3.1. Computational domain

The computational domain is illustrated in figure 4. The coordinate system is defined as follows: $ x$ denotes the streamwise direction, $ y$ the wall-normal direction and $ z$ the spanwise direction. In this work, the following velocity notation is used: $ U, V, W$ : time-averaged streamwise, wall-normal and spanwise velocity components, respectively; $ u, v, w$ : instantaneous streamwise, wall-normal and spanwise velocity components; $ u' = u - U, \; v' = v - V, \; w' = w - W$ : velocity fluctuations relative to the mean.

The liner is placed in the middle of the channel at the top wall of a duct with a rectangular cross-section. Each cavity has a square cross-section of $l=8.46d$ and a depth of $\zeta = {32.56d}$ , where $d=1.17$ mm is the orifice diameter. Each cavity has eight orifices, partition walls of thickness $w_p=1.08d$ and a face sheet thickness of $\tau = 0.46d$ . These dimensions result in a percentage of open area for the entire sample of 5.5 %. The cross-section of the channel has a height of $H = 2h = 40$ mm and a width equal to $l+w_p$ . In the upstream region of the channel, a zig-zag trip was added on both the top and bottom walls. Its size and position were manually adjusted to match the experimental velocity profile upstream of the liner. The zig-zag trip was placed at $x = -1367d$ upstream of the liner, where $x=0$ marks the start of the liner. The zig-zag trip had a height of $0.21d$ and a length of $1.71d$ .

To achieve a quasi-anechoic condition and prevent acoustic reflections at the channel’s termination, the fluid viscosity was significantly increased using sponge regions, as shown in figure 4. In these regions, the viscosity was increased by a factor of one hundred following an exponential law. All walls in the computational domain were treated as adiabatic. A uniform velocity boundary condition was applied at the inlet, corresponding to a Mach number of M = 0.3, which results in a centreline velocity of $U_0 = 110$ m s⁻¹ (M = 0.32) in the lined section. A pressure boundary condition was set at the outlet.

The computational domain was designed to replicate the experimental set-up of the UFSC Liner Test Rig (Bonomo et al. Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022). The design of the sample, although resembling the one presented in a previous numerical study (Pereira et al. Reference Pereira, Bonomo, Quintino, Da Silva, Cordioli and Avallone2023), exhibits variations in terms of the face sheet thickness, orifice diameter and shape of the edges of the orifice, which were slightly rounded as discovered from the three-dimensional scanning of the tested liner sample (Quintino et al. Reference Quintino, Bonomo, Cordioli, Jones, Howerton, Nark and Avallone2025). Due to manufacturing limitations, the scan also identified variability in these parameters along the sample. Therefore, averaged values of orifice diameter and face sheet thickness were used to construct the liner sample for the numerical simulations. However, a few differences between the real and simulated liners should be noted. First, the simulated liner is represented by a single row of eleven cavities, while the one tested in the experiments features an 8 $\times$ 33 cavity grid, as shown in figure 3. However, both configurations maintain the same number of orifices per cavity, ensuring consistency in porosity. A second key difference is that, in the UFSC Liner Test Rig, the duct has a rectangular cross-section of $100 \times 40$ mm $^2$ . In contrast, the simulation assumes periodic boundary conditions on both sides of the duct. Previous studies (Tam et al. Reference Tam, Pastouchenko, Jones and Watson2014) have shown that this choice has minimal influence on the acoustic response, supporting the validity of the comparison with the experimental results.

Figure 4.

Schematic representation of the full computational domain.

3.2. Computational grid design

A variable resolution (VR) scheme was adopted. Details of the VR regions are provided in figure 5. The VR regions are symmetric with respect to the centre of the channel. The finest resolution, VR = 7, was used to discretise the entire face sheet, the orifices and portions of the backing cavities. Each subsequent resolution level was defined by doubling the cell size of the previous level. Within the orifice, the minimum grid spacing was $\Delta z_{\textit{min}} = \Delta y_{\textit{min}} = \Delta x_{\textit{min}} = 0.0207d$ , yielding a resolution of $\approx 48$ vx/ $d$ for the fine mesh. This results in a spacing expressed in wall units equal to $\Delta x^+= \Delta y^+ = \Delta z^+=6.9$ . Where $\Delta x^+ = \Delta x u_{\tau } /\nu$ and $u_{\tau } = \sqrt {\tau _w / \rho } = 4.2$ m s⁻¹ refers to the friction velocity of the smooth reference surface.

Figure 5.

Schematic representation of the VR regions.

3.3. Simulation strategy and test cases

In the presence of the grazing flow, the flow developed spatially in the channel flow until statistical convergence was achieved. After achieving statistical convergence, the acoustic simulations were performed. Starting from the flow-only converged solution, an instantaneous flow field was saved and modified by superimposing a plane acoustic wave with a specified frequency and amplitude using the OptydB toolkit. This modified flow field was used as the initial condition for the subsequent simulations, which include the acoustic wave (Avallone et al. Reference Avallone, Manjunath, Ragni and Casalino2019).

While this approach effectively reduced computational costs, especially when analysing multiple configurations, it introduced a change in the initial condition that required a few acoustic cycles for the solution to reach a statistically steady state. For each configuration, the plane acoustic wave was at least 10 wavelengths long. The downside of this approach is that the length of the channel must be long enough to accommodate 10 acoustic wave wavelengths for the lowest frequency of interest. However, this approach is also beneficial since it allows the presence of acoustic sponges, thus minimising the impact of acoustic reflections at the boundary of the computational domain. Details on the computational cost are reported in Appendix A.2.

Twenty different simulations were performed and grouped into seven sets, as shown in table 1. Sets (A) and (B) consist of simulations with the acoustic liner, without grazing flow and with plane acoustic waves with amplitudes equal to 130 and 145 dB, and three frequencies (800, 1400 and 2000 Hz). The acoustic wave was located upstream of the liner. These frequencies were chosen to investigate the acoustic liner’s behaviour near the resonance frequency and at frequencies above and below it. The SPL was calculated using the standard reference pressure of $20 \times 10^{-6}$ Pa.

Table 1.

List of the simulations carried out.

