2.1. Governing equations and numerical method

To represent the propagation of sound waves through slits, the flow motion is modelled using the compressible Navier–Stokes equations, encompassing the continuity, momentum and energy equations

(2.1) \begin{align} \frac {\partial \rho }{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }\left (\rho \boldsymbol u \right )&=h_s h_{\delta }/c, \\[-12pt]\nonumber \end{align}

(2.2) \begin{align}\frac {\partial }{\partial t} \big (\rho \boldsymbol u \big) + \boldsymbol{\nabla }\boldsymbol{\cdot }\big( \rho \boldsymbol{u} \boldsymbol u^\mathsf{T} \big) + \boldsymbol{\nabla }\!p- \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{T} &=h_s h_{\delta } \boldsymbol{n}, \\[-12pt]\nonumber \end{align}

(2.3) \begin{align}\frac {\partial }{\partial t} \left (\rho E\right ) + \boldsymbol{\nabla }\boldsymbol{\cdot }\left [ (\rho E + p)\boldsymbol{u} - \boldsymbol{T} \boldsymbol{\cdot }\boldsymbol{u} \right ] &= 0,\end{align}

where $\boldsymbol u = [u,\;v]^{\mathsf{T}}$ is the velocity vector field, $c$ is the speed of sound, $\rho$ is the density, $p$ is the pressure, and $E$ is the total energy, $\boldsymbol{n}$ is the unit vector directed from the sound source toward the slit and

(2.4) \begin{gather} \boldsymbol{T} \equiv \frac {1}{\textit{Re}} \left [ \boldsymbol{\nabla }\boldsymbol{u}+(\boldsymbol{\nabla }\boldsymbol{u})^\mathsf{T} -\frac {2}{3}\left (\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}\right )\boldsymbol{I}\right ] \end{gather}

is the viscous stress tensor, with the bulk viscosity assumed negligible and thus omitted. The one-way plane wave source is modelled using two non-conservative source terms (Maeda & Colonius Reference Maeda and Colonius2017), where $h_s$ is the time-dependent amplitude of the monopole and dipole sources, and $h_{\delta }$ represents the Gaussian monopole support function provided by Maeda & Colonius (Reference Maeda and Colonius2017). The source terms in (2.1) and (2.2) represent a planar monopole and dipole source, respectively. The source terms cancel each other in one direction, resulting in a one-way planar sound source. All sound waves are assumed to be harmonic, implying that the time-dependent amplitude

(2.5) \begin{align} h_s = \widehat {A}\sin (2\pi f_s t), \end{align}

is sinusoidal with time, where $\widehat {A}$ is the pressure amplitude and $2\pi f_s$ is the angular frequency of the sound source. Closure is achieved using the ideal gas equation of state

(2.6) \begin{align} p = \left (\gamma - 1\right ) \rho e, \end{align}

which relates the internal energy, $e$ , to the total energy of the fluid, $E = e + \|\boldsymbol{u}\|^2/2$ . Here, $\gamma =1.4$ is the ratio of specific heats. Equations (2.1)–(2.6) are non-dimensionalised by the speed of sound, $c = {343}\,\textrm{m}\,\textrm {s}^{-1}$ and the slit opening, $o$ , and thickness, $d = o = {0.8}\,\textrm{mm}$ , and are parameterised by the Strouhal number $\textit{St} = 2 \pi f_{s} o / c$ and the Reynolds number $\textit{Re}=\rho c o/\mu$ . Here, $\rho$ is the density of air and $\mu$ is the dynamic viscosity, which varies with the Reynolds number across different simulations. The nominal Strouhal number and Reynolds number are defined as $\textit{St}_0 = 2 \pi f_{s,0} o / c$ and $\textit{Re}_0 = \rho _0 c o / \mu _0$ , respectively, where $f_{s,0} = {0.5}\,\textrm{kHz}$ is the reference frequency. The reference density and dynamic viscosity are taken as near-sea-level air properties, $\rho _0={1.19}\,{\textrm {kg} \,\textrm {m}^{-3}}$ and $\mu _0 = {1.84\times {10}^{-5}}\mathrm{Pa\,s}$ , respectively. All the length scales, $x$ , are non-dimensionalised by $d$ , time $t$ by $c/d$ , velocity $u$ by $c$ , pressure $p$ by $\rho _0 c^2$ , the source terms in the (2.1) $h_s h_{\delta }/c$ by $d/\rho _0c$ and source terms in the (2.2) $h_s h_{\delta } \boldsymbol n$ by $d/\rho _0c^2$ .

We perform DNSs using MFC, a GPU-accelerated compressible flow solver (Bryngelson et al. Reference Bryngelson, Schmidmayer, Coralic, Maeda, Meng and Colonius2021; Radhakrishnan et al. Reference Radhakrishnan, Berre, Wilfong, Spratt, Rodriguez, Colonius and Bryngelson2024; Wilfong et al. Reference Wilfong2025a , Reference Wilfong, Radhakrishnan, Berre, Prathi, Abbott and Bryngelsonb , Reference Wilfongc ). Equations (2.1)–(2.4) can be written compactly as

(2.7) \begin{gather} \frac {\partial \boldsymbol{q}}{\partial t} + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{F}(\boldsymbol{q}) = \boldsymbol{s}(\boldsymbol{q}), \end{gather}

where

(2.8) \begin{align} \boldsymbol{q} &= \begin{bmatrix} \rho \\[5pt] \rho \boldsymbol{u} \\[5pt] \rho E \\ \end{bmatrix}\!, \quad \boldsymbol{F} = \begin{bmatrix} \rho \boldsymbol{u} \\[5pt] \rho \boldsymbol{u}\otimes \boldsymbol{u} + p\boldsymbol{I} - \boldsymbol{T} \\[5pt] (\rho E + p) \boldsymbol{u} - \boldsymbol{T} \boldsymbol{\cdot }\boldsymbol{u} \\ \end{bmatrix} \quad \text{and} \quad \boldsymbol{s} = \begin{bmatrix} h_s h_{\delta }/c \\[5pt] h_s h_{\delta } \boldsymbol n \\[5pt] 0 \\ \end{bmatrix}\!, \end{align}

represent the state vector, flux tensor and the source vector, respectively. The spatial discretisation of (2.7) and (2.8) uses a finite-volume method, which is well suited to problems involving irregular geometries and complex compressible flows. A fifth-order accurate weighted essentially non-oscillatory scheme (Coralic & Colonius Reference Coralic and Colonius2014) is used to reconstruct the fluxes and viscous terms, which handles sharp gradients in the high-amplitude solution without introducing spurious oscillations. We perform temporal discretisation using a total variation diminishing third-order accurate Runge–Kutta scheme (Gottlieb & Shu Reference Gottlieb and Shu1998).

