Obsidian/Zotero/001_artiklid/bonomoeffectsshearedflowvelocity2025.md

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category: literaturenote citekey: bonomoeffectsshearedflowvelocity2025 title: Effects of the sheared flow velocity profile shape on impedance eduction in a 2D duct authors: "Bonomo, Lucas A; Brambley, Edward J; Cordioli, Julio A" year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.10129v1" zotero_key: 737A6XBH zotero_storage: ENZ23DUC collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Bonomo, Lucas A"

status: converted

Effects of the sheared flow velocity profile shape on impedance eduction in a 2D duct

Lucas Araujo Bonomoa,b, Edward James Brambleyb,c, Julio Apolinário Cordioli^a

^aDepartment of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, 88040-900, SC, Brazil ^bMathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom ^cWMG, University of Warwick, Coventry, CV4 7AL, United Kingdom

Abstract

We assess the impact of the flow profile shape on acoustic propagation in a two-dimensional duct within the typical operating range of impedance eduction facilities. Firstly, a numerical experiment is proposed in which the Pridmore–Brown equation is assumed to represent the true physical behaviour, and is used with both simplified flow profiles commonly used in the literature and a realistic representation of a turbulent boundary layer using a van Driest universal law of the wall model. The data from these numerical experiments are then used with a traditional impedance eduction process, and the resulting variation in obtained impedances are investigated. Secondly, we apply a less-traditional impedance eduction method that incorporates the sheared velocity profile to data obtained from real-world experiments. The results suggest that the Ingard–Myers boundary condition remains a good approximation to a realistic boundary layer profile, such as the universal law of the wall, at least in the two-dimensional case. However, it is also shown that the simplified flow profiles often used in the literature can lead to significant deviations from the results obtained using a realistic velocity distribution.

Keywords: acoustic liners, duct acoustics with flow, impedance eduction

1. Introduction

Acoustic liners are acoustic treatments applied to the walls of aircraft turbofan engine nacelles to mitigate fan noise. The simplest and most typical liner construction consists of a honeycomb structure with a hard backplate and a perforated facesheet [1]. An acoustic liner is typically characterised by its locally-reacting acoustic impedance, Z˜(ω) = θ + iχ, where θ is the resistance and χ is the reactance. This frequency-dependent parameter can be used as a boundary condition in simulations of aircraft engine noise, avoiding the still prohibitive computational cost of explicitly modelling an acoustic liner.

The impedance of an acoustic liner is known to depend on its geometry [2], as well as on operational conditions such as the grazing flow velocity and profile [3], and the incident Sound Pressure Level (SPL) [4]. Therefore, for proper liner characterisation, experiments must replicate the conditions inside a turbofan engine. For the experimental characterisation of acoustic liners with grazing flow, the early in-situ technique (or Dean's method) [5] and impedance eduction methods [6–8] are commonly used. While the in-situ technique provides a local value of the liner impedance, impedance eduction techniques give an averaged impedance as seen by the acoustic field and is experimentally much simpler than the in-situ technique. As a result, impedance eduction has been preferred by the academic community [e.g. 9, 10]. Impedance eduction methods rely on an acoustic propagation model in ducts, inferring the impedance from the model by best fitting the model results to the experimental data. Uniform flow and the Ingard–Myers boundary condition are the most common modelling assumptions. However, recent findings have sparked academic debate regarding the application of such assumptions.

One of the fundamental issues recently observed is that the educed impedance depends on the wave propagation direction relative to the mean flow, which constitutes a violation of the locally reacting hypothesis. This behaviour has been systematically captured by different laboratory facilities using both inverse and direct methods, as summarised by Bodén et al. [11]. Renou and Aurégan [12] were the first to demonstrate such discrepancies and attributed them to a failure of the Ingard–Myers boundary condition. Since then, other studies have suggested that an additional degree of freedom is necessary to fully characterise acoustic liners in the presence of flow, such as a thin boundary layer [13], shear stress at the wall [14, 15], or viscosity [16]. In general, this additional degree of freedom is fitted to experimental data in a way that removes the impedance dependence on the propagation direction. However, the proposed experimental procedures would yield this result for any additional degree of freedom, irrespective of its physical meaning. Spillere et al. [17] subsequently proposed an experimental routine to evaluate the accuracy of these novel boundary conditions. Results obtained suggested that none of the boundary conditions currently available in the literature are capable of correctly predicting the acoustic wavenumber beyond the data they were fitted to. In a different approach, Nark et al. [18] found that by multiplying the mean flow velocity by a fitting coefficient, it is possible to reduce the discrepancy between the upstream and downstream propagation direction educed impedances, but differences remain.

Another line of research involves substituting the traditional uniform flow hypothesis with a shear flow profile. Despite the three-dimensional nature of internal flows, most authors consider a two-dimensional representation of the computational domain, which implies a one-dimensional flow profile [15, 19–21]. Weng et al. [15] investigated the effect of the uniform flow hypothesis by performing impedance eduction with the Linearised Navier– Stokes Equations (LNSE) under sheared mean flow, which therefore also includes the effect of viscosity. Results suggest that including viscous effects and one-dimensional sheared velocity profiles is not sufficient to collapse the upstream and downstream educed curves. More recently, Yang et al. [21] conducted an extensive parametric study on the impact of the uniform flow assumption compared to sheared velocity profiles. Test data were numerically synthesised using the Linearised Euler Equations, with varying boundary layer thicknesses and flow velocities. They found that the influence of sheared flow becomes more significant for large Helmholtz numbers. It was concluded that the uniform mean flow is a valid approximation for small ducts at low frequencies, which is the case for most experimental facilities.