For the cases with grazing flow, the centreline Mach number was set to 0.32, as in the reference experiments. This corresponds to a bulk Reynolds number of $\textit{Re}_b = 2.7 \times 10^{5}$ and a friction Reynolds number of $\textit{Re}_{\tau } \approx 3500$ . Set (C) includes two simulations: one of a turbulent channel flow with smooth walls and the other of a turbulent channel flow with the liner mounted in the upper wall. All other sets account for the presence of grazing turbulent flow and acoustic waves. Set (D) contains simulations with different frequencies, at an SPL of 130 dB and an upstream-located acoustic source, i.e. propagating in the same direction as the mean flow. Set (E) includes simulations with different frequencies at a fixed SPL of 145 dB and an upstream acoustic source. Sets (F) and (G) include simulations with different frequencies, and downstream acoustic sources at an SPL of 130 and 145 dB, respectively.

3.4. Validation of the numerical approach

To validate the computational results, first the mean flow profile for the smooth-wall configuration was compared with the experimental data from Vallikivi, Hultmark & Smits (Reference Vallikivi, Hultmark and Smits2015) and Bonomo et al. (Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022) using three different mesh resolutions: coarse (12 vx/ $d$ ), medium (24 vx/ $d$ ) and fine (48 vx/ $d$ ). Literature data were also used because velocity fluctuation measurements were not available from the experiments conducted by Bonomo et al. (Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022).

Data are presented in wall units ( $y^+ = yu_{\tau }/\nu$ and $u^+=U/u_{\tau }$ ). In the experiments, the friction velocity was computed using the Clauser chart technique (Clauser Reference Clauser1954). In the simulations, $u_\tau$ was computed from the wall shear stress calculated using the extended wall model described in § 2. Figure 6(a) shows the mean velocity profile. The data from experiments by Bonomo et al. (Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022) and simulations at the three resolutions are compared with the law-of-the-wall using constants $k=0.37$ and $B=3.7$ , as recommended for turbulent channel flows (Nagib & Chauhan Reference Nagib and Chauhan2008). A good agreement between experiments and simulations is found.

Figure 6.

(a) Mean velocity profile comparison with experimental data by Bonomo et al. (Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022); (b) streamwise velocity variance compared with experiments by Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015).

Figure 6(b) presents measurements of the streamwise Reynolds stress derived from simulations, compared with experimental data reported by Vallikivi et al. (Reference Vallikivi, Hultmark and Smits2015) at a friction Reynolds number of $\textit{Re}_{\tau } \approx 3300$ . Values were made non-dimensional using the local $u_{\tau }^{2}$ . The profile of $u'^2$ for the fine simulation is similar to the experiments at a comparable Reynolds number. Further validation of the simulation was performed by comparing the friction coefficient $C_f$ , computed from the streamwise pressure drop, against the experimental data of Schultz & Flack (Reference Schultz and Flack2013). The comparison, presented in Appendix A, confirms the reliability of the numerical predictions.

Based on these results, the fine-resolution grid was selected for all subsequent analyses.

4.1. Sound pressure level decay

The analysis starts with the SPL distributions at the centreline of the channel (figure 7), where the acoustic pressure fluctuations are almost not affected by the flow-induced ones. The SPL distribution is plotted with respect to the incident $\textrm{SPL}_i$ for each case. Although the SPLs were set equal for both simulations, minor deviations of approximately 2 dB ( $\approx 50$ Pa) at the onset of the liner are present, which might be caused by different reflections of the acoustic waves at the wall impedance discontinuity depending on the propagation direction in the presence of flow (Saverna, Aurégan & Pagneux Reference Saverna, Aurégan and Pagneux2019). The shape of the SPL curves depends on the presence of grazing flow, the direction of acoustic wave propagation and its frequency. A relevant observation is that the SPL decays more when the source is located downstream, i.e. propagates against the mean flow, independently of the frequency of the acoustic wave. Furthermore, in this case, the SPL curves show a more monotonic behaviour compared with cases where the acoustic wave propagates in the same direction as the mean flow.

Figure 8 shows the distribution of SPL on the liner surface for the cases with an incoming acoustic plane acoustic wave with a frequency of 1400 Hz and SPL of 145 dB, with and without grazing flow. The SPL on the surface results from two main contributions: pressure fluctuations induced by the acoustic wave and those generated by the turbulent grazing flow. The interaction of the grazing flow with the liner’s orifices affects both wall-normal and streamwise velocity fluctuations near the orifices, as will be detailed in § 5.4. This results in localised variations of the surface pressure fluctuations, which contribute to the spatial distribution of SPL shown in the figure. This might explain why the local SPL can exceed that of the imposed acoustic wave alone, consistent with the findings of Roncen (Reference Roncen2025a ), who emphasised that turbulence-induced pressure fluctuations cannot be neglected.

The SPL surface distributions differ from those at the channel centreline. Without grazing flow, there is an almost monotonic SPL reduction on the surface. On the other hand, in the presence of a grazing flow, there is a reduction of the SPL upstream of the orifice and an increase downstream of it, independently of the propagation direction of the acoustic wave. This trend persists across all tested frequencies and SPL levels, including the lower SPL case (130 dB), although results are omitted here for brevity. The cause of this redistribution might lie in the near-wall hydrodynamic field around the orifices.

Figure 9.

(a) Resistance and (b) reactance along the liner’s surface obtained with in situ for the case with plane acoustic wave with SPL = 145 dB and $f$ = 1400 Hz under three conditions: without grazing flow, with grazing flow and upstream acoustic source and with grazing flow and downstream acoustic source.

4.2. Surface distribution of impedance components obtained with the in situ method

Resistance and reactance are computed using the in situ method as described above. The reference microphone at the backplate is always located at the centre of each cavity backplate, while the face sheet microphone is moved to generate contour plots. These contours are shown in figure 9 for an incident acoustic plane wave with SPL = 145 dB and $f$ = 1400 Hz, with and without grazing flow. To highlight the location-dependent impedance, the figures display the changes in resistance and reactance relative to the mean value calculated on the entire surface.

In the absence of the grazing flow, the resistance decreases in the propagation direction of the acoustic wave because of a reduction of the SPL. The resistance decreases along each cavity. Then, a jump in the resistance value is present when changing the cavity because the reference microphone at the backplate changes. Conversely, in the presence of a grazing flow, the spatial distribution of the resistance sees larger variations over each cavity; there is a strong reduction of the resistance when changing cavity; around each orifice, the resistance is always lower upstream of the orifice and higher downstream. The variation of resistance is up to a factor of 3, suggesting that the selection of the sampling location is crucial. This behaviour might be caused by the local surface SPL, which depends on the local amplitude of the acoustic wave and the surface pressure fluctuations altered by the presence of the orifices, as it will be described later. The pattern remains unchanged with respect to the direction of acoustic wave propagation, although the mean value over the surface slightly varies.