Figure 2.

Schematic of the flow configuration in the $x$ – $y$ domain simulated in this work.

2.2. Simulation configurations

Figure 2 shows the configuration of the 2-D computational domain, $\varOmega$ , discretised with a uniformly structured mesh with equal grid spacing. Rightward-going acoustic waves were emitted from a planar acoustic source $224 d$ from the incident face of the slit resonator ( $28 d$ away from the left domain boundary), where $d = {0.8}\,\textrm{mm}$ . The domain height is $H$ , where $H= 28 \, d$ . Ghost-cell non-reflective boundary conditions are used on the left and right boundaries, where primitive variables in the ghost cells are extrapolated from the interior to satisfy $\partial (\boldsymbol{\cdot })/\partial n \approx 0$ , and $n$ represents the direction perpendicular to the boundary. In practical systems, the boundary layers at the top and bottom domain walls may contribute to overall absorption. However, the present simulations intentionally isolate the dissipation associated with the slit mouth itself, so we neglected dissipation due to sound at the top and bottom domain boundaries and treated these surfaces as characteristic slip walls, following Thompson (Reference Thompson1990). The no-slip condition was enforced on the incident face of the slit and on all of its interior walls using the immersed-boundary method (Tseng & Ferziger Reference Tseng and Ferziger2003). Since heat conduction is not included in the governing equations, no thermal boundary condition is prescribed at the solid walls. The temperature field is obtained diagnostically from the equation of state using the local thermodynamic variables, and the immersed-boundary treatment therefore only enforces the velocity, pressure and density boundary conditions at the walls.

The immersed-boundary treatment uses ghost cells on a Cartesian finite-volume mesh. The boundary condition is enforced through an image-point reconstruction in which the flow variables at each ghost cell are interpolated between the body-intercept point on the immersed surface and a corresponding image point in the fluid domain, consistent with the ghost-cell immersed-boundary methodologies (Tseng & Ferziger Reference Tseng and Ferziger2003; Mittal & Seo Reference Mittal and Seo2023). This reconstruction involves a second-order interpolation, so the formal accuracy of the boundary closure is second order, even though the underlying finite-volume discretisation retains its high-order accuracy away from the immersed surface.

We use the ISPL to represent the decibel sound level at a specific frequency, defined as

(2.9) \begin{equation} \textit{ISPL} = 20 \log _{10}\left (\widehat {p}/\widehat {p}_{\textit{ref}}\right )\!, \end{equation}

where $\widehat {p} = \widehat {A}/\sqrt {2}$ is the dimensional root-mean-squared pressure of the sound, and $\widehat {p}_{\textit{ref}} = {20}\,{\unicode{x03BC}}\,\textrm{Pa}$ is the standard reference pressure in air.

The 2-D slit under consideration is a wall-bounded problem, and the oscillatory boundary layer is a near-wall feature characterised by steep gradients, rather than a propagating wave. Previous work has found that the acoustic-driven flow bounded by small slit resonators, as small as ${1}\,\textrm{cm}$ , is laminar at a high incident level of ${155}\,\textrm{dB}$ (Tam & Kurbatskii Reference Tam and Kurbatskii2000). The laminar condition with an oscillatory wave gives rise to a viscous Stokes layer inside the slit opening whose thickness is much smaller than the acoustic wavelength of our range of interest (Tam et al. Reference Awasthi, McCreton, Moreau and Doolan2005, Reference Cummings2010). The wavelength of the oscillatory Stokes layer $\lambda _s$ is defined as

(2.10) \begin{align} \lambda _s \equiv 2 \pi \delta _s= 2 \sqrt {{\pi \nu }/{f_s}}, \end{align}

where $\delta _s$ is the thickness of the Stokes layer, $\nu = \mu /\rho$ is the kinematic viscosity and $f_s$ is the acoustic frequency (White Reference White1991; Tam et al. Reference Tam, Ju, Jones, Watson and Parrott2005).

Figure 3.

Schematic of the 3-D simulation configuration. The slit is embedded in an $x$ – $y$ domain and extended uniformly in the spanwise direction $z$ with periodic boundary conditions. A spanwise-uniform acoustic source drives the flow. Two spanwise extents are considered, $L_{z,1}=11d$ and $L_{z,2}=22d$ . All other parameters are identical. The companion 2-D simulation in figure 2 uses the identical $x$ – $y$ configuration.

To examine the 3-D dynamics, the 2-D slit configuration in figure 2 is extended uniformly in the spanwise direction, $z$ , with periodic boundary conditions, as shown in figure 3. The acoustic forcing is extended consistently in the spanwise direction. Two spanwise extents are considered, with $L_{z,1}=11d$ and $L_{z,2}=22d$ . This configuration provides a more realistic setting, enabling the assessment of 3-D effects absent in 2-D simulations.

For varying acoustic frequencies, simulations are performed using at least ten cells per Stokes-layer wavelength ( $\lambda _s / \varDelta _x \geq 10$ ) to ensure negligible numerical errors when the Stokes boundary layers are resolved, where $\varDelta _x$ is the grid spacing. Grid convergence is confirmed by the negligible change in simulated acoustic power absorption coefficients when the cell count is doubled. Simulations are performed over multiple acoustic cycles with a constant time step to ensure the system reaches a dynamic steady state before data collection.

2.3. Direct numerical simulation verification

We verify our numerical simulations using a 2-D simulation of a discrete tone ( $\textit{ISPL} = {150}\,\textrm{dB}$ ) propagating as a plane wave through a 2-D slit resonator. The opening, $o$ , and thickness, $d$ (with $o = d$ ), of this slit are each scaled to be $1/28$ of the domain height to reproduce the configuration of Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001). A similar verification simulation has been conducted by Yu et al. (Reference Yu, Ahuja, Sankar and Bryngelson2024, Reference Yu, Ahuja, Sankar and Bryngelson2025a ). This verification set-up is analogous to the cases examined in this study, with the computational domain having a non-dimensional length of $294d$ , as illustrated in figure 4(a). In this verification case, we use non-reflective boundary conditions at the left horizontal boundary and impermeable slip walls for the remaining domain boundaries. The simulation time and grid size match those of the 2-D cases in table 1. Thus, the only differences between this verification simulation and the 2-D cases in table 1 are the termination boundary condition and the $x$ -direction extent of the computational domain.

Table 1.

