{0}}) \bar{(\frac{p}{ρ_{0}} + u M c_{0})}}

(33)

which leads to

P = \frac{π R^{2} k}{ρ_{0} c_{0}} \sum_{m = - \infty}^{\infty} \sum_{n = 1}^{N_{c, m}} \frac{{| A_{m n^{}}^{-} |}^{2}}{{(k - M k_{m n}^{-})}^{2}} (k_{m n}^{-} (1 - M^{2}) + M k)

(34)

where only upstream cut-on modes are considered. For this reason, the liner transmission loss can be evaluated by

Δ PWL = 10 \log_{10} (\frac{P_{rigid}}{P_{lined}})

(35)

where

P_{rigid}

and

P_{lined}

are the axial acoustic powers in rigid and lined configurations, respectively. It should be noted that this transmission loss is calculated at a single frequency, in this case the first BPF.

Results

The experimental transmission loss is shown in Figure 18. An attenuation peak of approximately 25 dB is observed at 3800 rpm, which corresponds to the frequency of interest when designing the optimal liner for the first BPF. For that reason, the liner barrel performed as expected. It should be noted, however, that computation of OASPL would indicate a lower attenuation since the second BPF becomes the dominant noise source in the lined configuration.

Figure 18.

Experimental liner transmission loss and comparison with numerical predictions using in-situ and educed impedances.

Also shown in Figure 18 are the numerical predictions using experimentally obtained impedances, including both in-situ and eduction techniques with upstream and downstream acoustic sources. The impedances were linearly interpolated from measurement with $M = 0.151$ and without flow at the UFSC Liner Test Rig for each frequency. In general, the simulations predicted an attenuation peak at 3800 rpm within 1.5 dB to the experimental result, except when using the educed impedance with downstream source. In particular, the educed impedances led to a better prediction at lower shaft rotation speeds, whereas none of the impedances were able to reproduce the insertion loss at higher shaft rotation speeds. In view of the numerous assumptions, however, the overall agreement is satisfactory.

The predictions using in-situ impedances obtained with upstream and downstream acoustic sources led to very similar results. On the other hand, the upstream and downstream educed impedances are sufficiently different to affect the transmission loss prediction (almost 5 dB at the attenuation peak). For that reason, the discrepancies between educed impedances using upstream and downstream acoustic sources deserve further investigation.

Conclusions

In this work, a liner design methodology has been applied to the EESC-USP Fan Noise Test Rig, located at the University of São Paulo. Steps included i) computation of optimal impedance for a given operating condition, ii) specification of liner geometry to achieve the optimal impedance, iii) fabrication and measurement of a flat test sample at the UFSC Liner Test Rig in order to obtain liner impedance with flow, iv) fabrication of a liner barrel and measurements at EESC-USP Fan Noise Test Rig to assess liner modal attenuation and axial power transmission loss, and v) comparison with a numerical model of the EESC-USP Fan Noise Test Rig using the experimentally obtained impedances. The main conclusions are summarized as follows.

Cremer’s impedance resulted in a reasonable prediction of the optimal impedance at EESC-USP Fan Noise Test Rig. In fact, liner attenuation at the first BPF was considerably high, such that the second BPF became the dominant noise source. An improved liner design could be achieved by considering not only the second BPF, but also sound scattering at the liner edges and duct transitions. In this case, the numerical model of the test rig could be used, together with a suitable cost function and optimization routine.

The effective parameters and non-linear behaviour of SDOF liners have been successfully obtained using the Brüel & Kjær impedance meter. It has been shown that liner barrel and flat sample had a similar cavity height, but a slightly different effective percentage of open area. On the other hand, normal incidence impedance measurements below 1200 Hz were unreliable due to sound scattering at the edges of the flanged termination, which correspond to the frequency range of interest at the EESC-USP Fan Noise Test Rig. Measurements at the UFSC Liner Test Rig do not suffer from this limitation, although results are only available for the flat test sample. Nevertheless, these can be used in a finite element model to satisfactorily predict the liner transmission loss at the EESC-USP Fan Noise Test Rig.

Measurements without flow at the UFSC Liner Test Rig also showed a systematic difference between in-situ and educed impedances, although the former appears to closely follow the prediction given by a modified Guess model. In the presence of flow, the impedance result depends not only on the measurement technique, but also on the acoustic source location. Predictions of the liner insertion loss at the EESC-USP using the experimentally obtained impedances indicate that the difference regarding the acoustic source location is negligible for in-situ impedances, but significant for educed impedances. Furthermore, the overall agreement between experimental results and numerical predictions is satisfactory, and the peak attenuation was, in most cases, correctly predicted.

Footnotes

Acknowledgements

Authors A. S. and L. B. acknowledge Paul Murray for the instructions on the use of the BK portable impedance meter.

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: The work reported in this article was supported by FINEP (Funding Authority for Studies and Projects), CNPq (National Council for Scientific and Technological Development) and Embraer S.A.

ORCID iDs

André MN Spillere

Lucas A Bonomo

Paulo C Greco Junior

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