2.1 RANS-based duct flow simulation

The Pitot data available within the GFIT is sufficient to estimate a large portion of the grazing flow profile but several additional parameters are needed from which to construct a viable computational model. To provide additional parameter estimates, a Reynolds-averaged Navier–Stokes (RANS) simulation was conducted of a large portion of the GFIT, including the upstream plenum and a significant portion of the exit ductwork. The NASA-provided CAD files were used to construct a structured grid on which the commercial CFD package Fluent was used to simulate the steady flow within the GFIT using the standard $k$ – ${\it\epsilon}$ turbulence model. Only hardwall conditions were used in the RANS prediction with adiabatic thermal conditions and the rectangular symmetry of the tunnel was used to reduce the grid requirements by using symmetry plane boundary conditions. The known plenum pressure and atmospheric temperature were used to estimate the total pressure and total temperature coming into the plenum, while the exit pressure and laminar-to-turbulent transition location were adjusted until the predicted and measured velocity profiles matched to a reasonable degree, as shown in figure 3. At the Pitot location the local Mach number was approximately 0.6 while at the liner location it was closer to Mach 0.5. As table 1 shows, several parameters were estimated from both experimental and simulation datasets and found to have differences less than 10 %. Other than to ensure grid convergence, no attempts were made to assess the sensitivity of the RANS predictions to the particular model chosen; the present results are similar to those found in Tam et al. (Reference Tam, Pastouchenko, Jones and Watson2014) using an entirely different method.

The RANS results provide confidence that the basic thermodynamic state and flow field throughout the GFIT are sufficiently characterized to permit the detailed direct numerical simulations to proceed. However, they also show that the Reynolds number based on the momentum thickness is approximately 23 000 at the liner test section, which is too large for a direct numerical simulation to be viable. Because prior experiments (Goldman & Panton Reference Goldman and Panton1976) suggest that the ratio of the momentum thickness ${\it\delta}^{\dagger }$ to the orifice diameter $d$ is an important parameter, it was decided to artificially lower the Reynolds number to $Re_{{\it\delta}^{\dagger }}\approx 2300$ while keeping the ratio ${\it\delta}^{\dagger }/d$ and the Mach number fixed at 2.0 and 0.5, respectively. The consequence of this modelling assumption is that the energy in the scales of motion smaller than the orifice diameter are reduced. However, in wall units for the reduced Reynolds numbers, the orifice diameter is $d^{+}\approx 114\gg 1$ so that it is reasonable to assume that only the larger scales of motion within the turbulent boundary layer are able to effectively interact with the orifice, and that smaller-scale motions are dynamically unimportant. There are not currently means to perform a posteriori tests on this modelling assumption.

Figure 3.

Velocity profile comparison between the experimental data and simulation results. Legend: ——, simulation data at the exit of the plane of the duct; –▵–, experimental data at the exit plane of the duct; - - - -, simulation data at the centre of the test section of the duct.

Table 1.

Comparisons of the turbulent boundary-layer features between the simulation results and experimental data.

2.2 Geometric model

The liner geometry tested within the GFIT has a facesheet thickness of ${\it\tau}=0.64$  mm (refer to figure 4(a) for symbol definitions), an orifice diameter of $d=0.99$  mm, a cavity depth of $D=38.1$  mm and a cavity width of $\ell =5.49$  mm. The open-area porosity is 8.7 % which corresponds to, on average, 8–9 orifices per cavity; the spacing of the orifices is not centred over the cavities so, at times, an orifice may span two or three cavities and thus be partially blocked by the cavity side walls. Simplification of the geometry was used to reduce the computational cost by including only one orifice per cavity and increasing the inter-orifice spacing so as to reduce their interaction. The resulting geometry is shown in figure 4(b) where the streamwise and spanwise lengths of the exterior domain, $L_{x}\approx 240d$ and $L_{z}\approx 80d$ , respectively, result in an assumed periodicity of the liner as shown in figure 4(c). Reducing the orifice-per-cavity count from 8–9 to 1 reduces the effective open-area porosity to 0.99 %. Finally, note that the present simulations use a temporally evolving boundary-layer model where the thickness of the layer grows with time rather than in the downstream direction. Consequences of these modelling assumptions will be discussed later in this paper.

Figure 4.

Several views of the computational model. (a) Important liner dimensions (adapted from Zhang & Bodony (Reference Zhang and Bodony2012)); (b) single liner with isolated orifice under a grazing flow; and (c) assumed periodicity.