Data extracted at the centreline of the liner sample are shown in figure 10 for both upstream (a,b) and downstream (c,d) acoustic wave locations. In this figure, the resistance predicted using the semi-empirical model by Yu et al. (Reference Yu, Ruiz and Kwan2008) is reported using the local SPL and the value of the boundary layer displacement thickness $\delta ^*$ obtained from the smooth-wall simulation upstream of the liner and the local one obtained from the lined simulation (which will be detailed in the next section). The line plot in figure 10(a,c) highlights that the semi-empirical prediction from Yu et al. (Reference Yu, Ruiz and Kwan2008) approaches the mean value of resistance if the local values of $\delta ^*$ are used. The reason will be explained in the following.

Similar observations can be made for the reactance, for which the cavity-averaged values (black dots in the figure) weakly depend on the acoustic wave propagation direction. These results contrast with findings reported in the literature for impedance educed using model-fitting inference methods (Spillere et al. Reference Spillere, Bonomo, Cordioli and Brambley2020) and may be explained by the fact that the in situ approach relies on surface pressure measurements and does not employ boundary conditions to model the near-wall flow–acoustic interaction.

Figure 10.

Streamwise distribution of resistance and reactance compared with the semi-empirical model by Yu et al. (Reference Yu, Ruiz and Kwan2008) using different values of $\delta ^{*}$ and the local SPL. Data shown for SPL = 145 dB and frequency of 1400 Hz. (a,b) Upstream acoustic source, (c,d) downstream acoustic source.

These findings emphasise that the local amplitude of the acoustic and flow profile must be considered when measuring with the in situ method. However, the semi-empirical model does not capture the high resistance values observed in the numerical data at the beginning and end of each cavity.

4.3. Average impedance and comparison with experimental data

While numerical simulations allow for mapping the impedance distribution on the entire liner sample, the average value is often used to characterise the acoustic liner. Therefore, the impedance obtained using the MM method is reported in figure 11(a,b) for the cases with SPL equal to 145 dB and the three frequencies investigated, with and without grazing flow and both acoustic wave propagation directions. Experimental results from Bonomo et al. (Reference Bonomo, Quintino, Spillere, Cordioli and Murray2022) are also reported. In figure 11(c,d), the in situ results are compared with the experimental data at a similar sampling location (figure 3 b); the bars indicate the minimum and maximum values obtained over the liner’s surface. Numerical and experimental results show a reasonable agreement with similar trends. However, as pointed out above, the length of the liner sample differs between experiments and simulations, thus challenging the comparison at the lowest frequency investigated.

Figure 11.

Comparison of (a,c) resistance and (b,d) reactance components of impedance for acoustic wave amplitude equal to 145 dB. (a,b) The MM and (c,d) in situ results.

While the two methods yield roughly similar resistance values without flow, the resistance values obtained in the presence of flow, both in the experiments and numerical simulations, differ. The largest value obtained from the in situ method is lower than the average one obtained with the MM method. However, both methods show that the resistance increases with the grazing flow and is lower when the acoustic source is located upstream.

The reactance shows smaller variation with and without flow than the resistance for both techniques. This result agrees with previous findings in the literature (Spillere et al. Reference Spillere, Bonomo, Cordioli and Brambley2020) and can be explained by the fact that, in the mass–spring analogy, the reactance corresponds to the balance between mass and stiffness, which is more directly related to the cavity depth than to the presence of low (Panton & Miller Reference Panton and Miller1975; Jones et al. Reference Jones, Tracy, Watson and Parrott2002). According to the Goodrich model (Yu et al. Reference Yu, Ruiz and Kwan2008) the reactance is therefore predominantly influenced by the cavity depth and by the SPL. Grazing flow affects the reactance only via the orifice end correction – i.e. the oscillating fluid volume in and around the orifice – thereby causing the observed minor variations in the reactance curve. However, the differences between cases are more evident in the results from the MM method. In particular, the MM method shows that the frequency at which the reactance is equal to zero shifts to higher values in the presence of the flow, and this does not depend on the propagation direction of the acoustic wave. This frequency shift is attributed to the flow-induced blockage effect, which effectively reduces the face sheet porosity (Tam & Kurbatskii Reference Tam and Kurbatskii2000a ). Furthermore, in the low-frequency range, the reactance values are higher for the upstream acoustic source than for the downstream one. It is important to note that the slope of the reactance curve differs between the MM and the in situ methods. This highlights that the educed impedance is inherently dependent on the eduction technique employed, as each method relies on distinct physical assumptions and boundary conditions.

Figure 12.

(a,c,e) Resistance and (b,d, f) reactance obtained from the MM and in situ techniques. The in situ results have been obtained as an average over the entire liner’s surface. Results refer to a plane acoustic wave at varying SPLs, with and without grazing flow. (a,b) Incident acoustic wave with $f = 800$ Hz, (c,d) $f = 1400$ Hz and (e, f) $f = 2000$ Hz.

Finally, the effect of the acoustic wave SPL is reported for the frequencies equal to 800, 1400 and 2000 Hz in figure 12. In this case, the in situ results are reported as the average over the entire liner surface for the sake of comparison with the MM method.

As expected, without the grazing flow, increasing the SPL results in a rise in resistance. This reflects a transition in the liner’s behaviour from an almost linear to a nonlinear regime (Melling Reference Melling1973; Tam & Kurbatskii Reference Tam and Kurbatskii2000b ). In the presence of flow, the resistance varies less by varying the SPL for both the MM and in situ techniques. Similar observations can be made for the reactance. This observation is consistent with previous findings in the literature, particularly with the trend in resistance predicted by the semi-empirical model proposed by Yu et al. (Reference Yu, Ruiz and Kwan2008). This effect may be attributed to the increase in pressure fluctuations caused by the turbulent flow, as reported by Roncen (Reference Roncen2025b ), which is not accounted for in the eduction techniques. Indeed, both flow- and SPL-induced effects are primarily related to the wall-normal particle velocity; increases in turbulent fluctuations or SPL both contribute to an increase in the liner’s resistance (Roncen Reference Roncen2025b ).

Notably, when considering the mean resistance obtained with the in situ method, at the highest SPL, values in the presence of grazing flow vary less on varying the direction of propagation of the acoustic wave with respect to the ones obtained with the MM technique. This might be caused by the fact that impedance obtained from the in situ technique is not dependent on any flow-based boundary condition. Finally, this figure shows that for both SPLs, the surface-averaged in situ results are different from the MM ones.