Simulation configuration of seven different acoustic frequencies. The following simulations include cases with three Reynolds numbers, $\textit{Re}/\textit{Re}_0 = 1$ , $1/2$ and $1/3$ , and two sound pressure levels: $\textit{ISPL} = {120}\,\textrm{dB}$ and ${150}\,\text{dB}$ . The three varying parameters result in 42 unique combinations. The total number of cells in the $x$ -direction ( $N_x$ ) and the $y$ -direction ( $N_{\!y}$ ) describes the grid size of the computational domain. The number of cells per Stokes’ wavelength $\lambda _s$ describes the cell size. We use the number of time steps per acoustic cycle to describe the time step size.

Table 1. Long description

A table comparing simulation configurations for different acoustic frequencies. The table has seven rows and seven columns. Column headers are Frequency, St/St0, Domain size, Cell size, Time step, Duration, and Analysis range. Row 1: Frequency, 0.5 kHz; St/St0, 1; Domain size, 20 160 x 1120; Cell size, 30; Time step, 8.0 10^4 cycle^-1; Duration, 25 cycles; Analysis range, 10-25 cycles. Row 2: Frequency, 1 kHz; St/St0, 2; Domain size, 20 160 x 1120; Cell size, 20; Time step, 4.0 10^4 cycle^-1; Duration, 50 cycles; Analysis range, 19-50 cycles. Row 3: Frequency, 2 kHz; St/St0, 4; Domain size, 20 160 x 1120; Cell size, 15; Time step, 2.0 10^4 cycle^-1; Duration, 75 cycles; Analysis range, 12-75 cycles. Row 4: Frequency, 3 kHz; St/St0, 6; Domain size, 20 160 x 1120; Cell size, 12; Time step, 1.3 10^4 cycle^-1; Duration, 113 cycles; Analysis range, 18-113 cycles. Row 5: Frequency, 4 kHz; St/St0, 8; Domain size, 20 160 x 1120; Cell size, 10; Time step, 1.0 10^4 cycle^-1; Duration, 150 cycles; Analysis range, 23-150 cycles. Row 6: Frequency, 5 kHz; St/St0, 10; Domain size, 25 200 x 1400; Cell size, 12; Time step, 0.8 10^4 cycle^-1; Duration, 139 cycles; Analysis range, 20-139 cycles. Row 7: Frequency, 6 kHz; St/St0, 12; Domain size, 25 200 x 1400; Cell size, 10; Time step, 0.7 10^4 cycle^-1; Duration, 167 cycles; Analysis range, 24-167 cycles.

Figure 4.

A 2-D slit resonator for verification showing (a) the computational domain with acoustic source (not to scale) and (b) comparison of power absorption-coefficient spectra among DNS and experiment (Expt.) of Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001) with discrete tones at $\textit{ISPL}= {150}\,\textrm{dB}$ .

We use the power absorption coefficient as the metric to quantify the acoustic performance. The absorption coefficient that represents the percentage of the sound power dissipated by the slit resonator is defined as

(2.11) \begin{equation} \alpha \equiv 1 - |\widehat {R}|^2 - |\widehat {T}|^2, \end{equation}

where $|\widehat {R}|^2$ and $|\widehat {T}|^2$ are the power reflection and transmission coefficients, respectively. Following Tam et al. (Reference Tam, Ju, Jones, Watson and Parrott2005) and Leung et al. (Reference Leung, So, Wang and Li2007), these coefficients are calculated from the recorded time histories using the transfer function method of Chung & Blaser (Reference Chung and Blaser1980). In the present verification case with a reflective termination, all the transmitted waves will be reflected by the termination, yielding $|\widehat {T}|^2 \approx 0$ . The power absorption coefficient is approximated as $\alpha \approx 1 - |\widehat {R}|^2$ . For discrete-tone analysis, the power reflection coefficient at a given frequency is determined via linear interpolation between the two adjacent spectral points that bracket the target frequency.

We collect pressure data at two points along the domain’s horizontal centreline at each simulation time step. The two points are $21.875d$ and $57.5d$ upstream of the slit incident face, where higher-order modes are evanescent. The sampling frequency of the recorded time history is as large as the inverse of the simulation time step. Figure 4(b) compares the present absorption-coefficient spectra with those reported by Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001) at a nominal incident sound intensity of 150 dB. The discrepancy between this work and Tam’s DNS is within $12\,\%$ across the full frequency band, with the largest discrepancy $11.7\,\%$ occurring at 2 kHz. In Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001), KE loss is estimated by identifying and integrating the energy of individual large-scale shed vortices, and thus excludes contributions from other scales. In contrast, the absorption coefficient in the present study is obtained using the transfer-function method, which measures the net acoustic energy loss from the reflected and transmitted wave fields. Since these two approaches do not quantify the same quantity, a slight discrepancy within $12\,\%$ between the present results and those of Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001) is to be expected. Nevertheless, the overall agreement between the two DNS datasets supports the accuracy of the present numerical method for simulating slit resonators under acoustic excitation.

In the following, the MFC solver is used to generate a database for large-aspect-ratio acoustic slits over the parameter ranges detailed in § 3. The comparison between the 2-D and 3-D simulations in § 4.4 suggests that the flow remains effectively two-dimensional.

3.1. Set-up

The dominant factors influencing the performance of the acoustic slit include ISPL, acoustic frequency and Reynolds number. To isolate the effect of each parameter, simulations are performed across various combinations of these three variables. Specifically, two ISPLs, ${120}$ and ${150}\,\textrm{dB}$ , are considered to represent cases of weak and strong sound excitations, respectively. The acoustic frequency is varied from ${0.5}$ to ${6}\,\textrm{kHz}$ to represent a broad range of acoustic behaviours. For a fixed speed of sound and slit width, three Reynolds numbers are examined to investigate the effect of the varying dynamic viscosity. Table 1 summarises the considered parameters and the corresponding simulation configurations.

All simulations were performed using MFC on the NCSA Delta supercomputer. Each case was executed on three NVIDIA A100 GPUs. The overall GPU utilisation exceeds 90 % (Radhakrishnan et al. Reference Radhakrishnan, Berre, Wilfong, Spratt, Rodriguez, Colonius and Bryngelson2024). The required wall time of each simulation is approximately 40 h and depends weakly on the acoustic frequency and the corresponding grid resolution. Numerical stability is verified under a Courant–Friedrichs–Lewy number of less than $0.5$ . For the simulations documented in table 1, the raw data for the analysed cycles are publicly available in the Georgia Tech Digital Repository, comprising 42 MATLAB-formatted mex files. For each case, the full velocity and density fields are stored over the analysis range indicated in the table, comprising 640 time snapshots. The public flow-field data are down sampled by a factor of 2 in each coordinate direction. Each file is approximately 4.5 GB in size and corresponds to a unique parameter combination.