A notable exception to the trend of simplifying the 3D nature of ducts to a 2D problem is the work of Roncen et al. [22]. They proposed a numerical experiment in which the solution of the acoustic field using the uniform flow hypothesis (the Convected Helmholtz Equation, CHE) was compared to the solutions of the Linearised Euler Equations (LEE), which considered both one-dimensional and two-dimensional flow profiles. Results from this numerically synthesised experiment suggest significant differences in the wavenumbers obtained for the eigenvalue problem of each case. Furthermore, it was found that the uniform flow hypothesis, combined with the Ingard–Myers boundary condition, introduces a bias error that follows the same trend observed in the upstream-downstream discrepancy. The authors subsequently performed a numerical impedance eduction, using the wavenumbers computed for the 2D flow profile as input, but applying the 1D flow profile in the impedance calculation routine. Although earlier work by Jing et al. [20] suggests that simplifying the 2D flow profile to a 1D flow profile requires re-scaling the bulk Mach number, Roncen et al. [22] did not correct the flow profile in their numerical eduction routine, arguing that this is common practice. The results of their analysis suggest that the experimentally observed upstream-downstream mismatch may be caused by the bias error introduced when simplifying the 2D flow profile. The authors also examined the impact of using the 2D-LEE solver in impedance eduction. They concluded that at low frequencies, the mismatch is quantitatively similar whether considering the two-dimensional flow profile or a uniform flow. However, at higher frequencies, the 2D-LEE eduction cancels most of the impedance mismatch, akin to observations made by Nark et al. [18], through the adjustment of the Mach number. Another observation from their analysis is that in the case of 2D-LEE eduction, the mismatch trend at lower frequencies is opposite to that observed with the traditional uniform flow eduction approach.

A common practice in the studies mentioned above is the use of simplified formulations for the shared boundary layer shape. Since the early work of Nayfeh et al. [23], it has been widely accepted that the acoustic field is not significantly impacted by the actual flow profile, provided that the boundary layer displacement thickness remains the same. This conclusion was drawn by analysing the acoustic attenuation across a two-dimensional lined duct, which is directly proportional to the imaginary component of the wavenumber. However, this observation was valid only for downstream propagation. For upstream propagation, their results suggest that acoustic attenuation (and therefore the wavenumbers) also depends on the flow shape factor. Jing et al. [20] analysed the effect of the flow profile on acoustic impedance eduction with sheared flow, but in their study only downstream propagation was considered, and in this case, good agreement was found by matching the average Mach number and the boundary layer displacement thickness.

To the best of the authors' knowledge, there are no reports in the literature on the influence of the boundary layer shape on impedance eduction when considering sheared flow with different propagation directions and realistic flow profiles. This effect may help

Figure 1: Schematic duct and coordinates system assumed in this work.

explain if the assumption of a uniform flow has a role in the current debate regarding the upstream and downstream discrepancy debate. We propose investigating realistic turbulent boundary layer profiles to compare them with the simpler formulations commonly used in the literature, examining how the small divergences in wavenumbers reported by Nayfeh et al. [23] affect the educed impedance. The key difference here compared with early contribution from Nayfeh et al. [23] is the consideration of duct geometries and frequency ranges typical of impedance eduction facilities. On the other hand, as opposed to the more recent studies, we propose to consider a realistic representation of the turbulent boundary layer, namely, the universal law of the wall.

A numerical experiment approach is proposed. The solution of the Pridmore–Brown equation [24] is assumed as the exact solution for the acoustic field propagating over a sheared mean flow, from which the axial wavenumbers in an infinite two-dimensional duct are obtained. These wavenumbers are then used in the traditional straightforward impedance eduction routine [7], assuming uniform flow and the Ingard–Myers boundary condition. In addition, a parametric study is conducted within the scope of this numerical experiment to evaluate the impact of the test rig duct width and the average Mach number on the accuracy of the Ingard–Myers boundary condition. Finally, we employ an iterative impedance eduction method on experimental data, similar to the one used by Roncen et al. [22], to analyse the impact of different flow velocity profile distributions on impedance eduction, comparing these results with the estimates provided by the Ingard–Myers boundary condition.

This document is organised as follows. Section 2 presents the governing equations for duct acoustics with grazing flow. Section 3 describes the flow velocity distributions considered. The setup for the numerical experiments is detailed in Section 4. The main theoretical results are discussed in Section 5, while application to experimental data is presented in Section 6. Finally, the primary conclusions are summarised in Section 7.

2. Governing equations

For the purpose of this study, we consider the infinite 2D duct depicted in Fig. 1. The duct cross-section has width W. An axial flow with velocities profile u⁰ = U0(x)kˆ is assumed, where kˆ is the unitary vector in z axis, which implies that the flow profile has no dependence on the axial direction. The wall located at x = −W/2 has a locally-reacting frequencydependent impedance Z(ω), while the other wall at x = W/2 is acoustically rigid.