5.1. Instantaneous flow

Before digging into the mean flow quantities, a qualitative description of the instantaneous flow features is reported in figure 13. More in detail, figure 13(a,b) present the contour plots of the instantaneous streamwise velocity component, and figure 13(c,d) the wall-normal velocity component in a wall-parallel plane at $y/h=0.003$ , corresponding to $y^+ \approx 15$ in the smooth-wall case. Comparisons are made between the smooth wall and the lined surface. The streamwise velocity component exhibits low- and high-speed streaks, as for the smooth-wall case, but with higher intensity fluctuations and larger spanwise size. Higher wall-normal velocity fluctuations are observed where the liner orifices are located, caused by the flow recirculation within the orifices (Shahzad et al. Reference Shahzad, Hickel and Modesti2023b ). This is more evident in the wall-normal plane at $z/l = 0.5$ shown in figure 13(e, f). The higher velocity fluctuations, caused by the flow within the orifices, are not only localised close to the surface but also extend towards the centre of the duct. This suggests that the acoustic liner modifies the flow near the wall and in the outer portion of the flow.

Figure 13.

(a,b) Instantaneous streamwise $u^{\prime }$ (c,d) and wall-normal $v^{\prime }$ velocity components at $y/h = 0.003$ . (e, f) Instantaneous vertical velocity $v^{\prime }$ component at $z/l = 0.5$ . (a,c,e) Represent the smooth-wall case while (b,d, f) represent the lined-wall case. All figures show the lined-wall case without incident sound.

5.2. Mean velocity profiles

Figure 14 compares the smooth-wall mean streamwise velocity profiles at $x/L=0$ with the ones at three streamwise positions over the liner, $x/L=0,0.45$ and $1$ . At the beginning of the liner ( $x/L=0$ ), the introduction of the lined section changes the pressure gradient compared with the smooth-wall configuration, primarily affecting the outer-layer velocity profiles. The velocity profile at $x/L=0.45$ reveals a pronounced reduction of the velocity in the wall-normal region ( $10^{-2} \lt y/h \lt 5 {\times }10^{-2}$ ). The average streamwise velocity reduction in this region, relative to the inlet profile, is $\Delta U =- (U_{x/L = 0.45} - U_{x/L=0})/U_{x/L=0} = 14\,\%$ . This result is consistent with previous findings for turbulent flows over rough surfaces (Jiménez Reference Jiménez2004) and acoustic liners (Shahzad et al. Reference Shahzad, Hickel and Modesti2023b ). The streamwise velocity near the wall continues to reduce moving at $x/L = 1$ , but the rate of reduction is less strong than for the first half of the lined section ( $\Delta U =- (U_{x/L = 1} - U_{x/L=0.45})/U_{x/L=0.45} = 3\,\%$ ).

Figure 14.

Time average of streamwise velocity profile in outer coordinates sampled at three locations over the liner compared with the smooth case. No incident sound wave.

The impact of the grazing acoustic wave on the streamwise development of the channel flow is shown in figure 15. Here, the mean flow velocity profiles over the liner with acoustics are compared with those without acoustics. Each column reports a streamwise location, moving downstream from left to right. Each row describes the impact of a different parameter of the acoustic wave. At the beginning of the lined section, $x/L=0$ , the grazing acoustic waves weakly affect the mean velocity profiles, independently of the SPL (figure 15 a), the frequency of the plane wave (figure 15 d), and the propagation direction of the acoustic wave (figure 15 g). The acoustic waves modify the velocity profile in the region in the range 0.01 $\lt y/h \lt$ 0.04; these differences are more evident, even if weak, i.e. $\Delta U \lt 4\,\%$ , when increasing the SPL and near the resonance frequency of the liner, estimated without flow. Furthermore, the upstream acoustic source causes a slightly higher velocity deficit with respect to the downstream case.

The most relevant impact of the acoustic waves on the mean flow is found at $x/L = 0.45$ . Here, the mean flow profile varies depending on the characteristics of the acoustic wave. The trends are similar to those found at $x/L = 0$ but enhanced. The observed dependence on the excitation frequency may be explained by the different proximity of the excitation to the liner’s resonance frequency in the presence of grazing flow (figure 11). The dependency on the propagation direction is more evident at this streamwise location, with a larger velocity deficit for the upstream source.

At the end of the lined section, the mean velocity profiles do not depend on the characteristics of the plane acoustic wave. This might be due to the flow recovering immediately from the perturbations caused by the acoustic-induced field.

Figure 15.

Time-averaged velocity profiles sampled at (a,d,g) $x/L = 0$ , (b,e,h) $x/L = 0.45$ and (c, f,i) $x/L = 1$ . The impact of the plane wave properties: (a,b,c) SPL, (d,e, f) frequency and (g,h,i) propagation direction of the acoustic wave is described.

5.2.1. Streamwise distribution of the boundary layer integral quantities

Previous analyses have shown that the presence of the liner affects the mean streamwise velocity profiles in the region $10^{-2} \lt y/h \lt 5 {\times }10^{-2}$ . To investigate this effect in more detail, the streamwise evolution of the boundary layer thickness $\delta$ and the boundary layer displacement thickness $\delta ^*$ is described. The two quantities are defined as

(5.1) \begin{align} U(y=\delta ) = 0.99 U_{0}, \\[-12pt]\nonumber \end{align}

(5.2) \begin{align} \delta ^* = \int _0^{h} \left (1 - \frac {U(y)}{U_{0}}\right ) \, {\rm d}y. \end{align}

The boundary layer thickness, in this context, is used to measure the impact of the liner on the velocity profile. The reasons are: (i) the flow within the channel is not symmetric with respect to the centreline because the acoustic liner is present only on one side; (ii) the flow is not yet fully developed, as also happens in the reference experiment. However, since $\delta$ , is by definition an arbitrary quantity, we quantified also the streamwise variation of the displacement thickness $\delta ^*$ , which better indicates near-wall flow gradients and is an important parameter for characterising the flow profile in the semi-empirical models used for computing impedance (Nayfeh et al. Reference Nayfeh, Kaiser and Shaker1974; Yu et al. Reference Yu, Ruiz and Kwan2008). These quantities are reported in figures 16 and 17 and are made non-dimensional with respect to the corresponding value for the smooth wall at $x/L=0$ .

Figure 16.

Streamwise development of $\delta$ over the liner. (a) Effect of acoustic waves’ amplitudes at a fixed frequency of 1400 Hz; (b) effect of the acoustic waves’ frequency at a fixed amplitude of 145 dB; (c) effect of different propagation directions at a fixed frequency of 1400 Hz and amplitude of 145 dB.