2.3 Grid design

As noted in § 1, the simulations include a turbulent boundary layer grazing a circular orifice with a hexagonal backing cavity in the presence of an externally imposed incident acoustic field. The liner geometry is identical to that used previously (Zhang & Bodony Reference Zhang and Bodony2012) without a grazing flow but with an enlarged domain above the liner facesheet. Four meshes are arranged in an overset – sometimes called Chimera – setting where multiple grids can overlap and their common information is communicated by interpolation (Bodony et al. Reference Bodony, Zagaris, Reichert and Zhang2011; Reichert, Heath & Bodony Reference Reichert, Heath and Bodony2012). Details about the individual grids are given in table 2 where grid 1 is used to hold the turbulent boundary layer while grids 2–4 are primarily to represent the circular orifice and its backing cavity. The individual grids are relatively simple to generate while their interpolation data was determined using a NASA-developed pre-processing tool (Suhs, Rogers & Dietz Reference Suhs, Rogers and Dietz2003) coupled to one developed at the Air Force Research Laboratory to produce sixth-order interpolation stencils (Sherer, Visbal & Galbraith Reference Sherer, Visbal and Galbraith2006). The minimum grid spacing in the orifice is ${\rm\Delta}r_{min}=0.0075d$ at the orifice walls. In the $z$ -direction, a uniform grid spacing ${\rm\Delta}z_{min}=0.01d$ is used $1d$ offset above and below the orifice. At $10d$ away from the orifice, the resolution decreases smoothly but ensures ${\rm\Delta}z_{min}\leqslant 0.1d$ . Also note that the size of the upper rectangular box is set to be sufficiently large for fully uncorrelated statistics, as noted in appendix A. The in-orifice flow has been shown to be fully resolved in prior work (Zhang & Bodony Reference Zhang and Bodony2012).

Table 2.

Grid information used in the numerical simulation.

2.4 Numerical method

On each of the four grids we solve the compressible Navier–Stokes equations in a non-orthogonal coordinate system ${\bf\xi}={\it\bf\Xi}(\boldsymbol{x})$ , with ${\bf\xi}=({\it\xi},{\it\eta},{\it\zeta})^{\text{T}}$ , written in conservative form

where $\text{J}=|\partial \boldsymbol{x}/\partial {\bf\xi}|$ is the Jacobian of the grid transformation, $\hat{{\it\xi}}_{j,i}=\text{J}^{-1}\partial {\it\xi}_{j}/\partial x_{i}$ are the Jacobian-weighted metrics and $U_{i}=u_{j}\hat{{\it\xi}}_{i,j}$ are the contravariant velocities. A Newtonian fluid with Fourier’s law of heat conduction is assumed such that with ${\it\lambda}={\it\mu}_{B}-2{\it\mu}/3$ and ${\it\mu}_{B}=0.6{\it\mu}$ for air. The temperature dependence of ${\it\mu}$ and $k$ are given by power laws

with $n=0.666$ . When the equations are non-dimensionalized with respect to the ambient density ${\it\rho}_{\infty }$ , the ambient speed of sound $a_{\infty }$ , the orifice diameter $d$ and the ambient viscosity ${\it\mu}_{\infty }$ the equation of state for a calorically perfect ideal gas is

with ${\it\gamma}=C_{p}/C_{v}$ , the ratio of specific heats, fixed at 1.4. The equations are discretized using a node-centred finite difference approach of the form

where $\{\unicode[STIX]{x1D64B},\unicode[STIX]{x1D64C}\}$ are explicit summation-by-parts matrices (Strand Reference Strand1994) that are eighth-order accurate in the interior and fifth-order accurate globally. The required second derivatives use repeated first derivatives and do not provide dissipation for under-resolved flow scales so that a high-order spectral like filter is used to preserve numerical stability (Visbal & Gaitonde Reference Visbal and Gaitonde2002). Time integration uses the standard fourth-order Runge–Kutta method.

The boundary conditions listed in figure 4 and the no-slip, isothermal condition applied to all solid surfaces use the simultaneous-approximation-term formulation developed by Svärd, Carpenter & Nordström (Reference Svärd, Carpenter and Nordström2007) and Svärd & Nordström (Reference Svärd and Nordström2008). A ‘sponge’ zone (Bodony Reference Bodony2006) is used above the facesheet to accurately enforce the incident acoustic wave while simultaneously absorbing reflected acoustic waves. The incident acoustic field is planar, monochromatic and of the form,

with assumed real part, where $\boldsymbol{q}_{inc}^{\prime }=[{\it\rho}_{inc}^{\prime },\boldsymbol{u}_{inc}^{\prime },p_{inc}^{\prime }]^{\text{T}}$ , and $A=10^{\text{SPL}/20-9.701}$ for air at standard temperature and pressure. The wavenumber $\boldsymbol{k}$ and frequency ${\it\omega}$ are related by $|\boldsymbol{k}|={\it\omega}/c_{\infty }$ and the direction of $\boldsymbol{k}$ is normal to the facesheet. These waves are imposed only at top of the domain (sponge zone) and propagate to the orifice by the fully nonlinear equations of motion in the direction normal to the facesheet. The sponge zone also imposes the mean grazing flow. In doing so the present simulations differ from their experimental motivation in which a grazing acoustic field is used.