For the configurations listed in table 1, the acoustically induced flow remains at low Mach number ( $\textit{Ma}$ ), with $\textit{Ma} \leq 0.05$ throughout the computational domain. Velocities and their fluctuations are thus small compared with the ambient speed of sound. Consequently, compressibility effects in the induced flow are weak, and the pressure fluctuations associated with acoustic propagation are only mildly affected by local density variations, which remain below 0.5 % of their initial value. This behaviour is consistent with previous studies of low-Mach-number oscillatory flows in resonant cavities, which have shown that the velocity field is predominantly solenoidal and that compressibility enters only as a higher-order correction (Landau & Lifshitz Reference Landau and Lifshitz1987). Despite the relatively low Mach numbers, the flow outside the slit exhibits a complex, irregular dynamics, whereas the flow inside the slit remains laminar. We emphasise, however, that compressibility cannot be neglected in the simulations, as compressibility is essential for accurately resolving the propagation of acoustic waves.

3.2. Data

To illustrate the rich physics in the database, figure 5 shows the normalised instantaneous vorticity fields

(3.1) \begin{align} \boldsymbol{\omega }(\mathsf{{x}};t_i) = \frac { \left [\partial _x \boldsymbol{v}-\partial _{\!y} \boldsymbol{u}\right ](\mathsf{{x}};t_i) }{ \max \{\left [\partial _x \boldsymbol{v}-\partial _{\!y} \boldsymbol{u}\right ](\mathsf{{x}};t_i) \} }, \end{align}

for both 2-D and 3-D simulations at $\textit{ISPL} = {150}\,\textrm{dB}$ . Further 2-D results spanning a range of $\textit{St}$ – $\textit{Re}$ combinations are included in figure 30. Here, $\mathsf{{x}}$ denotes the discretised spatial coordinate vector over the computational domain. Across all cases, the snapshots reveal well-resolved boundary layers generated as the flow interacts with the slit, followed by the formation of periodic vortex shedding. These results indicate that a portion of the incident acoustic energy is converted into vortical motion through interactions with the slit. These vortical structures intensify and periodically detach from the slit, forming coherent vortices that dominate the near-slit region. Once shed, the vortices convect both upstream and downstream, undergoing complex interactions that give rise to large- and small-scale motions. As the acoustic frequency increases, the vortical structures gradually become confined to the vicinity of the slit, suppressing long-range interactions and limiting the generation of far-field vorticity. For $L_{z,1}=11d$ , hairpin-like structures emerge in the vorticity field away from the slit. For the larger spanwise extent, $L_{z,2}=22d$ , the vorticity field shows the presence of line vortices with only weak spanwise variation, indicating that the dominant flow dynamics remains effectively two-dimensional. This observation is consistent with the results of Xu et al. (Reference Xu, Li and Guo2014) for a spanwise extent of $L_{z}=40d$ , where the vorticity fields likewise showed limited three-dimensionality and the absorption coefficients from 2-D and 3-D simulations were in favourable agreement. Further analysis of the present 3-D simulations is provided in § 4.4.

Figure 5.

Instantaneous 2-D (a, b) and 3-D (c, d) vorticity fields, $\boldsymbol{\omega }(\mathsf{{x}};t_i)$ , at $\textit{ISPL} = {150}\,\textrm{dB}$ : (a) $\textit{Re}=\textit{Re}_0$ and $\textit{St} = \textit{St}_0$ ; (b) $\textit{Re}=\textit{Re}_0/3$ and $\textit{St} = 12\textit{St}_0$ ; (c) $\textit{Re}=\textit{Re}_0/3$ , $\textit{St} = 2\textit{St}_0$ and $L_{z,1}=11d$ ; and (d) $\textit{Re}=\textit{Re}_0/3$ , $\textit{St} = 2\textit{St}_0$ and $L_{z,2}=22d$ . The instantaneous 2-D VL fields (integrated across the $y$ direction), $\boldsymbol{D}(\boldsymbol{x};t_i)$ , and their temporal average, $\overline {\boldsymbol{D}}(\boldsymbol{x})$ , are included in (a, b) for comparison. Additional 2-D results spanning a range of $\textit{St}$ – $\textit{Re}$ combinations are included in figure 30.

In the present analysis, we quantify viscous dissipation associated with these vortical structures using the irreversible viscous dissipation density $\boldsymbol{T}: \boldsymbol{\nabla }\boldsymbol{u}$ , which is non-negative for the Newtonian fluids we assume in this study and a measure of thermodynamic dissipation. Figure 5 further examines the instantaneous irreversible VL fields integrated in the $y$ -direction

(3.2) \begin{align} \boldsymbol{D}(\boldsymbol{x};t_i)=\sum _{y\in [-14d,14d]} \left [ \mathsf{{T}}: \boldsymbol{\nabla }\mathsf{{u}}\right ](\mathsf{{x}};t_i)\varDelta _{\!y}, \quad \text{with} \quad \mathsf{{T}} =\frac {1}{\textit{Re}}\Big ( \boldsymbol{\nabla }\mathsf{{u}}+ \boldsymbol{\nabla }\mathsf{{u}}^{\mathsf{T}}-\frac {2}{3}(\boldsymbol{\nabla }\boldsymbol{\cdot }\mathsf{{u}})\boldsymbol{I}\Big ), \end{align}

where $\varDelta _{\!y}=\varDelta _x$ under uniform grid spacing. The corresponding temporal averages over $N_t$ snapshots are

(3.3) \begin{align} \overline {\boldsymbol{D}}(\boldsymbol{x}) = \frac {1}{N_t} \sum _{i=1}^{N_t}{\boldsymbol{D}}(\boldsymbol{x};t_i). \end{align}

Across all cases, the VL fields peak near the slit and attain values larger than those in other regions, indicating that the dominant energy loss is associated with velocity fluctuations near the slit and their conversion into vorticity or viscous dissipation within the boundary layer. In contrast, the detached vortices contribute comparatively little. At the lower acoustic frequency of $\textit{St} = \textit{St}_0$ , VL is larger near the inlet of the slit than at the outlet. This trend is consistent with the corresponding flow fields, in which more pronounced upward-travelling vortex shedding occurs at lower frequencies. At lower Reynolds numbers within the range considered, viscous effects become more pronounced relative to inertial ones, leading to greater VL while still sustaining vortex shedding.

Figure 6.