The in-duct acoustic propagation can be described by the Pridmore-Brown Equation

(PBE) [24], such that

$$(i\omega + \boldsymbol{u_0} \cdot \boldsymbol{\nabla}) \left( \frac{1}{c_0^2} (i\omega + \boldsymbol{u_0} \cdot \boldsymbol{\nabla})^2 \tilde{p}' - \nabla^2 \tilde{p}' \right) + 2 \frac{\partial}{\partial z} (\boldsymbol{\nabla} \tilde{p}' \cdot \boldsymbol{\nabla} U_0) = 0,$$ (1)

where p˜ ′ is the acoustic pressure, ω is the frequency (exp(iωt) dependence assumed), c⁰ the speed of sound, i = ^√ −1 the complex imaginary unity and ∇ = (∂/∂x, ∂/∂z). Given the axial invariance of the problem, we can assume an axial modal solutions on the form p˜ ′ (x, z) = ˜p ′ (x) exp(−ikzz), where k^z is the axial wavenumber, so that Eq. (1) can be written as

$$\left(\nabla{\perp}^{2} + \frac{\omega^{2}}{c{0}^{2}}\right)\tilde{p}' - k{z}\left(\frac{U{0}}{\omega}\nabla{\perp}^{2} - \frac{2}{\omega}\nabla{\perp}U{0}\cdot\nabla{\perp} + \frac{3\omega U{0}}{c{0}^{2}}\right)\tilde{p}' - k{z}^{2}\left(1 - \frac{3U{0}^{2}}{c{0}^{2}}\right)\tilde{p}' - k{z}^{3}\left[\frac{U{0}}{\omega}\left(\frac{U{0}^{2}}{c_{0}^{2}} - 1\right)\right]\tilde{p}' = 0, \quad (2)$$

where ∇^⊥ = (∂/∂x, 0). As boundary conditions, at rigid walls the normal acoustic velocity u ′ vanishes, such that

$$\mathbf{u'} \cdot \hat{\mathbf{n}} = 0, \tag{3}$$

where ˆn is a unitary normal vector pointing into the wall. Since non-slip flows are assumed, the locally reacting impedance boundary condition can be written as

$$Z = \frac{1}{\rho_0 c_0} \frac{\tilde{p}'}{\mathbf{u}' \cdot \hat{\mathbf{n}}},\tag{4}$$

where the air characteristic impedance ρ0c⁰ is used as a normalisation factor and ρ⁰ is the air density. For the two-dimensional duct assumed in this work, ˆn = ˆi at x = W/2, where ˆi is the unitary vector in x axis, and ˆn = −ˆi at x = −W/2. For later convenience, we also introduce the distance to the wall ξ = W/2 − |x|.

2.1. Eigenvalue problem

In this section, we seek to describe the governing equations as a generalized eigenvalue problem. One can rewrite the PBE (Eq. (2)) in a discrete version as

$$(A_0 + A_1 k_z + A_2 k_z^2 + A_3 k_z^3) \tilde{\mathbf{p}} = \mathbf{0}, \tag{5}$$

where the A^j terms involve differentiation in x and multiplication by the frequency ω and the mean flow U⁰ and its x-derivatives. In the present work, we follow a strategy similar to Boyer et al. [25], where the problem is discretised by projecting onto a Gauss–Lobatto grid using Chebyshev polynomials as basis, with (5) required to hold at each grid point (a pseudo-spectral method). Finally, to solve the cubic generalised eigenvalue problem given by Eq. (5), auxiliary variables are introduced of the form ˜p^p = kz˜p^p−¹ for p > 0, and the resulting linear eigenvalue problem is solved using the QZ algorithm [26, p. 129].

In order to apply a lined wall boundary condition to the generalised eigenvalue problem, we rewrite Eq. (4) as

$$\frac{\mathrm{d}\tilde{p}'}{\mathrm{d}x}n_x + \frac{\mathrm{i}\omega}{c_0Z}\tilde{p}' = 0,\tag{6}$$

where n^x = −1 at x = −W/2. For the hard wall opposite to the liner, the corresponding boundary condition is

$$\frac{\mathrm{d}\tilde{p}'}{\mathrm{d}x} = 0. \tag{7}$$

If a uniform flow is assumed, i.e. U⁰ ≡ Mc⁰ = constant, where M is the bulk (average) Mach number, the PBE reduces to the Convected Helmholtz Equation (CHE),

$$\nabla^2 \tilde{p}' + \left(k_0 - iM \frac{\partial}{\partial z}\right)^2 \tilde{p}' = 0.$$ (8)

where k⁰ ≡ ω/c⁰ is the free-field wavenumber. For lined walls, the slip velocity at the wall is taken into account by means of the Ingard–Myers Boundary Condition (IMBC) [27, 28], leading to

$$\frac{\partial \tilde{p}'}{\partial x} = \frac{1}{ik_0 Z} \left( ik_0 + M_w \frac{\partial}{\partial z} \right)^2 \tilde{p}', \tag{9}$$

where M^w is the slipping velocity at the wall, and so M^w = M for a uniform flow. For a non-slip flow, M^w = 0, and Eq. (9) reduces to Eq. (6).

2.2. Impedance eduction

In this work, we consider the traditional straightforward wavenumber based impedance eduction first proposed by Jing et al. [7]. Applying the IMBC (Eq. (9)) on the lined wall and Eq. (6) on the rigid walls of the CHE solution leads to the eigenvalue problem

$$k_x \tan(k_x W) - \frac{1}{ik_0 Z} (ik_0 - iMk_z)^2 = 0,$$ (10)

where k^x is the transverse wavenumber given by the dispersion relation

$$k_x^2 = (k_0 - Mk_z)^2 - k_z^2. (11)$$

Once the axial wavenumber is known, it is straightforward to calculate the liner impedance from Eqs. (10) and (11).