The simulation was executed at a constant time step with a CFL number of approximately 0.48, which corresponds to $10^{-6}$ of one acoustic cycle at 3 kHz. Data are collected every 1000 iterations.

Figure 5.

Momentum thickness of the temporally evolving turbulent boundary layer under a normal incident sound at different SPLs.

5.1 Mid-facesheet velocity distributions

The impact of the grazing flow on the in-orifice flow can be quickly appreciated by considering the profiles of the orifice-normal velocity taken in the orifice, along the facesheet mid-plane, as shown in figure 10. Conditions without grazing flow and with laminar and turbulent boundary layers are shown for 3 kHz, 130 dB incident sound. The no-flow cases illustrate the azimuthal axisymmetry the velocity profile has when a grazing flow is absent and the profile can be well approximated by hyperbolic tangent curves with a fitted boundary-layer thickness or with periodic pipe flow root-mean-square (r.m.s.) profiles (Zhang & Bodony Reference Zhang and Bodony2012). When a grazing flow is present in the $+x$ direction, the azimuthal symmetry is clearly broken and the inflow and outflow cycles are not qualitatively or quantitatively similar, as shown in figure 10(b,c,e,f). Note also that the peak velocity increases by a factor of 2 when the grazing flow is turbulent, and that it is concentrated on the downstream side of the orifice during the inflow cycle. During the outflow cycle the flow is more azimuthally symmetric but still not as much as for the no-flow case.

5.2 Orifice velocity time histories

To further quantitatively analyse the biased velocity profiles three probes, labelled A, B and C in figure 11, were placed in the mid-plane of the orifice to record the time history of the velocity fluctuations after the initial transient period. Figure 12 demonstrates the time history of the velocity fluctuations at 130 dB case with different grazing flow conditions. Since the velocity profile has an axisymmetric feature at a quiescent condition, the time history of the velocity fluctuations at A–C collapse. With a laminar boundary layer present, the velocity fluctuations at location A, B and C still remain sinusoidal but with different magnitudes and phases. At Probe A, the velocity fluctuation is nearly always positive, indicating an outflow-dominated motion even during the inflow phase of the incident sound. Although Probe B sits in the centre of the orifice, the velocity fluctuation observed has a positive constant mean value that suggests a biased outflow motion on top of which sits the sound-induced fluctuations. Probe C sits $0.36d$ downstream of the origin and reveals a strong motion that is roughly symmetric about $v/a_{\infty }=0$ . For the turbulent boundary-layer-induced flow in figure 12(c), the same qualitative picture emerges as in the laminar case but with significant added stochasticity. Again, on average, Probes A and B show an outflow-only and biased outflow, respectively, with roughly the same velocity magnitude peaks, albeit with an increased level of high-frequency (relative to the forcing frequency) fluctuations arising from the boundary layer. Probe C now shows a net inflow character with peak velocities that are roughly 2 times those for the laminar boundary-layer case.

Figure 13.

Velocity fluctuations at different SPL with turbulent grazing flow for simulations C1–C4 (3 kHz). Legend: - - - -, Probe A; ——, Probe B; — ⋅ —, Probe C: (a) 130 dB, (b) 140 dB, (c) 150 dB, (d) 160 dB.

Figure 14.

Velocity fluctuations at different frequencies for simulations B1–B3 (130 dB). Legend: - - - -, Probe A; ——, Probe B; — ⋅ —, Probe C: (a) 2.2 kHz, (b) 2.6 kHz, (c) 3.0 kHz.

Figures 13 and 14 demonstrate the effect of changing the sound amplitude or the frequency, respectively, on the velocity history recorded at Probes A, B and C. The velocity fluctuation magnitude increases as the amplitude of the incident sound goes up. The maximum magnitude of the fluctuation reaches roughly 10 %, 18 % and 30 % of the ambient speed of sound for 140 dB, 150 dB and 160 dB, respectively. For reference, a planar acoustic wave would induce velocities of 0.1 %, 0.4 %, 1 % and 4 % of the ambient speed of sound, respectively, indicating the importance of the hydrodynamic field. The additional fluctuations caused by the turbulent boundary layer play a less significant role as the acoustic field becomes stronger and, eventually, the acoustic field overwhelms the hydrodynamic field at 160 dB. Also note that at higher SPLs, the qualitative differences between the fluctuations at different probe locations lessens, indicating a more azimuthally symmetric in-orifice flow. At fixed 130 dB amplitude (see figure 14) the frequency of the incident sound does not appear to influence the in-orifice flow by a large amount.