Instantaneous VL (integrated across the $y$ direction), $\boldsymbol{D}(\boldsymbol{x};t_i)$ , and vorticity fields, $\boldsymbol{\omega }(\mathsf{{x}};t_i)$ , at $\textit{ISPL} = {120}\,\textrm{dB}$ and $\textit{St}=\textit{St}_0$ : (a) $\textit{Re}=\textit{Re}_0$ and (b) $\textit{Re}_0/3$ .

For comparison, figure 6 shows the results at $\textit{ISPL} = {120}\,\textrm{dB}$ . The flow fields with higher Strouhal numbers are similar, so we focus on a representative Strouhal number of $\textit{St}_0$ . At this weaker sound excitation, the vortex shedding remains nearly symmetric, with vorticity confined to boundary layers adjacent to the slit walls. No evidence of large-scale vortex pairing or chaotic modulation is found. Compared with high-intensity acoustic excitation, these cases exhibit substantially weaker VL, approximately 3 orders of magnitude lower.

3.3. Mapping between imposed control parameters and induced in-slit response

As the quiescent background flow does not provide a characteristic velocity scale for the slit flow, we distinguish between the parameters for the imposed operating condition and those induced by the resulting flow response. The imposed parameter group, $(\textit{Re},\textit{St},\textit{ISPL})$ , is treated as the set of controllable factors: these quantities are prescribed directly through the fluid properties, geometry and acoustic source. The in-slit shear number

(3.4) \begin{equation} \textit{Sh}=\frac {o}{\delta _s}, \end{equation}

which measures the slit width, $o$ , relative to the viscous boundary-layer thickness, $\delta _s=\sqrt {\mu /(\pi f_s \rho _0)}$ , also depends only on the imposed configuration. By contrast, the effective in-slit Strouhal and Reynolds numbers cannot be directly specified a priori as the relevant in-slit velocity scale is itself an outcome of the simulation or experiment. We therefore define

(3.5) \begin{equation} \textit{St}_{\textit{slit}}=\frac {2\pi f_s o}{u_{\textit{slit,pk}}},\qquad \textit{Re}_{\textit{slit}}=\frac {\rho _0 u_{\textit{slit,pk}}\,o}{\mu }, \end{equation}

based on the peak in-slit velocity

(3.6) \begin{align} u_{\textit{slit,pk}}\equiv \max _{t}\max _{(x,y)\in \varOmega _{\textit{slit}}} |u(x,y,t)|. \end{align}

These imposed and induced groups are related through

(3.7) \begin{equation} \textit{Re}_{\textit{slit}}\,\textit{St}_{\textit{slit}}= \textit{Re} \,\textit{St} = 2\,\textit{Sh}^2, \end{equation}

so the slit dynamics occupies the 2-D $(\textit{St}_{\textit{slit}},\textit{Re}_{\textit{slit}})$ plane, on which lines of constant $\textit{Sh}$ are hyperbolae.

Figure 7.

Mapping from the imposed dimensionless control parameters to the induced in-slit response groups: (a) imposed incident plane $(\textit{St},\textit{Re})$ , defined using the sound speed $c_0$ , showing a rectangular grid formed by three viscosities, $\mu _0$ , $2\mu _0$ and $3\mu _0$ (corresponding to $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ ), and seven forcing frequencies, $f_s=$ 0.5–6 $\textrm {kHz}$ ; (b, c) corresponding induced response planes $(\textit{St}_{\textit{slit}},\textit{Re}_{\textit{slit}})$ , defined using the peak in-slit velocity, $u_{\textit{slit,pk}}$ , for (b) $\textit{ISPL}={150}\,\textrm{dB}$ and (c) ${120}\,\textrm{dB}$ .

Figure 7 illustrates how the imposed parameter space maps onto the induced in-slit parameter space at $\textit{ISPL}={150}$ and ${120}\,\textrm{dB}$ . The imposed parameter space in figure 7(a) forms a rectangular grid, comprising three rows of constant $\textit{Re}$ (constant $\mu$ ) and seven columns of constant forcing frequency. This grid becomes distorted in the induced plane, and lines of constant $\textit{Sh}$ provide an undistorted reference, as shown in figures 7(b) and 7(c). Further details on the in-slit velocity scales are provided in Appendix A. For $f\lesssim {4}\,\textrm{kHz}$ , the in-slit Strouhal number remains nearly independent of $\textit{Re}$ , indicating that the induced velocity response scales approximately with the forcing frequency over this range. The in-slit Reynolds number largely retains its scaling with the imposed $\textit{Re}$ while decreasing overall as the acoustic frequency increases. These trends show that the induced Strouhal and Reynolds numbers are ordered by the imposed Strouhal and Reynolds numbers. Because the local slit dynamics is governed by the induced in-slit parameters, we report trends in both the induced and the imposed variables; the imposed parameters additionally serve as labels for the overall trends, as they are the directly prescribed operating conditions.

This mapping is a calibrated response relation for the present slit geometry and porosity, not a universal predictor. In particular, as $u_{\textit{slit}}(f,\mu ,\textit{ISPL})$ is geometry specific and the acoustic impedance depends on ISPL, the induced in-slit conditions must be determined from the realised flow response for each prescribed operating condition.

The acoustic frequencies considered here span a broad range of vortex-formation regimes (Ingard & Labate Reference Ingard and Labate1950; Aulitto Reference Aulitto2023). At $\textit{ISPL}={150}\,\textrm{dB}$ , the lowest frequency, $f={500}\,\textrm{Hz}$ , gives $\textit{St}_{\textit{slit}}\approx 0.04$ , so that the slit width is much smaller than the oscillatory displacement. This corresponds to a strongly nonlinear regime, in which vortices are shed from the slit and convect away. At the highest frequency considered, $\textit{St}_{\textit{slit}}=O(1)$ , and the vortices that form remain localised near the slit openings. At $\textit{ISPL}={120}\,\textrm{dB}$ , $\textit{St}_{\textit{slit}}$ ranges from $O(1)$ to approximately $50$ , consistent with the observed suppression of vortex formation. Over the same band, the shear number spans $\textit{Sh}\approx 5$ – $28$ , so the Stokes layer remains thin relative to the slit ( $\textit{Sh}\gg 1$ ) and a distinct edge shear layer is present throughout, in contrast to the viscous-dominated micro-slit regime ( $\textit{Sh}\approx 1$ – $4$ ) of Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022).