3. Velocities profile shape functions

The simplest formulation considered in this work is the sinusoidal flow profile, as presented by Gabard and Astley [29]. In this case,

$$\frac{U_0(x)}{c_0} = \begin{cases} M_s \sin \frac{\pi \xi}{2\delta_s}, & 0 \le \xi \le \delta_s \ M_s, & \xi > \delta_s, \end{cases}$$ (12)

where M^s is the free-stream Mach number, and δ^s is the boundary layer thickness.

Another commonly employed formulation in the literature is the hyperbolic tangent profile introduced by Rienstra and Vilenski [30], which was used, for instance, by Roncen et al. [22] in their work on 2D flow profile effects on impedance eduction. This profile is given by

$$\frac{U_0(r)}{c_0} = M_c \left[ \tanh\left(\frac{1-r}{\delta_t}\right) + \left(1 - \tanh\left(1/\delta_t\right)\right) \left(\frac{1 + \tanh\left(1/\delta_t\right)}{\delta_t}r + \left(1 + r\right)\right) (1-r) \right],\tag{13}$$

where M^c is the centreline Mach number, r is the radial position, and δ^t is a shape factor. In this work, we use the coordinate transformation r = 2|x|/W to obtain the flow profiles in the x coordinate system.

We also aim to consider a more realistic representation of a turbulent boundary layer velocity profile. One may express the boundary profile over a smooth wall using a universal wall law, which, according to van Driest [31], is given by

$$U^{+} = \int_{0}^{y^{+}} \frac{2}{1 + \sqrt{1 + 4\kappa^{2}y^{+2}(1 - \exp(-y^{+}/A^{+}))^{2}}} dy^{+} + \Pi, \tag{14}$$

where U ⁺ ≡ U0/u^τ is the flow profile normalised by the friction velocity u^τ , κ ≈ 0.42 is the von Kármán constant, A⁺ ≈ 27 is the van Driest constant, and y ⁺ = ξuτ/ν is the distance from the wall, ξ, normalised to viscous lengths, with ν being the air kinematic viscosity. As will be discussed later, for the small ducts considered in this study, the boundary layer can extend the entire half-duct width. To ensure that the derivative of the profile is continuous at the duct centreline, we propose adding a cubic term to Eq. (14), denoted by Π, which is given by

$$\Pi = \frac{2}{1 + \sqrt{1 + 4\kappa^2 y{\text{max}}^+^2 (1 - \exp(-y{\text{max}}^+/A^+))^2}} (y{\text{max}}^+ - y^+) \left(\frac{y^+}{y{\text{max}}^+}\right)^2, \tag{15}$$

where y + max = W uτ/ν/2 is the distance from the wall to the centreline in viscous lengths. One of the simplest formulations for the boundary layer shape is the assumption of a linear variation. However, for the purposes of this work, the linear flow profile is not advantageous, as, although it is continuous, its first derivative is not continuous, which compromises the convergence of the pseudospectral method used in this study [26, p. 32]. Therefore, we will not use it. Another common simplification for the flow profile shape is the inverse power law, which provides a good approximation of a turbulent boundary layer [21, 23]. However, the inverse power law formulation has a problem since the velocity gradient near the wall tends to infinity. One could avoid this problem either by solving for acoustic displacement, as done by Yang et al. [21], which implies that the gradient of the mean flow does not appear explicitly in the equation, or by adding a linear sub-layer near the wall [e.g. 23], which would result in an additional discontinuity in the derivative. Both solutions add complexity to this problem, and neither is as accurate as using the turbulent wall law formulation from Eq. (14)

.

Figure 2: Reference impedances for numerical experiments. (a) SDOF-like; (b) digitalization of CT57 from Roncen et al. [22].

4. Numerical setup

In this work, we consider a small rectangular duct, representative of traditional liner impedance eduction facilities [9, 10, 12, 15]. Initially, we consider the dimensions of the Liner Impedance Test Rig from the Federal University of Santa Catarina (LITR/UFSC), which has a rectangular cross-section with a width of W = 40 mm.

For the lined wall impedance, two reference impedances are considered. First, we use the impedance given by

$$Z_{\text{SDOF}}(\omega) = 2 - i \left( \cot (k_0 h) - (0.03 k_0)^2 \right),$$ (16)

with h = 35 mm, which is representative of a typical Single-Degree-Of-Freedom (SDOF) liner in the considered frequency range, and is shown in Figure 2a. Additionally, one of the reference impedances considered by Roncen et al. [22] is replicated in this work. The impedance modelling the ceramic liner CT57 was digitized for use in this study and is denoted ZCT57, as presented in Figure 2b. The typical range for impedance eduction, from 500 to 3000 Hz, with a 50 Hz step, is used for ZSDOF, while a reduced range from 500 to 1800 Hz is used for ZCT57 due to the range of frequencies for which data is available.