5.3 Phase-averaged velocity profiles

The in-orifice velocity probes indicated qualitatively different behaviour when a grazing flow was present, with quantitative differences arising from the state of the external boundary layer. They also suggest that in- and outflow cycles arising from the incident sound may be very different. Any future fluid dynamic-based model of the in-orifice flow must take into account these flow asymmetries so, to make the details more accessible for modelling, we construct phase-averaged data conditioned on the incident sound to better describe how the incident sound, orifice wall boundary layers and the grazing boundary layer interact. Our choice is motivated by prior experimental data collected in a water tunnel (Hersh & Rogers Reference Hersh and Rogers1976).

Figure 15.

Contour plots of phase-averaged velocity $v/a_{\infty }$ with streamlines taken at different phase angles. All cases have a turbulent boundary-layer grazing flow and an incident sound of 3 kHz, cases C1–C4, with amplitude indicated: (a) 130 dB, (b) 140 dB, (c) 150 dB, (d) 160 dB.

Figure 16.

Contour plots of phase-averaged velocity $v/a_{\infty }$ with streamlines taken at different phase angles. All cases have an incident sound of frequency 3 kHz with amplitude and boundary-layer state indicated, cases D1–D4: (a) 130 dB laminar BL, (b) 130 dB turbulent BL, (c) 160 dB laminar BL, (d) 160 dB turbulent BL.

Referring to figure 11, phase-averaged data are extracted along a plane with normal in the transverse direction $+z$ , corresponding to ${\it\varphi}=0^{\circ },\,180^{\circ }$ . The plane cuts through the middle of the orifice, $x=0$ . The phase angle ${\it\phi}$ of the incident sound is taken relative to the sound as it impacts the plate with ${\it\phi}=0$ being the beginning of the inflow cycle and ${\it\phi}={\rm\pi}$ being the beginning of the outflow cycle.

The images in figure 15 show how the phase-averaged flow varies with ${\it\phi}$ and between cases of different amplitude at a 3 kHz constant frequency, cases C1–C4. Only the region in the orifice’s immediate vicinity is shown with a small portion of the facesheet visible. Considering first the 130 dB incident case in figure 15(a) it is clear that during the inflow cycle the near-orifice flow shares resemblance to a driven cavity flow wherein, for sufficiently low Reynolds number, a vortex sits in the upstream side and drives a circulatory flow within the orifice. The analogy is imprecise in that the strength of the induced flow depends both on the grazing boundary-layer-applied shear and the incident pressure field, as noted in the other phase angles between ${\it\phi}={\rm\pi}/4$ and ${\it\phi}={\rm\pi}$ . During the outflow cycle, where ${\rm\pi}\leqslant {\it\phi}\leqslant 2{\rm\pi}$ , the upstream vortex disappears and the driven cavity analogy fails. Instead, the in-orifice flow more resembles that expected from our earlier work without a grazing flow present; it is only when the ejected flow passes the plane of the facesheet that the grazing flow appears to be effective. At higher acoustic sound amplitudes the influence of the upstream vortex reduces and the analogy further degrades. It is always true that, during the inflow cycle, the fluid that is able to get into the cavity enters from the downstream side while, during the outflow cycle, the downstream bias is nearly absent. The origin of this asymmetry is, of course, the asymmetry in the flow exterior to the orifice – either the on the facesheet side or within the cavity – but its presence is crucial to effectively modelling the flow. More quantitative measures of the asymmetry will be given later in §§ 5.5.2 and 5.6.

Figure 16 again looks at the phase-averaged view of the near-orifice flow but focuses on the influence of the state of the external boundary layer for the 130 and 160 dB incident sound cases, all at 3 kHz. During the inflow cycle for 130 dB sound the laminar and turbulent boundary-layer state has a strong influence on the size and effective strength of the upstream vortex, in analogy to the driven cavity, where the stronger mean shear from the turbulent boundary layer creates a stronger induced flow within the orifice. At 160 dB incident the inflow cycle depends only weakly on the boundary-layer state, an observation that carries over to the outflow cycles at both sound amplitudes.

5.4 Dimensionless parameters

In the foregoing analysis it was shown that the in-orifice flow is influenced by the combined effects of the incident acoustic field and the grazing flow. We now examine how to scale the data.