Figure 17.

Streamwise development of $\delta ^*$ over the liner. (a) Effect of different acoustic waves’ amplitude at a fixed frequency of 1400 Hz; (b) effect of acoustic waves’ frequency at a fixed amplitude equal to 145 dB; (c) effect of different propagation directions at a fixed frequency of 1400 Hz.

The streamwise development of $\delta$ in the smooth-wall case (figure 16) confirms that the flow is not yet fully developed in the lined section. As a matter of fact, $\delta$ shows a plateau from $x/L\approx 0.75$ . The presence of the liner increases $\delta$ by approximately 25 %. The streamwise development of $\delta$ varies weakly when the acoustic wave is present. It is interesting to notice that the effect of the SPL, frequency and direction of the acoustic waves are visible up to $x/L\approx 0.8$ or $x/\delta _{\textit{smooth}} \approx 10$ . Beyond this point, the curves tend to converge, suggesting that the influence of the acoustic perturbation may diminish once the turbulent flow starts adapting to the lined surface. The weak dependence of $\delta$ on the features of the acoustic wave aligns with the previous observations (figure 15) that the presence of the acoustic liner dominates the mean flow variation.

The streamwise development of $\delta ^*$ (figure 17) shows a more pronounced increase with respect to the smooth wall up to 30 % and depends on the features of the acoustic plane wave. More in detail, it increases in the presence of the acoustic wave, it grows more with higher SPL (figure 17 a), at frequencies close to the resonance frequency of the liner (figure 17 b) and it depends on the propagation direction of the acoustic wave (figure 17 c). In contrast to $\delta$ , $\delta ^{*}$ emphasises the differences between upstream and downstream acoustic waves. This behaviour highlights the influence of wave direction on near-wall velocity deficit.

The step-like behaviour of $\delta ^*$ , in correspondence with each cavity of the acoustic liner, is due to the near-wall flow modifications induced by the orifices. This effect is further enhanced by the acoustic-induced flow field. These flow features will be visualised and described in the next section.

5.3. Impact of flow development on the near-wall and in-orifice flow field

The observed changes in boundary layer integral parameters are driven mainly by the presence of the liner. The introduction of the acoustic liner modifies the near-wall flow, which is reflected in an increase in the boundary layer displacement thickness. To further investigate this aspect, figure 18 presents contour plots of the streamwise and wall-normal velocity components at the first and last cavities in the absence of an acoustic plane wave.

Figure 18.

Contour of mean velocity component (a,c) and wall-normal velocity component (b,d) for the flow-only case. (a,b) First cavity, (c,d) last cavity. (e) Difference in the streamwise velocity defined as $\Delta U = (U_{\textit{last cav.}} - U_{\textit{first cav.}})$ . (f) Difference in the wall-normal velocity defined as $\Delta V = (V_{\textit{last cav.}} - V_{\textit{first cav.}})$ . All results refer to the lined-wall case without acoustic sources.

In both cavities, the wall-normal velocity contours reveal a quasi-steady vortex occupying the upstream portion of each orifice, consistent with earlier observations (Tam & Kurbatskii Reference Tam and Kurbatskii2000b ; Avallone et al. Reference Avallone, Manjunath, Ragni and Casalino2019). The vortex strength, however, slightly differs between cavities and between successive orifices. Looking at streamwise velocity components over the two cavities (figure 18 a,c), for both cases, downstream of the orifices, the flow is first displaced away from the wall and then tends to reattach. The displacement in the downstream direction causes a reduction of the near-wall streamwise velocity gradient from the most upstream to the most downstream location. This is quantified in figure 18(e) where the difference between the flow field over the cavities is quantified as $\Delta U = (U_{\textit{last cav.}} - U_{\textit{first cav.}})$ . Over the last cavity, near the wall, the velocity decreases by up to 12 % of $U_0$ . As a consequence, over the orifices, the streamwise velocity component gradient in the wall-normal direction ${\rm d}U/{\rm d}y$ is also less strong for the most downstream cavity, thus resulting in localised high amplitude $\Delta U/U_0$ regions within the orifice of approximately 10 %. The latter might affect the strength of the quasi-steady vortex that is formed within each orifice, as suggested by an increase in the wall-normal velocity in the upstream part of the last cavity’s orifice by $\Delta V = V_{\textit{last cav.}} - V_{\textit{first cav.}} = 0.03\,U_0$ (figure 18 f).

To quantify this effect, figure 19 examines the local shear-related scale $(\nu |{\rm d}U/{\rm d}y|)^{1/2}$ , which has the same units as the friction velocity, for the first and last cavities in the flow-only configuration. The contours reveal a streamwise attenuation of shear intensity over the orifice in the downstream direction, as clearly shown in figure 19(c), which presents this quantity at $y/h=0$ . In the first cavity, the upstream orifice exhibits the strongest shear layer; at the downstream orifice, the magnitude is roughly halved, and the high-intensity shear region extends less far downstream. This trend is even more pronounced in the last cavity: the shear over its first orifice is already weaker than that in the first cavity, and it diminishes sharply over the second orifice. Moreover, $\nu (|{\rm d}U/{\rm d}y|)^{1/2}$ over the last cavities experiences a reduction in magnitude of up to approximately 30 % compared with the first cavity. The stronger shear layer at the first cavity obstructs the penetration of the acoustic wave into the orifice with respect to what happens at the last cavity. This variation can be interpreted as a spatial asymmetry in the flow field, caused by the growth of the boundary layer displacement thickness. The acoustic wave propagating in the direction opposite to the mean flow will see different near-wall flow gradients and different shears over the orifices with respect to the wave which propagates in the same direction as the flow. Furthermore, the development of the flow highlights the relevance of near-wall flow features and the need to correctly account for the boundary layer displacement thickness.

Figure 19.

Contour of $\nu (|{\rm d}U/{\rm d}y|)^{1/2}$ for (a) the first and (b) the last cavities for the flow-only case. (c) Line plots of $\nu (|{\rm d}U/{\rm d}y|)^{1/2}$ sampled at $y/h=0$ above the orifices, comparing the first and last cavities and highlighting the downstream weakening of the shear layer. Plots show the case without incident sound waves.