For the velocity profiles, we consider the three formulations presented in Section 3. The turbulent universal wall law, given by Eq. (14), with ν = 1.48 × 10−⁵ m²/s and u^τ = 3.956 m s−¹ , is selected as the baseline case. This corresponds to the fit of Eq. (14) to experimental data gathered at the LITR/UFSC, allowing for the comparison of different flow profile formulations with a realistic velocity distribution in a typical liner test rig duct. This leads to an average Mach number of M = 0.279, a boundary layer thickness of δ99 % = 15.72 mm, and a boundary layer displacement thickness of δ ^∗ = 1.70 mm.

We aim to reproduce the study of Nayfeh et al. [23] in the context of impedance eduction. To do so, we first need to find the parameters for the hyperbolic tangent and sinusoidal flow profiles that match the same average Mach number M and boundary layer thickness δ99.9 % as the baseline case. This results in different boundary layer displacement thicknesses δ ∗ for each flow profile formulation, which is expected to lead to different acoustic fields. The parameters obtained for both the hyperbolic tangent and sinusoidal flow profiles are summarized in Table 1, along with the resulting δ ∗ for each case. The different flow profiles

Table 1: Resulting parameters for velocities profile formulations fit to baseline case average Mach number and δ99 %. Baseline case corresponds to universal wall law with ν = 1.48 × 10−⁵ m2/s and u^τ = 3.956 m s−¹

Velocities Profile	Adjusted Parameters		Resulting δ ∗	δ99 %
Hyperbolic Tangent	Mc = 0.363	δt = 0.3546	4.66 mm	15.72 mm
Sinusoidal	Ms = 0.406	δs = 17.3 mm	6.28 mm	15.72 mm

are compared to the experimental data gathered at the facility in Figure 3, where the derivative of the velocity distribution is also presented.

Next, we consider the case where the different velocity distributions share the same average Mach number and boundary layer displacement thickness δ ∗ . The parameters obtained for both the hyperbolic tangent and sinusoidal flow profiles under this new condition are summarized in Table 2, along with the resulting δ99 %. The resulting flow profiles are once again compared to the experimental data in Figure 4.

The number of points in the computational domain used for the pseudospectral solver was determined based on the critical case, which, for this study, corresponds to the universal wall law due to its high gradient near the walls.

5. Theoretical results and discussion

5.1. Effects of flow profiles on axial wavenumbers

First, we examine the wavenumbers obtained from the Pridmore–Brown equation for different velocity profile shapes, all with the same boundary layer thickness, δ99 %, and average Mach number M. These results are compared with the wavenumbers derived from the Convected Helmholtz equation with the Ingard–Myers boundary condition, which models the refraction at the boundary layer considering the average Mach number. For brevity, we focus initially on the case of ZSDOF. The wavenumbers for the least attenuated mode, for both upstream (k z ) and downstream (k + z ) propagation are presented in Figure 5. Results suggest good agreement between all considered velocity profiles and the predictions from

Table 2: Resulting parameters for velocities profile formulations fit to baseline case average Mach number and δ ∗ . Baseline case corresponds to universal wall law with ν = 1.48 × 10⁻⁵ m²/s and u^τ = 3.75 m s⁻¹

Velocities Profile		Adjusted Parameters	Resulting δ99 %	δ ∗
Hyperbolic Tangent	Mc = 0.305	δt = 0.1227	6.50 mm	1.70 mm
Sinusoidal	Ms = 0.305	δs = 4.7 mm	4.26 mm	1.70 mm

Figure 3: Flow velocities profiles in linear (a) and logarithmic (b) scales, and velocity gradients (c) considered in the first step of this work. Hyperbolic tangent and sinusoidal flow profiles are best fitted to match the bulk Mach number and boundary layer thickness of the law of the wall.

the CHE-IMBC for the real component of the wavenumber in both propagation directions. However, for the imaginary component of the wavenumber, which corresponds to the attenuation rate, significant differences are observed between the wavenumbers obtained for the hyperbolic tangent and sinusoidal flow profiles, compared to those obtained for the more realistic distribution given by the universal wall law, particularly for upstream propagation. On the other hand, good agreement is observed between the CHE-IMBC solution and the PBE solutions. This suggests that the IMBC may provide a good approximation for the typical range of frequencies and duct dimensions of impedance eduction. A possible explanation is that, due to the high gradient near the wall, even though the velocity distribution extends nearly across the entire duct half-width, the region where refractive effects are significant is confined to a much thinner region, making the infinitely thin hypothesis of the IMBC more appropriate than originally suggested.

Next, we consider the case where the different velocity distributions are adjusted to match the average Mach number and the boundary layer displacement thickness, δ ∗ , rather than

Figure 4: Flow velocities profiles in linear (a) and logarithmic (b) scales, and velocity gradients (c) considered in the first step of this work. Hyperbolic tangent and sinusoidal flow profiles are best fitted to match the bulk Mach number and boundary layer displacement thickness of the law of the wall.

the boundary layer thickness, δ99 %. This approach is expected to improve the agreement between the acoustic attenuation predicted by the different flow profiles, particularly for downstream propagation [23]. The wavenumbers for the least attenuated mode in both upstream and downstream propagation are shown in Figures 6 and 7, for the impedances ZSDOF and ZCT57, respectively.

As expected, a better agreement is observed for the wavenumbers obtained for the different velocity distributions, especially for downstream propagation. However, for upstream propagation, the wavenumbers for the hyperbolic tangent and sinusoidal flow profiles agree well with each other, but still diverge noticeably from the solution for the wall law and the prediction for the CHE-IMBC. These initial results suggest that assuming a uniform flow and compensating for the refraction within the boundary layer using the Ingard–Myers boundary conditions provides better estimates of the acoustic field in impedance eduction facilities under typical test conditions, compared to solving for an explicit velocity distribution that is not representative of realistic conditions.