5.5 Discharge coefficient

5.6 Pressure statistics

In estimating the discharge coefficient in § 5.5, it was observed that the qualitative time histories of point measurements taken within the orifice changed dramatically with sound amplitude. To quantify this observation, time traces of the pressure were taken within the orifice and their autocorrelations, $\mathscr{R}(t)$ , were used to compute the integral time scale according to

Because the data are not phase averaged, the integral time scale indicates whether the probe sees a pressure field that is dominated by the turbulent hydrodynamic field (with a relatively short time scale) or the incident acoustic field (with a relatively long time scale) and could be used as an alternative measure for estimating the discharge coefficient. The data are shown in figure 24, normalized by the acoustic period, where the impact of the sound field is clear: as the sound amplitude increases, its influence over the in-orifice dynamics increases and imposes its time scale upon the flow until, eventually, the impact of the turbulent boundary layer on the in-orifice flow is confined to a decreasingly small region near the upstream orifice wall. Similar results hold for the vertical velocity component but are not shown for space.

Figure 24.

Contour plot of the integral time scale of the pressure, relative to the acoustic period, over the orifice for the case of 3.0 kHz incident sound with a turbulent boundary layer at the indicated sound amplitudes: (a) no sound, (b) 130 dB, (c) 140 dB, (d) 150 dB, (e) 160 dB.

6.1 Wall shear stress

We focus first on the facesheet wall shear stress. Figure 26 shows the distribution of the time-averaged wall shear stress in the drag direction near the orifice for five cases with the orifice and cavity present. At no sound and low sound amplitude conditions the change in the wall shear stress is minimal and confined to a very small region immediately downstream of the orifice. At sound amplitudes 150 dB and above there is a noticeable increase in the wall shear stress downstream of the orifice, but also upstream of it. The latter observation is a consequence of the sound-induced pressure gradient causing the boundary layer to locally thin prior to entering the cavity. Changes to the boundary-layer state affect the change in the surface wall stress as shown in figure 27. At 130 dB incident sound amplitude the laminar boundary layer exhibits a very limited shear stress increase downstream of the orifice while at 160 dB there are two ‘wings’ of increased wall shear stress that surround a locally reversed flow region reminiscent of a jet-in-crossflow scenario (Mahesh Reference Mahesh2013).

Figure 26.

Wall shear stress distribution with the presence of the orifice and turbulent boundary layer: (a) no sound, (b) 130 dB, (c) 140 dB, (d) 150 dB, (e) 160 dB.

Figure 27.

Wall shear stress distribution with the presence of the orifice and laminar/turbulent boundary layer: (a) 130 dB laminar, (b) 130 dB turbulent, (c) 160 dB laminar, (d) 160 dB turbulent.

6.2 Surface drag

With the pointwise wall shear stress distribution evaluated with and without the presence of the orifice, we may compute the surface drag via evaluating the integrals in (6.1) and (6.2), respectively; the results are shown in figure 28. Without any incident sound the overall surface drag reduces when an orifice is punched into the facesheet, a simple consequence of the reduced surface area over which the grazing flow can generate forces. Once sound is added the facesheet drag monotonically increases. It is interesting to note that at 130 dB incident conditions, which correspond to the Langley testing conditions, the surface drag for both facesheets is approximately equal; at higher sound levels the facesheet drag increase is noticeable and on the order of 10–15 %.

Figure 28.

Comparisons of the surface drag with and without the orifice under a turbulent boundary layer under different SPLs and 3 kHz frequency.

6.3 Total drag

Including the cavity and orifice walls in the drag estimation in (6.2) has important consequences on the drag estimation as shown in comparing figure 29 to figure 30. With the additional surfaces the pressure forces are sufficiently large to cause a measurable drag increase for all cases, even when sound is absent, with the drag doubling at 160 dB. Important is the observation that, without sound, the surface stress only contribution shows a decrease in the drag while an increase is found when the cavity and orifice surfaces are included. This observation implies that attempts to assess linear drag using surface shear stress techniques (oil film, interferometry, in situ) only may report incorrect trends. Whether the boundary layer is laminar or turbulent changes the drag quantitatively, of course, but does not alter the general conclusions (see figure 30). Changing the incident sound frequency has a minor effect on the liner-induced drag, as shown in figure 31, but it is observed that the drag increase reduces for lower frequencies. Our database does not include frequencies lower than 2.2 kHz so we do not know whether the trend continues. Finally, figure 32 shows that as the sound amplitude increases, the cavity and orifice contribute more to the overall drag profile of the liner such that at 160 dB conditions their summation is approximately the same as the surface shear stress. If we consider the data in figures 30–32 together, it seems clear that the liner-induced drag is likely a function of sound amplitude and that surface stress measurements may not be the correct indicator.