3.4. Acoustic absorption

Following the method outlined in § 2.3, we compute and present the power absorption coefficient in figure 8 to compare the total energy absorption at varying parameter combinations. At $\textit{ISPL} = {150}\,\textrm{dB}$ , for larger $\textit{Re}$ , the $\alpha$ is larger. Power absorption at $\textit{Re} = \textit{Re}_0$ is approximately $0.01$ and $0.02$ larger than at $\textit{Re} = \textit{Re}_0/2$ and $\textit{Re} = \textit{Re}_0/3$ , respectively. On the contrary, for larger $\textit{Re}$ (i.e. lower viscosity), $\alpha$ is smaller at $\textit{ISPL} = {120}\,\textrm{dB}$ . Power absorption at $\textit{Re} = \textit{Re}_0$ is approximately $0.015$ and $0.03$ smaller than at $\textit{Re} = \textit{Re}_0/2$ and $\textit{Re} = \textit{Re}_0/3$ , respectively.

Figure 8.

The power absorption coefficients $\alpha \equiv 1 - |\widehat {R}|^2 - |\widehat {T}|^2$ across all the $\textit{St}$ – $\textit{Re}$ combinations listed in table 1, subject to (a) $\textit{ISPL} = {150}\,\textrm{dB}$ and (b) $\textit{ISPL} = {120}\,\textrm{dB}$ . Probe locations are given in figure 2. The shaded zone in (a) demonstrates the range of $\alpha$ at $\textit{ISPL} = {120}\,\textrm{dB}$ , which is plotted in (b). Insets in (a) show the normalised instantaneous vorticity fields, $\boldsymbol{\omega }(\mathsf{{x}};t_i)$ , of cases (a, i), (a, iii) and (a, iv) in figure 30 to highlight the change in vorticity field across the range of $\alpha$ observed as $\textit{St}$ increases for $\textit{ISPL} = {150}\,\textrm{dB}$ and $\textit{Re} = \textit{Re}_0$ .

The dependence of $\alpha$ on both $\textit{Re}$ and $\textit{Re}_{\textit{slit}}$ remains significantly weaker, by at least an order of magnitude, than its dependence on $\textit{St}$ at $\textit{ISPL}={150}\,\textrm{dB}$ , where the variation of $\alpha$ across the frequency sweep exceeds $0.4$ . On the contrary, at $\textit{ISPL}={120}\,\textrm{dB}$ , the $\textit{Re}$ -dependence becomes comparable to the $\textit{St}$ -dependence, reflecting that, in the absence of strong vortex shedding, the absorption is dominated by the VL, and not by the frequency-dependent dynamics. In sum, figure 8 shows two distinct absorption regimes: a vortex-dominated regime at high ISPL, where $\textit{St}$ is the primary dominating parameter, and a viscosity-dominated regime at low ISPL, where $\textit{Re}$ plays a comparable role.

Figure 9.

Incident acoustic energy-loss characteristics across the $\textit{St}$ – $\textit{Re}$ – $\textit{ISPL}$ combinations listed in table 1: (a) power reflection, $|\widehat {R}|^2$ ; and (b) power transmission coefficients, $| \widehat {T}|^2$ .

To complement the absorption spectra and to provide designers with information on the partition between reflected and transmitted acoustic power, we report the power reflection $|\widehat {R}|^2$ and transmission coefficients $|\widehat {T}|^2$ in figure 9. Across all cases, $|\widehat {R}|^2$ increases monotonically with $\textit{St}$ , while $|\widehat {T}|^2$ decreases. At low $\textit{St}/\textit{St}_0\approx 1$ , the $\textit{ISPL}={120}\,\textrm{dB}$ cases are predominantly transmitting the power ( $|\widehat {T}|^2\approx 0.8$ ) with weak reflection ( $|\widehat {R}|^2\approx 0.1$ ), whereas at $\textit{ISPL}={150}\,\textrm{dB}$ transmission is substantially reduced. At $\textit{St}/\textit{St}_0 \gt 8$ , the slit is predominantly reflecting comparable power for both ISPLs, with $|\widehat {R}|^2 \gt 0.8$ and $|\widehat {T}|^2 \lt 0.1$ . The ISPL dependence is weaker, especially for the transmitted power, at higher forcing frequencies ( $\textit{St}/\textit{St}_0 \gt 8$ ). Variations with $\textit{Re}$ are comparatively modest over the range considered, indicating that $\textit{St}$ and ISPL primarily set the partition between reflected and transmitted power in this configuration.

The present DNS serves as a design tool for a slit–cavity configuration. We compare the slit impedance under different imposed operating conditions, even under the constraint of the anechoic termination.

Following Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022), we characterise the slit transfer impedance based on the complex acoustic pressure drop across the slit measured by the three probes divided by the volume flux through it, referenced to the in-slit velocity through the porosity

(3.8) \begin{equation} \widehat {Z}_{\textit{slit}}=\frac {\widehat {p}_{\textit{left}}-\widehat {p}_{\textit{right}}} {\rho c\,\widehat {u}_{\textit{slit}}}=\sigma \,(\widehat {Z}_{s}-\widehat {Z}_{\textit{ter}}), \end{equation}

with the porosity $\sigma =o/H$ , the sample-loaded surface impedance, $\widehat {Z}_{s}=(1+\widehat {R})/(1-\widehat {R})$ , obtained from the reflection coefficient (Chung & Blaser Reference Chung and Blaser1980) and the downstream termination impedance, $\widehat {Z}_{\textit{ter}}$ . As the computational domain is terminated anechoically, $\widehat {Z}_{\textit{ter}}=1$ and

(3.9) \begin{equation} \widehat {Z}_{\textit{slit}}=\sigma \left (\frac {1+\widehat {R}}{1-\widehat {R}}-1\right )\!, \end{equation}

which references the impedance to the slit, rather than the slit surface, and removes the radiation load of the downstream anechoic termination. This is the empirical slit transfer impedance of Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022), here evaluated at the driving frequency in the compressible finite-amplitude regime.

The resistance is a small quantity relative to the reactance, so evaluating it from the difference $\textrm{Re} (\widehat {Z}_s)-1$ is poorly conditioned at high shear number. Following Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022), we instead define the resistance through the time-averaged dissipated power, $\langle P_{\textit{slit}}\rangle =({1}/{2})\, \textrm{Re} (\widehat {Z}_{\textit{slit}})\,\rho c\,|\widehat {u}_{\textit{slit, pk}}|^2$ . Equating this to the absorption coefficient $\alpha =1-|\widehat {R}|^2-|\widehat {T}|^2$ gives

(3.10) \begin{equation} \textrm{Re} (\widehat {Z}_{\textit{slit}})=\sigma \,\frac {\alpha }{|1-\widehat {R}|^2} =\sigma \,\frac {1-|\widehat {R}|^2-|\widehat {T}|^2}{|1-\widehat {R}|^2}, \end{equation}

which is independent of the downstream reference plane.