Figure 5: Wavenumbers obtained for the SDOF-like impedance with different velocities distributions for the same M and δ99.9 %.

It is worth noting that the difference between the wavenumbers obtained for the wall law velocity distribution and the estimation obtained with CHE-IMBC is notably higher at the lower frequency range for the impedance ZCT57, as shown in Figure 7d. To investigate this, we propose analysing the error between the estimation from the CHE-IMBC, kz,CHE, and the exact solution from the PBE, kz,PBE, defined as

$$error = \frac{|k{z,PBE} - k{z,CHE}|}{k_{z,PBE}},$$ (17)

in the complex impedance plane. We fix the frequency at 550 Hz and consider the resistance range θ ∈ [0, 5] and the reactance range χ ∈ [−5, 5]. The three flow profile formulations are considered, and the corresponding contour plots are shown in Figure 8.

Results suggest that the error function is almost zero for the majority of the considered

Figure 6: Wavenumbers obtained for the SDOF-like impedance with different velocities distributions for the same M and δ ∗

impedance plane, with the notable exception of the region defined by resistances smaller than 1 and reactances between -2 and 0. This region has previously been identified as a potential area where double roots for the surface modes may occur [32], which could compromise the stability of the Ingard–Myers boundary condition [33].

5.2. Effects of flow profiles on impedance eduction

The next step, which is the main goal of this study, is to evaluate the impact of considering different flow velocity profiles on the evaluation of the acoustic field in impedance eduction. As discussed in Section 2.2, we use the wavenumbers obtained for the least attenuated mode, considering the different velocity profiles, in the classical straightforward impedance eduction routine. This routine assumes uniform flow and the Ingard–Myers boundary condition to model the slip velocity at the wall. For the sake of brevity, from this point on, we will

Figure 7: Wavenumbers obtained for the digitized CT57 impedance with different velocities distributions for the same M and δ ∗

focus solely on the impedance ZSDOF, as it is a more representative case of typical acoustic liners' impedance. The impedances educed with the proposed numerical experiment using Eqs. (10) and (11), along with the wavenumbers obtained for the different velocity profiles in the PBE, are shown in Figure 9.

The impedances educed using the wavenumbers obtained from the solution of the hyperbolic tangent and sinusoidal flow profiles exhibit a similar trend regarding the mismatch observed experimentally between upstream and downstream acoustic sources (downstream and upstream propagation, respectively). At lower frequencies, the upstream source case results in a lower resistance, with the opposite trend observed at higher frequencies. This behaviour is similar to what has been observed by Roncen et al. [22]; however, in our case, the reference impedance is not the midpoint between the two curves. For the most realistic flow profile, the wall law, the conclusions differ significantly. At the lower frequency end,

Figure 8: Contour plots of the error of assuming the Ingard–Myers boundary condition as a simplification of different velocities profile formulations. Red "X" denote the value of the CT57 impedance at 550 Hz.

the assumption of uniform flow with the IMBC introduces a small bias for both acoustic source positions, with good agreement observed between them. At higher frequencies, the curves diverge, with the upstream source (downstream propagation) surprisingly showing a greater deviation from the reference impedance.

The results obtained so far in this work suggest that the shape of the velocity profile considered when solving for the acoustic field in small ducts with lined walls plays a significant role. Additionally, the Ingard–Myers boundary condition provides better estimations, when compared to the exact solution for simplified velocity distributions. However, a single duct geometry and a single average Mach number have been considered. In what follows, we propose a parametric study on the duct geometry and bulk Mach number impact in our conclusions.

5.3. Parametric study

In this section, we analyse the sensitivity of the IMBC accuracy to variations in the average Mach number and duct width through a parametric analysis. We consider the wavenumbers obtained by solving the PBE and the flow profile described by the universal law of the wall. Furthermore, the thickness of the boundary layer may influence the accuracy of the Ingard–Myers boundary condition. However, since it has been observed that the

Figure 9: Educed impedances obtained for the wavenumbers evaluated for the PBE considering different velocities profile shapes with the impedance ZSDOF at the lined wall. Velocities profiles match the universal wall law with u^τ = 3.95 m s⁻¹ , in average Mach number and boundary layer displacement thickness δ ∗ . US - Upstream Source (downstream propagation), and; DS - Downstream Source (upstream propagation).

Table 3: Friction velocities u^τ considered for the parametric analysis as a function of the average Mach number M for a duct width of W = 40 mm.

Figure 10: Parametric study on the effect of the average Mach number. Impedances educed with the IMBC for the wavenumbers obtained for the exact solution of the PBE with a realistic flow profile. Solid lines: downstream acoustic source; dashed lines: upstream acoustic source.

universal law of the wall formulation used in this work can extend to the entire half-width of the duct, the boundary layer thickness in this case is a function of both the duct width and the viscosity. To produce significant variations in δ by changing the viscosity ν, non-realistic values would need to be considered. For this reason, we have decided not to include the boundary layer thickness as a parameter in this parametric analysis.