Figure 29.

Drag increase associated with the resonator, under a turbulent boundary layer, for 3 kHz incident sound frequency at the indicated sound amplitudes.

Figure 30.

Drag increase associated with the resonator for 3 kHz incident sound at the indicated amplitude and boundary-layer state.

Figure 31.

Frequency dependence of the drag increase for 130 dB incident sound and turbulent boundary layer.

Figure 32.

Relative contributions of the facesheet (–▫–), cavity (–○–), and orifice walls (–▵–) to the drag for 3 kHz incident sound on a turbulent boundary layer.

The scaling of the drag increment is now considered using the data presented in § 5.4.1. We select case A in table 1 as the reference because there is no sound involved in which case the definition of the drag component increments become straightforward if we compute the difference between any case from set B to D with the reference case (case A),

Scaling the drag increment by the velocity scales $v_{ac}^{\prime }$ and $v_{peak}^{\prime }$ introduced in § 5.4.1 are shown in figure 33. Using $v_{ac}^{\prime }$ (figure 33 a) yields drag increments that are only roughly constant over the 130–160 dB amplitudes investigated. Using $v_{peak}^{\prime }$ on the other hand, shown in figure 33(b), shows that ${\it\delta}F/({\it\rho}_{\infty }v_{peak}^{\prime }U_{\infty }d^{2})\rightarrow$ 0.08 and 0.1 for the facesheet shear stress and cavity pressure component. The orifice wall pressure component shows a convincing quadratic scaling ${\it\delta}F\sim (v_{peak}^{\prime })^{2}$ and clearly indicates the importance of the hydrodynamic origin of the pressure drag.

Figure 33.

Different drag increment component scaled by different in-orifice velocities at different SPLs. (a) Scaled by the acoustic velocity $v_{ac}^{\prime }$ ; (b) Scaled by the simulation database $v_{peak}^{\prime }$ . Legends: –▪–, wall shear stress component; –●–, cavity pressure component and –▴– orifice wall pressure component (3 kHz incident sound).

7.1 Direct impedance prediction

Attention now focuses on extracting the impedance values for the simulated single degree-of-freedom liner that was also tested at the NASA Langley Research Center at the same grazing flow and incident sound conditions (Jones, Watson & Nark Reference Jones, Watson and Nark2010). There are several differences between the real and simulated liner that must be mentioned. First, the porosities of the two liners are different since the experimental liner has 8–9 orifices per cavity, randomly located, while the simulated liner has only one orifice located top-dead-centre over the cavity, thus changing the porosity ${\it\sigma}$ from 8.7 % to 0.99 %. Under the assumption that the individual orifices do not interact (a point we will return to shortly) then a porosity-scaled impedance measure makes sense. Theoretical support for this scaling is given in Melling (Reference Melling1973), Hersh et al. (Reference Hersh, Walker and Celano2003) and in Singh & Rienstra (Reference Singh and Rienstra2014), for example, in the linear and weakly nonlinear sound amplitude regimes. Further support for the use of porosity-scaled impedance was given in Zhang & Bodony (Reference Zhang and Bodony2012) using normal impedance tube experimental data and more sophisticated semi-empirical models (Ingard Reference Ingard1953; Melling Reference Melling1973; Jones et al. Reference Jones, Smith, Parrott and Smith2004). Note that the impedance is normalized by the characteristic impedance ${\it\rho}_{\infty }a_{\infty }$ .

A second difference arises because of the single orifice assumption in the simulations. The experimental liner test sample used in Jones et al. (Reference Jones, Watson and Nark2010) contained many hexagonal cavities, each with 8–9 orifices, mounted on one side of the GFIT and spatially extended in the flow direction. It stands to reason that the downstream orifices see a different environment than do the upstream orifices and, potentially, are characterized by a different impedance. The impedance eduction methods used thus represent a liner aggregate value. In contrast the simulations, with their single, isolated orifice, represent a localized value. It is not known what consequences these differences have on the values of $Z$ reported, but will be the focus of future studies.

The last differences are due to the idealization of the flow and acoustic environment between the GFIT and the simulations. The GFIT is a $2^{\prime \prime }\times 2.5^{\prime \prime }$ rectangular cross-section duct with grazing acoustic waves that have been influenced by the duct flow while the simulations assume an infinitely large facesheet with normal acoustic incidence. It is unknown how important these differences are but the results of Tam et al. (Reference Tam, Pastouchenko, Jones and Watson2014), which include duct effects and grazing incidence, suggest they are not dominant.