Figure 10.

Slit transfer impedance, $\widehat {Z}_{\textit{slit}}$ , across the cases of table 1: (a) resistance $\textrm{Re} (\widehat {Z}_{\textit{slit}})$ , evaluated from (3.10), and (b) reactance $\textrm{Im} (\widehat {Z}_{\textit{slit}})$ , from (3.9).

Figure 10 reports $\textrm{Re} (\widehat {Z}_{\textit{slit}})$ and $\textrm{Im} (\widehat {Z}_{\textit{slit}})$ versus the peak in-slit velocity $u_{\textit{slit,pk}}$ for both excitation levels and the three Reynolds numbers. The resistance increases by roughly an order of magnitude as $u_{\textit{slit,pk}}$ grows between $\textit{ISPL}={120}$ and ${150}\,\textrm{dB}$ , consistent with the nonlinear-resistance growth of Ingard & Labate (Reference Ingard and Labate1950) and with the amplitude dependence measured by Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022).

Figure 11.

Slit resistance $\textrm{Re} (\widehat {Z}_{\textit{slit}})$ as a function of (a) the slit peak velocity $u_{\textit{slit,pk}}$ and (b) slit Strouhal number $\textit{St}_{\textit{slit}}$ at $\textit{ISPL} = {150}\,\textrm{dB}$ , and (c) the slit peak velocity $u_{\textit{slit,pk}}$ and (d) slit Strouhal number $\textit{St}_{\textit{slit}}$ at $\textit{ISPL} = {120}\,\textrm{dB}$ .

Figure 11 shows the slit resistance against the peak velocity, $u_{\textit{slit,pk}}$ , and slit Strouhal number, $\textit{St}_{\textit{slit}}=2 \pi f_s o/u_{\textit{slit, pk}}$ . At 150 dB, figure 11(a) indicates the peak velocity is larger for higher $\textit{Re}$ , which leads to slightly larger resistance. As $\textit{St}_{\textit{slit}}$ decreases while the in-slit velocity grows, figure 11(b) shows the resistance rises by roughly a factor of four over the range sampled. At 120 dB, figure 11(c) indicates the peak velocity is nearly independent of $\textit{Re}$ and the resistance, dominated by viscous attenuation, is larger at lower $\textit{Re}$ . Figure 11(d) shows the resistance varies weakly with $\textit{St}_{\textit{slit}}$ . The resistance rises by roughly an order of magnitude between $\textit{ISPL}={120}$ and 150 dB, consistent with the nonlinear-resistance growth reported by Ingard & Labate (Reference Ingard and Labate1950).

Figure 12.

Slit resistance $\textrm{Re} (\widehat {Z}_{\textit{slit}})$ as a function of the shear number $\textit{Sh}$ at $\textit{Re} = \textit{Re}_0$ . The resistance increases by roughly an order of magnitude from $\textit{ISPL} ={120}\,\text{dB}$ to $={150}\,\text{dB}$ at $\textit{Sh} \lt 20$ , recovering the nonlinear-resistance behaviour of Ingard & Labate (Reference Ingard and Labate1950) and the ISPL dependence measured by Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022).

Figure 12 shows the slit resistance versus shear number at $\textit{Re}=\textit{Re}_0$ . At 150 dB the resistance is smaller with larger $\textit{Sh}$ , whereas at 120 dB it is nearly independent of $\textit{Sh}$ . This contrast between a strongly amplitude- and frequency-dependent resistance at high ISPL and a weakly varying resistance at low ISPL is qualitatively consistent with the impedance-tube measurements of Aulitto et al. (Reference Aulitto, Hirschberg, Arteaga and Buijssen2022) at a shear number range of 1–4 and ISPL range of 83–118 dB for micro-slit absorbers.

These resistance trends are a direct expression of the vortex-formation regime that the operating conditions occupy, which the in-slit Strouhal number and shear number make explicit. The two ISPLs occupy complementary parts of this range. At 150 dB, $\textit{St}_{\textit{slit}}$ rises from $\approx 0.04$ at 500 Hz to $\approx 1$ at 6 kHz. According to Ingard & Labate (Reference Ingard and Labate1950), the slit moves from strongly nonlinear vortex shedding toward the onset of vortices, which has been demonstrated in figure 5. At 120 dB, $\textit{St}_{\textit{slit}}$ starts near unity and grows to $\approx 50$ , carrying the slit from the moderate vortex regime into the linear regime in which the resistance is nearly independent of $\textit{Sh}$ . Therefore, the band $f_s=$ 0.5–6 $\textrm {kHz}$ covers the full sequence of vortex-formation regimes, from strongly nonlinear shedding through the moderate regime to the linear regime dominated by viscous attenuation.

Figure 13.

Power absorption coefficient $\alpha \equiv 1-|\widehat {R}|^2-|\widehat {T}|^2$ versus the in-slit Strouhal number $\textit{St}_{\textit{slit}}$ for the three Reynolds numbers at (a) $\textit{ISPL}={150}\,\textrm{dB}$ and (b) ${120}\,\textrm{dB}$ . At high amplitude the three curves collapse onto a single function of $\textit{St}_{\textit{slit}}$ ; at low amplitude they retain a $\textit{Re}$ dependence characteristic of the viscous regime.

Figure 13 recasts the power absorption coefficient against the in-slit Strouhal number. At 150 dB, the three Reynolds numbers collapse onto a single curve, so the high-amplitude absorption is set by $\textit{St}_{\textit{slit}}$ : it is largest for $\textit{St}_{\textit{slit}}\lesssim 0.1$ , where vortex shedding is strongest, and falls toward $\textit{St}_{\textit{slit}}=1$ . At 120 dB, the curves do not collapse and retain a clear $\textit{Re}$ dependence, so in the viscous regime the shear number, not $\textit{St}_{\textit{slit}}$ , sets the loss.

For an acoustically compact slit ( $2 \pi f d/c \ll 1$ ), it can be considered as an elemental input within a lumped-impedance framework (e.g. in perforated plates by Melling (Reference Melling1973) and Aulitto (Reference Aulitto2023)) to estimate the response of a complex configuration. Given the constraint that slit resistance depends on the local velocity at the slit mouth, which in a resonator differs from the incident particle velocity, the present work can also support a self-consistent evaluation when the ISPL conditions bracket a range of in-slit amplitudes that a cavity-backed configuration may realise.