First, we examine the effect of the average Mach number. The duct width is set to W = 40 mm, and the air viscosity is ν = 1.48 m²/s. The friction velocity was adjusted to vary with the average Mach number, and the values considered are presented in Table 3. The impedances educed for the different average velocities are shown in Figure 10.

The results suggest that increasing the Mach number leads to higher errors in the educed impedance when using the Ingard-–Myers boundary condition instead of solving for the exact flow profile, particularly for the resistance. This aligns with the observations from the evaluation of the wavenumbers, where the largest differences are noted in the imaginary component of the wavenumber. As with the resistance, the imaginary component is related to acoustic dissipation. These findings are consistent with previous studies reported in the literature [21].

Next, we examine the effect of the duct width—and consequently, the boundary layer

Table 4: Friction velocities u^τ considered for the parametric analysis as a function of the duct width W for an average Mach number of 0.279.

Figure 11: Parametric study on the effect of the duct width. Impedances educed with the IMBC for the wavenumbers obtained for the exact solution of the PBE with a realistic flow profile. Solid lines: downstream acoustic source; dashed lines: upstream acoustic source.

thickness—on the accuracy of the IMBC for impedance eduction. This analysis considers the typical dimensions of traditional liner impedance eduction facilities, which typically feature duct widths smaller than 70 mm. Novel approaches, such as curved [34] and multimodal [35] duct configurations, are not considered in this study. The average Mach number is fixed at the same value as the baseline case, M = 0.279, and the friction velocity is adjusted for each considered duct width. The duct width values and corresponding friction velocities are summarised in Table 4, while the educed impedances for the different cases are presented in Figure 11.

The results suggest that the accuracy of the IMBC decreases with increasing duct width, particularly in predicting the resistive component of the impedance. For the reactance, larger errors are observed at the higher frequency range, especially for the downstream acoustic source (upstream propagation). These findings align with expectations, as an increase in duct width leads to a corresponding increase in the dimensional boundary layer thickness. This deviates further from the infinitely thin boundary layer assumption of the IMBC.

Table 5: Liner samples parameters. σ - percentage of open area; h - cavity height; d - holes diameter, and; t - perforate sheet thickness.

Parameter	σ [%]	h [mm]	d [mm]	t [mm]
Sample A	5	40	1.2	1
Sample B	12	25.4	0.835	1

6. Application to Experimental Data

Finally, we propose to extend the analysis of this study to experimental data gathered at the Liner Impedance Test Rig of the Federal University of Santa Catarina (LITR/UFSC). The test rig's test section consists of modular rectangular cross-sectioned ducts measuring 40×100 mm² (i.e., W = 40 mm). Quasi-anechoic terminations at the test rig inlet and outlet minimise acoustic reflections. Eight Beyma CP-855nD compression drivers are distributed both upstream and downstream of the liner test sample holder to generate sound fields up to 150 dB, propagating either with or against the flow towards the liner sample.

An external compressed air system provides the flow supply, capable of sustaining a cross-section averaged flow up to Mach 0.7. A Pitot tube located at the test rig inlet is used to control and monitor the flow Mach number during tests. The average Mach number in the lined section is derived from the Pitot tube measurement using a pre-calibrated factor determined through a quadrature method. The liner sample holder has an opening for liner samples with a maximum length of 420 mm.

An array of sixteen equally spaced flush-mounted B&K DeltaTron 4944 1/4" pressure field microphones is installed on the wall opposite the liner section for impedance eduction. The spacing between consecutive microphones is 20 mm. In this work, half of the microphones are skipped, resulting in an effective separation of 40 mm to reduce uncertainties in the lower attenuation range of the liner [36]. Signals are recorded using a National Instruments PXIe-4499 data acquisition (DAQ) module at a sampling rate of 25.6 kHz. Measurements are conducted using a harmonic excitation signal, which also serves as a reference for cross-spectrum estimation using Welch's method, with 30 averages of 25 600 samples and 75 % overlap. All hardware control, signal processing, and data post-processing are performed using in-house Python3 code.

Two liner samples are employed in this study, referred to as samples A and B. Both samples are typical single-degree-of-freedom liner constructions, each with a length of 420 mm. A summary of the relevant parameters for both samples is presented in Table 5.

Tests were conducted under three different flow conditions: in the absence of flow (M = 0); and with bulk Mach numbers (area-averaged) of M = 0.2 and M = 0.3. A stepped puretone excitation was employed in a frequency range from 500 Hz to 3000 Hz, with increments of 100 Hz. The sound pressure level was set to 130 dB for the plane wave amplitude propagating towards the liner, with the acoustic source positioned either upstream or downstream (one at a time) of the liner.

We consider four cases for impedance eduction using the experimental acoustic field. First, we examine the traditional straightforward method, which assumes that acoustic propagation is governed by the CHE and that the Ingard–Myers boundary condition applies to the lined walls. The other three cases involve solving the PBE while considering different flow velocity distributions. In the first of these, we assume that the flow profile can be approximated by the universal law of the wall, Eq. (14). The friction velocity is adjusted so that the average Mach number of the 1D profile matches the bulk Mach number of the 2D test section. This approach follows the conclusion of Jing et al. [20], who demonstrated that, when simplifying a 3D duct to a 2D duct, the average Mach number must remain constant.

The other two cases also solve the PBE but use a hyperbolic tangent velocity distribution. In the first of these, the boundary layer thickness δ99 % is matched to that of the universal law of the wall. In the final case, the boundary layer displacement thickness, δ ∗ , is matched instead.