5.4. Impact of SPL on the near-wall and in-cavity flow field

The preceding analysis demonstrated that the streamwise development of the grazing flow alters the near-wall and in-orifice velocity fields, primarily through a progressive reduction in the streamwise velocity gradient and weakening of the shear layer over the downstream orifices. These modifications influence the vortex dynamics and flow penetration within the cavities, even in the absence of acoustic excitation. To further understand the coupling between fluid dynamics and acoustics, this section describes how the amplitude of the acoustic wave affects the mean and fluctuating flow fields in the vicinity of the orifices. Analyses were conducted for the sixth cavity (i.e. $x/L$ = 0.45).

Figure 20.

Contour of the mean and standard deviation of the streamwise and wall-normal velocity components for the cases: flow only, lined (a,b,c,d), flow and upstream acoustic source with SPL = 130 dB and $f$ = 1400 Hz (e,f,g,h) and flow and upstream acoustic source with SPL = 145 dB and $f$ = 1400 Hz (i,j,k,l).

Figure 20 presents the contours of the mean streamwise and wall-normal velocity components, along with their standard deviations. The case with acoustic excitation is compared with the case without acoustic excitation. In the absence of acoustic waves (figure 20 a–d), the contour of the standard deviation of the streamwise velocity component $\sigma _u/U_0$ (figure 20 c) highlights the presence of high-intensity fluctuations over the first orifice and after the second one. These fluctuations are caused by vortices shed near the wall, which are also responsible for the displacement of the flow away from the wall. On the other hand, the turbulent flow causes high-intensity fluctuations of the vertical velocity component $\sigma _v/U_0$ (figure 20 d) at the centre of the orifice and the downstream edge of the face sheet. The location of the peak displaces slightly upstream in the second orifice with respect to the first one.

The presence of acoustic waves, with only one frequency reported for the sake of conciseness, weakly alters the mean flow, but largely the velocity fluctuations, as can be seen by comparing figures 20(c) and 20(m). As the amplitude of the acoustic wave increases, the flow is progressively displaced away from the wall, a behaviour analogous to what occurs in turbulent boundary layers subjected to wall-normal blowing or flow over permeable surfaces (Jiménez Reference Jiménez2004; Zhang & Bodony Reference Zhang and Bodony2016; Avallone et al. Reference Avallone, Manjunath, Ragni and Casalino2019; Spillere et al. Reference Spillere, Bonomo, Cordioli and Brambley2020). This results in a stronger velocity deficit near the wall and elevated streamwise fluctuations in the outer layer. Increasing the SPL to 145 dB, the amplitude of the streamwise velocity fluctuations near the wall increases; $\sigma _u/U_0$ shows a large amplitude further away from the wall, which increases at the downstream orifice. Also, $\sigma _v/U_0$ varies substantially with respect to the other two cases; the peak of $\sigma _v/U_0$ is localised at the downstream edge of both orifices, and it increases in amplitude at the more downstream edge.

5.5. Impact of different directions of propagation of the acoustic wave

This section investigates how the propagation direction of the acoustic wave affects the flow field. To isolate this effect, the mean velocity field is analysed at $x/L=0.54$ , corresponding to the seventh cavity. Here, the incident SPL is the same for both directions (SPL $\approx 137$ dB), yet the displacement thickness still differs by $\approx 3\,\%$ .

Contours of the time-averaged and standard deviation of both the streamwise and wall-normal velocity components are shown in figure 21 for the cases: flow only, flow and upstream acoustic source and flow and downstream acoustic source. Only the case with acoustic plane wave amplitude equal to 145 dB and $f=1400$ Hz is reported for the sake of conciseness, because similar observations can be made for the other cases.

Figure 21.

Contour of the mean and standard deviation of both the streamwise and the wall-normal velocities for the cases: flow only, lined (a,b,c,d), flow and upstream acoustic source with SPL = 145 dB and $f$ = 1400 Hz (e,f,g,h) and flow and downstream acoustic source with SPL = 145 dB and $f$ = 1400 Hz (i,j,k,l).

The propagation direction of the acoustic wave with respect to the mean flow has a minor impact on the time-averaged flow field when the local amplitude of the acoustic wave is similar. Contours of $U/U_0$ and $V/U_0$ do not show relevant differences between the two cases. The only visible difference is that, downstream of the second orifice, the flow is slightly less displaced away from the wall when the acoustic wave propagates in the opposite direction to the mean flow, which is consistent with the $\delta ^*$ evolution (figure 17 c). The strong similarities between the two cases explain the previous observation that the mean flow is affected more by the presence of orifices than by the acoustic field. However, the propagation direction of the acoustic wave has more impact on the amplitude of the velocity fluctuations than on their spatial distribution. For the upstream acoustic source, $\sigma _u/U_0$ is higher, and the region with high amplitude extends spatially further downstream and away from the wall than in the case with a downstream source. This might be caused by the convective nature of the flow that tends to drag downstream the velocity fluctuations induced by the acoustic-induced velocity within the orifice. For the downstream acoustic source, by contrast, it appears that the regions with high amplitude velocity fluctuations are confined near the wall.

The direction of propagation of the acoustic wave also impacts the amplitude of the vertical velocity fluctuations within the orifice. As illustrated in figure 22, the peak of $\sigma _v/U_0$ shifts towards the downstream edge of each orifice and penetrates deeper into the cavity when acoustic forcing is present. For the cases with acoustics, the peak value nearly doubles at $y/\tau =-1$ compared with the flow-only case. The distribution remains topologically similar for the two cases. Therefore, it can be concluded that, at similar SPL, the wave direction weakly alters the spatial distribution of the in-orifice flow fluctuations but modulates their intensity. The larger amplitude of the velocity fluctuations observed in the upstream-propagating wave case may be linked to the different boundary layer displacement thicknesses.

Figure 22.

Spatial distribution of $\sigma _{v}/U_0$ into two adjacent orifices of the seventh cavity at (a) $y/\tau = 0$ , (b) $y/\tau = -0.5$ , (c) $y/\tau =-1$ . Lined flow-only case, flow and acoustics with SPL = 145 dB $f = 1400$ Hz and upstream acoustic source, SPL = 145 dB $f$ = 1400 Hz and downstream acoustic source.

5.6. Streamwise evolution of the standard deviation of the velocity fluctuations

To assess how the liner surface geometry and acoustic forcing influence the outer layer of the turbulent flow, the standard deviations of the streamwise and wall-normal velocity components are examined at six streamwise stations along the liner, as shown in figure 23. The analysis focuses on cases with a fixed excitation frequency of 1400 Hz, and varying SPLs and propagation directions. Results are compared against both the flow-only and smooth-wall baselines. Similar trends were observed at other frequencies and are omitted here for brevity.