In addition to the physical differences in the experimental setup and conditions it is important to mention that the impedance eduction methods are also different. The simulations use a two microphone method (Dean Reference Dean1974) that connects pressure measurements taken on the liner facesheet and along the cavity end wall to estimate the local impedance that we showed to be useful when orifice effects are avoided (Zhang & Bodony Reference Zhang and Bodony2012). Using the same data collected in Jones et al. (Reference Jones, Watson and Nark2010), different hypotheses are available to link the mean flow and the pressure measurements taken within the GFIT to the estimated liner impedance. Watson and his co-workers have been examining the dependence the flow profile and eduction method has on the impedance prediction and summarized several results in Watson & Jones (Reference Watson and Jones2013) and in Jones et al. (Reference Jones, Watson, Howerton and Busse-Gerstengarbe2015) and the reader is referred to these two papers for additional details. What is relevant for the current discussion is that how the mean flow is modelled (as a plug flow or as a shear flow) and how the disturbances within the duct are modelled or approximated (as with a Pridmore–Brown equation with or without a Prony expansion) is critical to estimating the impedance.

Figure 34.

Normalized impedance prediction at 130 dB with turbulent boundary layer. Legend: data of Jones et al. (Reference Jones, Watson and Nark2010) using different impedance eduction methods: ▫, 1-D PB model; ▵, 1-D shear flow model; ▿, 1-D PB Prony model; $+$ , Plug flow Prony model. The present simulation data are denoted by ●. Details on the experimental eduction methods are given in the main text. ‘PB’ is short for Pridmore–Brown: (a) normalized resistance, (b) normalized reactance.

The predicted impedance is shown in figure 34 at 130 dB tonal amplitude at the frequencies indicated and compared with the impedance values educed using different methods in Jones et al. (Reference Jones, Watson and Nark2010). It is clear that the predictions are within the scatter of impedance values educed from the experimental data and that general trends are difficult to define, at least over the modest frequency range available. (In Jones et al. (Reference Jones, Watson and Nark2010) it is noted that the cut-on frequency for the GFIT at Mach 0.5 occurs between 2.6 and 3.0 kHz so that some of the scatter present in the experimental data at 3 kHz in figure 34 may be due to the presence of these higher-order modes. The data at 2.6 kHz and below are believed to be free of high-order mode effects.) The uncertainty in the predicted values of resistance and reactance are shown by the horizontal bars visible within the symbols and indicates that the results are converged (see Zhang Reference Zhang2014).

The fact that the present impedance prediction falls within the scatter of the experimental data suggests two points. First, the simulations represent faithfully the interaction of the sound with the cavity-backed circular orifice and led credibility to the predictions. Second, and more importantly, how the impedance is educed is as important as its value since, as figure 34 shows, the scatter in $Z$ can exceed a factor of two and indicate opposite trends.

When simulation-only impedance predictions at a fixed frequency of 3 kHz but increasing amplitude are shown in figure 35 are considered, it is clear that whether a boundary layer is present, and its state, affects the predicted impedance to a large degree for the 130 dB sound amplitude case but very little at a sound amplitude of 160 dB. It appears to be the case that the resistance increases monotonically with increasing sound amplitude, even with a grazing flow present, and that the reactance is much less sensitive to the grazing flow, observations that are consistent with empirical modelling presented in Motsinger & Kraft (Reference Motsinger, Kraft and Hubbard1995).

Figure 35.

Normalized impedance prediction at different SPLs and grazing flow conditions. Legend: $\blacklozenge$ , no grazing flow; ▪, laminar boundary layer; ●, turbulent boundary layer: (a) normalized resistance, (b) normalized reactance.

7.2 Time-domain impedance model

Our starting point is the time-domain model of the orifice-normal momentum equation studied in Zhang & Bodony (Reference Zhang and Bodony2012),

where ${\it\xi}(t)$ is interpreted as the orifice area-averaged displacement of the control volume at the centre point of the orifice and $C_{D}$ is the discharge coefficient of the orifice jet that models the effective geometry of the orifice. Further, the presence of the external turbulent boundary induces an unsteady, pseudo-random force that affects the in-orifice fluid and is modelled as the last term in the expression where $T_{j}$ represents the strength of the turbulent boundary-layer-induced pressure fluctuations at frequency ${\it\omega}_{j}$ and ${\it\phi}_{j}$ is the assumed-to-be random phase. We use the model of Goody (Reference Goody2004) to relate $T_{j}$ to the boundary-layer dynamic pressure and thickness. The cavity pressure $P_{c}$ is isentropically related to the change in density caused by the volume change $-{\it\xi}A$ .