For the impedance eduction considering the solution of the PBE, we follow an iterative procedure first presented by Roncen et al. [22]. The dominant axial wavenumber of sound in the lined duct section, kz,exp, is extracted from the equally spaced microphone array record using the KT algorithm, as detailed in Bonomo et al. [36]. The eduction routine minimises a cost function defined as

$$\mathcal{F}(Z) = |k{z,\text{exp}} - k{z,\text{PBE}}(Z)|, \qquad (18)$$

where kz,PBE is the wavenumber obtained by solving the eigenvalue problem of the PBE. To accelerate convergence, the impedance obtained by solving the convected Helmholtz equation with the Ingard–Myers boundary condition is used as an initial guess.

The results obtained for the four cases considered, using samples A and B, are shown in Figures 12 and 13, respectively. Greater differences between upstream and downstream educed impedances are observed with sample A, which exhibits stronger non-linear behaviour with respect to flow effects. However, the conclusions regarding the impact of assuming different 1D flow velocity distributions are consistent for both liners and can be summarised as follows. The differences in the educed resistances for the different velocity profiles are larger compared to the differences in the reactances, aligning with observations from the numerical experiment. Greater differences are also noted with increasing Mach number.

Additionally, the impact of assuming different flow profiles is more pronounced for a downstream acoustic source (upstream propagation), consistent with the larger biases observed in the wavenumbers obtained for the different formulations. Regarding the different formulations used for solving the PBE, significant differences are observed among the three cases. The case with matching δ ∗ shows the best agreement between the solution obtained for the hyperbolic tangent profile and that for the universal law of the wall.

As anticipated by the numerical experiment, good agreement is observed between the prediction using the Ingard–Myers boundary condition and the solution obtained for the universal law of the wall profile, particularly for an upstream acoustic source (downstream propagation). For impedances educed with a downstream acoustic source, the IMBC slightly over-predicts compared with the solution of the PBE with the law of the wall. This is consistent with the results of Weng et al. [15], who solved the linearised Navier–Stokes equation for a realistic flow profile comparable to the one considered in the present work.

Figure 12: Impedances educed for sample A. (a,b) M = 0.2; (c,d) M = 0.3. Dashed lines denote impedances educed upstream/downstream, and solid lines denote impedances educed downstream/upstream.

Figure 13: Impedances educed for sample B. (a,b) M = 0.2; (c,d) M = 0.3. Dashed lines denote impedances educed upstream/downstream, and solid lines denote impedances educed downstream/upstream.

7. Conclusion

In this work, the effects of the sheared flow profile shape on acoustic propagation in a 2D duct were revisited, extending the early work of Nayfeh et al. [23] to the context of impedance eduction techniques. Three velocity profiles were considered for solving the acoustic field in a duct with sheared grazing flow, using the Pridmore–Brown equation, with the profiles matching the average Mach number and either the δ99 % or δ ^∗ boundary layer thicknesses. The wavenumbers obtained from the PBE were then compared to those estimated by solving the case with uniform flow, i.e., the Convected Helmholtz equation, with the Ingard–Myers boundary condition handling the refractive effects within the boundary layer. Results suggest that the IMBC leads to lower errors relative to the solution for a sheared flow profile with a realistic wall law turbulent boundary layer velocity distribution, compared to the solution obtained with a simplified profile formulation. Consistent with the findings of Nayfeh et al. [23], it was observed that matching the boundary layer displacement thickness δ ∗ improves agreement between the different formulations, although noticeable differences remain.

Next, a numerical experiment was conducted to assess the accuracy of the IMBC in the typical impedance eduction range for small ducts, by assuming that the PBE is a exact representation of the real world. The wavenumbers obtained for the different flow profile formulations were used as input for the traditional straightforward impedance eduction routine. Results suggest that for non-realistic flow profile formulations, the simplification to the uniform flow assumption with the IMBC may lead to mismatches between results obtained for upstream and downstream propagating waves, particularly for the acoustic resistance. However, this mismatch does not occur when a realistic velocity distribution is used to simulate real-world conditions. A parametric study was then conducted to investigate the impact of the average Mach number and the duct width, and consequently, the boundary layer thickness. It was found that the error associated with the simplification to the IMBC increases with both the average Mach number and the duct width.

Finally, an iterative eduction routine was used to evaluate the impact of solving for the sheared flow profile rather than relying on the IMBC approximation with experimental acoustic data. This analysis allowed to investigate the effect of simplifying the flow profile representation when performing impedance eduction and solving for the sheared flow. The results obtained align with the conclusions of Weng et al. [15], who suggested that solving for the acoustic field using a realistic flow profile produces reasonable agreement with the IMBC solution.

The main conclusion of this work is that the Ingard–Myers boundary condition is a reasonable simplification in the context of low Mach number and small-duct impedance eduction, at least for 2D ducts. The natural continuation of this work is its extension to a realistic 3D duct, as proposed by Roncen et al. [22], while taking into account the importance of a realistic flow profile representation.

Acknowledgments

On behalf of L.A. Bonomo and J.A. Cordioli, this research was partially funded by CNPq (National Council for Scientific and Technological Development). L.A. Bonomo acknowledges that this study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES), Finance Code 001. E.J Brambley gratefully acknowledges the support of the UK Engineering and Physical Sciences Research Council (EPSRC grant EP/V002929/1).

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