At $x/L = 0$ (station 1), no appreciable differences are observed in the streamwise velocity standard deviation profiles between the smooth-wall and lined configurations. Similarly, acoustic forcing has no discernible effect on the flow when applied over a smooth wall. In contrast, the wall-normal velocity fluctuations exhibit a slight increase in the region $10^{-2} \lt y/h \lt 1$ compared with the smooth-wall case. However, differences become more visible at the second streamwise station ( $x/L=0.45$ ) where the lined flow-only case exhibits a distinct hump at $y/h \approx 4 {\times }10^{-2}$ . This outer-layer peak is consistent with previous observations of zero-pressure-gradient (ZPG) turbulent boundary layers over porous or perforated surfaces, where wall-normal transport promotes the formation of secondary motions (Scarano, Jacob & Gowree Reference Scarano, Jacob and Gowree2024). The hump at $y/h \approx 4 {\times }10^{-2}$ is amplified by the presence of the acoustic wave. The amplitude of the hump increases as the amplitude of the acoustic wave increases, and when the acoustic wave propagates as the mean flow does. For the case with the acoustic wave propagating in the direction opposite to the mean flow, no difference is visible with respect to the flow-only case. This is because the amplitude of the acoustic wave at this location has decreased to approximately 130 dB (figure 7). Closer to the wall, almost no variation is found with respect to the smooth-wall case. Considering the standard deviation of the wall-normal velocity component, a similar conclusion can be drawn. Specifically, an increase in $\sigma _v/U_0$ is observed at the same wall-normal location ( $y/h \approx 0.04$ ), consistent with the upward displacement of fluid from the near-wall region toward the outer layer. This displacement becomes more pronounced in the case with an upstream acoustic source at SPL = 145 dB, highlighting the role of acoustic forcing in enhancing momentum exchange and vertical transport within the near-wall flow. At $x/L = 0.73$ (i.e. the third streamwise station), the outer peaks of both the streamwise velocity and wall-normal velocity variance are enhanced with respect to the smooth-wall case. At this station, for both the streamwise and wall-normal velocity, it is more difficult to spot differences in the outer hump amplitude between the cases with varying both the amplitude and direction of the acoustic wave, meaning that the effect of the acoustic is less evident. The reason behind this behaviour is believed to be driven by two different causes: (i) the acoustic wave is attenuated when progressing over the liner, (ii) the turbulent flow is adapting to the change in the surface. This is further supported by the standard deviation profiles at station 4. As a matter of fact, at this location, almost no differences are found in the amplitude of the outer hump. The turbulent flow seems to be adapted to the lined surface. Even though the phenomenology of the set-up under investigation slightly differs from a ZPG turbulent boundary layer, the behaviour of the flow over the lined surface resembles the behaviour of turbulent flow over rough surfaces, where outer-layer features stabilise further downstream (Jaiswal & Ganapathisubramani Reference Jaiswal and Ganapathisubramani2024). A slight reduction in the inner peak amplitude is also observed, consistent with earlier studies of turbulent flow over acoustic liners (Shahzad et al. Reference Shahzad, Hickel and Modesti2023a ), and this inner-layer attenuation is modestly modulated by the acoustic forcing.

At $x/L=1.0$ (station 5), the flow encounters a sudden transition from the lined surface to a smooth wall. Here, the outer-layer hump is modulated once again by the amplitude and direction of the acoustic wave. The largest deviation from the lined flow-only case is found in the downstream case at SPL = 145 dB. This region is characterised by the formation of secondary motions induced by the abrupt surface change, which are further intensified by the acoustic forcing. As a result, both streamwise and wall-normal velocity fluctuations show increased amplitude modulation by the acoustic wave properties.

Finally, at the most downstream location, $x/L = 1.5$ , where the smooth condition is recovered downstream of the liner, variance profiles for both wall-normal and streamwise velocity components resemble the ones for the smooth wall. This indicates that after the surface transitions back to a smooth one, the flow tends to recover to a canonical state, regardless of whether acoustic forcing is present or not.

Figure 23.

Streamwise velocity standard deviation (top row) and wall-normal velocity standard deviation (bottom row) for the different test cases reported in the legend.

6.1. Acoustic-induced velocity field within the orifices

Building on the previous findings, it is evident that the combined effects of the lined surface and acoustic excitation influence both the time-averaged and fluctuating flow fields near the wall. These modifications, in turn, alter the flow dynamics within the orifices. The behaviour of the flow in and around the orifices might serve as a direct indicator of acoustic energy dissipation. In this section, details of the acoustic-induced velocity within the orifices are described. For this purpose, the acoustic-induced velocity fields have been obtained using the triple value decomposition approach described in § 2. They are shown in figure 24 for the case with SPL equal to 145 dB and for the three frequencies. For each case, the inflow and outflow phases are shown, corresponding to the maximum and minimum velocity within the upstream orifice.

Figure 24.

Contour plots of the acoustic-induced velocity at the peak of the inflow phase and peak of the outflow phase. All cases have an incident sound with amplitude equal to 145 dB and the three frequencies analysed.

The reconstructed velocity is normalised by the theoretical acoustic velocity proposed by Morse, Ingard & Morse (Reference Morse, Ingard and Morse1968) as done in previous studies (Zhang & Bodony Reference Zhang and Bodony2016; Léon et al. Reference Léon, Méry, Piot and Conte2019), even if valid in the absence of grazing flow

(6.1) \begin{equation} v^*_{ac}=\frac {|{p^\prime }|}{\rho \omega (\tau +0.8d)} \frac {1}{\sqrt { \left [\left(\frac {\omega _H}{\omega }\right)^2-1 \right ] + \left ( \frac {\omega _H}{\omega Q}\right )^2 }}, \end{equation}

where $p^\prime$ is the pressure fluctuation, obtained from the SPL, $\omega$ and $\omega _H$ are the forcing and resonant frequency, $Q$ is the quality factor and $\rho$ is the density. A constant value of $Q = 7$ was used for both the no-flow and flow conditions. This value was determined as the best fit to match the acoustically induced velocity in the no-flow case, resulting in velocity values of 6.76, 23.66 and 12.37 m s⁻¹ for acoustic wave amplitudes of 145 dB at frequencies of 800, 1400 and 2000 Hz, respectively.

In the absence of grazing flow, the velocity contours exhibit a slightly asymmetric spatial distribution for all frequencies and for both the inflow and outflow phases. The asymmetry concerning the centre of the orifice $x/d=0$ is due to the grazing acoustic wave causing a flow reversal region near the upstream edge. This is true for both the inflow and outflow cases. Additionally, the orifices exhibit mirrored velocity distributions, suggesting a mutual interaction between them. As expected, the amplitude of the acoustic-induced velocity is higher near the resonance frequency.