7.3 Modelling the discharge coefficient

As we showed in the absence of a grazing flow, modelling the discharge coefficient $C_{D}$ in (7.1) can be reduced to modelling the in-orifice boundary layers with success (Zhang & Bodony Reference Zhang and Bodony2012). However the grazing flow creates a significant flow complexity for which we have not found a suitable fully theoretical model. We are therefore forced to use a semi-empirical approach where $C_{D}$ is linked to the external flow field and incident acoustic field as much as possible.

The DNS data reveals that the discharge coefficient differs significantly between the inflow and outflow cycles; therefore, it is reasonable to model them separately. Rogers & Hersh (Reference Rogers and Hersh1975), as guided by their water tunnel tests, proposed that both the inflow and outflow discharge coefficient, namely $C_{Di}$ and $C_{Do}$ , are strong functions of $v_{i}/U_{\infty }$ and $v_{o}/U_{\infty }$ , the core inflow and outflow velocities, respectively, found within the orifice, and they proposed the ‘lid model’ with a power law relation between $C_{D}$ and the inflow/outflow velocities. These formulas work well when $C_{Di}$ and $C_{Do}$ are small but fail when $v_{i}/U_{\infty }$ or $v_{o}/U_{\infty }$ exceed 0.3 as the expressions yield a predicted discharge coefficient greater than one. Therefore, we propose the modified parameterization,

where $v_{i}/U_{\infty }$ and $v_{o}/U_{\infty }$ goes as $(v_{i}/U_{\infty })^{{\it\gamma}_{1}}$ or $(v_{o}/U_{\infty })^{{\it\gamma}_{2}}$ when $v_{i}/U_{\infty }\ll 1$ or $v_{o}/U_{\infty }\ll 1$ . When $v_{i}/U_{\infty }$ and $v_{o}/U_{\infty }$ is large, the prediction goes to a constant, ${\it\beta}_{1}$ or ${\it\beta}_{2}$ , meaning a maximum discharge coefficient is achieved. The four parameters $\{{\it\beta}_{i},{\it\gamma}_{i}\}_{i=1}^{2}$ were determined from the DNS databases by using a least-squares fit informed by conditionally averaging the inflow and outflow cycles and computing the core inflow and outflow areas and by measuring the area-averaged in-orifice velocity. The results are shown in figure 36 where the present DNS data is shown alongside the original data of Rogers & Hersh (Reference Rogers and Hersh1975). The new parameterization works reasonably well and is consistent with the originally measured data.

Figure 36.

Comparison of the measured discharge coefficient with experimental data at different $v_{i}/U_{\infty }$ . Legend: ○, experimental data; - - - -, empirical formula proposed by Rogers & Hersh (Reference Rogers and Hersh1975); ——, present empirical model. ▴, ▾, simulation data: (a) inflow discharge coefficient, (b) outflow discharge coefficient.

To link the discharge coefficient to the acoustic field more directly, we appeal to the phase-averaged streamlines shown in figures 15 and 16 where it is clear that the curvature of the streamlines during the inflow portion of the cycle decreases with increasing incident sound level. Assuming that the streamline curvature is a hydrodynamic effect induced by the acoustic pressure field, one can estimate from the steady Bernoulli’s equation that the induced velocity scale is,

where the incident sound was given in (2.9). To recover the outflow velocity $v_{o}$ , applying conservation of mass requires (for constant density) that,

which yields an implicit equation for $v_{o}$ . Comparing the predicted $C_{D}$ values with those measured from the DNS databases is shown in figure 37 where it is seen that, to a large degree, the model performs well.

Figure 37.

Comparison of measured (closed symbols) and modelled discharge coefficients (open symbols) for 3 kHz incident sound at the indicated amplitude.

7.4 Reduced-order model impedance prediction

The reduced-order model predictions of the liner impedance are shown in figure 38 and are compared with the educed values from the DNS databases. At 130 dB the model does a reasonable job of reproducing the qualitative trends and the quantitative values for the frequency range shown. At higher sound amplitudes the model is able to predict the resistance up to the highest amplitude case considered; however, it fails to predict the reactance accurately and exhibits a trend with increasing amplitude that is not present in the simulation data. Despite its limitations the model does properly identify the presence and state of the boundary layer in the impedance values.

Figure 38.

Impedance predictions from reduced-order model (7.1) (open symbols) compared to those deduced directly from the DNS databases at the conditions indicated (closed symbols). See figure 34 for legend: (a) 130 dB, (b) 130 dB, (c) 3.0 kHz, (d) 3.0 kHz.