When the grazing flow is introduced, the spatial distribution of the acoustic-induced velocity in the orifice changes. The quasi-steady vortex formed on the upstream edge of the orifice opposes the penetration of the acoustic velocity. Therefore, only the downstream half of each orifice is largely subjected to the periodic acoustic-induced motion. The spatial distribution is slightly affected by the frequency of the acoustic wave. Interestingly, for all the frequencies, the amplitude of the maximum velocity is very similar between the no-flow and the grazing flow cases. These observations agree with previous observations that one of the effects of the grazing flow is to reduce the effective porosity of the liner (Zhang & Bodony Reference Zhang and Bodony2016; Avallone & Casalino Reference Avallone and Casalino2021). Finally, even though the wavelength exceeds the size of the cavity, there is still a delay in the time at which maximum velocity is reached at the two successive orifices.

By comparing the spatial distributions of the acoustic-induced velocity, it can be noticed that, while for the inflow phase, the acoustic-induced velocity penetrates within the cavity, for the outflow phase, the acoustic-induced velocity weakly penetrates within the turbulent grazing flow, thus suggesting that the flow opposes the ejection of acoustic-induced velocity from the cavity.

As described earlier, the streamwise evolution of the flow modifies the shear layer over the orifices and, therefore, depending on the shear strength and the local SPL, the way the acoustic wave penetrates within the orifice might differ. For this reason, the acoustic-induced velocity fields are shown for the first and last of the cavities in figure 25. The contours of the acoustic-induced flow field are compared for the upstream- and downstream-propagating acoustic waves. Only the case with the acoustic wave amplitude equal to 145 dB and frequency equal to 1400 Hz is shown for the sake of brevity.

The figure highlights the role of the shear layer over the orifices. As a matter of fact, for a given SPL, over the last cavity, the shear-layer strength is lower, as it was shown in figure 19. Consequently, both during the inflow and outflow phases, the high-intensity region of the acoustic-induced velocity is more spread within the orifice. This behaviour is particularly evident when comparing the first and last orifices – i.e. those initially exposed to the acoustic wave. This suggests the streamwise development of the flow over the liner has an impact on how each cavity contributes to the acoustic dissipation.

Figure 25.

Contour plots of the acoustic-induced velocity inside the orifices for different acoustic source locations relative to the grazing flow. The top row represents the peak of the inflow phase, while the bottom row corresponds to the peak of the outflow phase. Acoustic source: SPL = 145 dB, f = 1400 Hz. The phase locking has been performed on the first orifice of each cavity with respect to the acoustic source propagation direction.

6.2. Mass flow rate within the orifices

To quantify the impact of the flow development, and in particular of the displacement of the near-wall flow over the liner, the phase-averaged mass flow rate $\dot {m} (\phi )$ through each cavity has been calculated as

(6.2) \begin{equation} \dot {m} (\phi ) = \int _{A_o} \rho \!\left (\boldsymbol{x,\phi }\right )\, \tilde {\boldsymbol{v}}\!\left (\boldsymbol{x, \phi } \right ) \boldsymbol{\cdot }\boldsymbol{n}\, \mathrm{d}A_o, \end{equation}

where $A_o$ is the area of the eight orifices. Therefore, the mass flow rate is computed as the integral of the mass flux through the eight orifices of the cavity.

Figure 26.

Comparison of the phase-locked mass flow rate over a cycle for a whole cavity; (a) SPL = 130 dB, (b) SPL = 145 dB.

Figure 26(a) shows the phase-averaged mass flow rate over one acoustic cycle for plane waves with SPL equal to 130 and 145 dB over the first and last cavities. For the lowest SPL, the mass flow is one order of magnitude lower than at 145 dB. However, for both values of SPLs, the mass flow decreases in the presence of grazing flow, because of the reduction of the effective area due to the quasi-steady vortex. At 130 dB, there is almost no difference between the upstream and downstream cases. Based on the previous findings, this is likely related to the fact that the amplitude of the acoustic source is of the same order as turbulent velocity fluctuations. With SPL = 145 dB, the mass flow rate is higher over the last cavity. The weaker shear layer over the last cavity leads to an increased capability of the acoustic wave to penetrate inside the cavity, as described earlier. The higher-frequency fluctuations visible for both cases in the presence of flow are due to the acoustic fluctuation associated with the cavity depth mode that triggers a shear-layer instability over the orifice, thereby generating a Rossiter-like mode. This feature is identified at both SPLs, and it also persists in the absence of acoustic forcing. This confirms that it is not driven by the external excitation but rather corresponds to an intrinsic aerodynamic mode of the system, influenced by the cavity geometry and orifice shape (Dacome, Siebols & Baars Reference Dacome, Siebols and Baars2024).

Theoretical background on such cavity modes is provided in Panton & Miller (Reference Panton and Miller1975), while recent experimental evidence of similar high-frequency tones has been reported by Dacome et al. (Reference Dacome, Siebols and Baars2024). Moreover, a modal decomposition analysis, performed on a subset of the present dataset (Scarano et al. 2025), revealed a mode corresponding to the self-excited tone at 14 kHz.

Figure 27 reports the streamwise distribution of mass flow computed as the integral over the inflow phase (0 to $\pi$ in figure 26 a–b). It is evident that, for a given SPL, i.e. 145 dB, the mass flow is higher by approximately 22 % at the last cavity for the downstream acoustic source with respect to the first cavity for the upstream acoustic source. This is more evident for the 145 dB case because the amplitude of the acoustic fluctuations is higher; it is less evident for the 130 dB case, but the trend is comparable.

Referring to the SPL decay presented in figure 7, the different slopes between the different acoustic source locations may be explained by the different shear strengths over the orifices along the liner. Therefore, when the acoustic wave propagates in the direction opposite to the mean flow, it first encounters a thicker boundary layer displacement thickness and orifices with weaker shear strength, and then ones with larger shear strength and smaller boundary layer displacement thickness. The opposite happens for the upstream-located acoustic source. This supports the idea that, in conventional facilities, the boundary layer cannot be assumed to be thin. Its development over the liner must be accounted for, as it plays a crucial role in the interaction between the acoustic field and the flow. Furthermore, this finding suggests that the flow development must be considered in eduction methods, which is not the case for the methods currently adopted.

Figure 27.

Streamwise evolution of the mass flow rate computed as the integral over the inflow phase.