The analysis domain is confined to $\varOmega = [x/d, y/d]\in [-14, 14] \times [-14, 14]$ for $\textit{St}/\textit{St}_0 \le 8$ and $\varOmega = [x/d, y/d]\in [-11.2, 11.2] \times [-14, 14]$ for $\textit{St}/\textit{St}_0 \ge 10$ . The horizontal distribution of VL identifies the region containing more than 99 % of the total VL. Within this region, more than 99 % of the total KE associated with the induced flow field is retained, giving the spatial extent used henceforth for spectral analysis.

This database is directly relevant to real-world engineering design scenarios involving sound at different ISPLs and frequencies in different media with different viscosities. The ISPL sets the magnitude of the imposed velocity oscillation through the plane wave approximation of acoustic impedance $u \approx p/\widehat {z}$ , where $\widehat {z} = \rho c$ is the characteristic acoustic impedance of air. The velocity magnitude scales the vorticity field Mach number, and thus the importance of compressible effects and nonlinear interactions. The acoustic frequency is introduced through the Strouhal number, which compares the convective time scale of the vorticity field with the flow oscillation period. A combination of both parameters determines the presence of sound sources with varying amplitudes and frequencies. The wavelength of the Stokes layer $\lambda _s$ depends on the Reynolds number with a similarity $\lambda _s \sim d/\sqrt {\textit{Re}\, \textit{St}}$ . Different slit thickness will change $\textit{Re}$ and $\textit{St}$ as well. To separate the effects attributed to $\textit{Re}$ and $\textit{St}$ , we vary the fluid property dynamic viscosity to adjust for varying $\textit{Re}$ .

4.1. Spectral proper orthogonal decomposition

Consider a time series consisting of $N_t$ snapshots. Each snapshot is represented by an observable state vector $\mathsf{{y}}\in \mathbb{C}^N$ , where $N$ is the dimension of the data. In line with common practice in the analysis of compressible turbulence (Kida & Orszag Reference Kida and Orszag1990; Wang et al. Reference Wang, Yang, Shi, Xiao, He and Chen2013), the observable is the fluctuating density-weighted velocity field, defined as

(4.1) \begin{align} \mathsf{{y}}(\mathsf{{x}};t) = \sqrt {\mathsf{{\beta }}(\mathsf{{x}};t)} \, \mathsf{{u}}(\mathsf{{x}};t), \end{align}

where $\beta = \rho /\rho _0$ is the dimensionless density ratio at each spatial location (a scalar), and $\mathsf{{\beta }}$ denotes its spatially discretised array representation, quantifying deviations from the nominal reference density, $\rho _0$ . With this definition, the total KE within the computational domain can be expressed using the energy norm $\|\boldsymbol{\cdot }\|_{\!y}$ , induced by the inner product, as

(4.2) \begin{align} K \equiv \int _\varOmega \frac {1}{2}\beta (\boldsymbol{x}) \boldsymbol u^{\textit{H}}(\boldsymbol{x})\boldsymbol{u}(\boldsymbol{x})\,\text{d} \boldsymbol{x} \approx \frac {1}{2} \mathsf{{y}}^{\textit{H}}\boldsymbol{W}_{\!y} \mathsf{{y}} =\frac {1}{2} \langle \mathsf{{y}},\mathsf{{y}}\rangle _{\!y} \equiv \frac {1}{2} \|\mathsf{{y}}\|_{\!y}^2. \end{align}

Here, $\boldsymbol{W}_{\!y}$ is a Hermitian positive–definite weight matrix for the discrete approximation of the volume integral, $\sum _{\varOmega }( \, \boldsymbol{\cdot }\, )\varDelta _x\varDelta _{\!y}$ , and $( \, \boldsymbol{\cdot }\, )^{\textit{H}}$ denotes the Hermitian transpose. The density ratio is introduced solely as a weighting function for the velocity field. For nearly incompressible flows, as in the present configuration, density variations are weak, and their weighting has only a minor effect ( $\beta \approx 1$ ). Nevertheless, we retain this formulation to account for the contributions of both velocity and density fluctuations to the dissipation.

We seek spatio-temporal structures that optimally represent the spectral content of $\mathsf{y}$ , expressed through its Fourier expansion

(4.3) \begin{align} \mathsf{{y}}(t) = \sum _{n=-\infty }^\infty \widehat {\mathsf{{y}}}_n \mathrm{e}^{\textit{i} 2\pi f_n t} . \end{align}

Assuming ergodicity in statistically stationary flows, we apply Welch’s method (Welch Reference Welch1967) to obtain convergent estimates of the spectral densities. Specifically, the time series is divided into $N_{\textit{blk}}$ blocks, each containing $N_{\textit{DFT}}$ consecutive snapshots separated by a constant time step $\Delta t$ . Adjacent blocks may overlap by $N_{\textit{ovlp}}$ snapshots. Guidance on the appropriate selection of these parameters is provided in Schmidt & Colonius (Reference Schmidt and Colonius2020). In practice, the $m$ th block, for $m=1,2,\ldots ,N_{\textit{blk}}$ , is constructed as

(4.4) \begin{align} \mathsf{{y}}^{(m)} &= \left [ \begin{matrix} \mathsf{{y}}_{1}^{(m)}& \mathsf{{y}}_{2}^{(m)} &{\cdots} & \mathsf{{y}}_{N_f}^{(m)} \end{matrix}\right ] \in \mathbb{C}^{N\times N_{\textit{DFT}}}, \end{align}

and its discrete Fourier transform (DFT) along the temporal domain is denoted by

(4.5) \begin{align} \widehat {{\boldsymbol{Y}}}^{(m)} &= \left [ \begin{matrix} \widehat {\mathsf{{y}}}_{1}^{(m)}& \widehat {\mathsf{{y}}}_{2}^{(m)} &{\cdots} & \widehat {\mathsf{{y}}}_{N_f}^{(m)} \end{matrix}\right ]\!. \end{align}

Here, only the non-negative Fourier components are collected due to the reality of the data, yielding $N_{f}=\lfloor N_{\textit{DFT}}/2 \rfloor +1$ frequency components. For each frequency index $n=1,2,\ldots ,N_f$ , the ensemble of $N_{\textit{blk}}$ Fourier realisations at frequency $f_n$ across all data blocks forms a data matrix

(4.6) \begin{align} \widehat {{\boldsymbol{Y}}}_n &= \left [ \begin{matrix} \widehat {\mathsf{{y}}}_{n}^{(1)}& \widehat {\mathsf{{y}}}_{n}^{(2)} &{\cdots} & \widehat {\mathsf{{y}}}_{n}^{(N_{\textit{blk}})} \end{matrix}\right ]\!. \end{align}

The corresponding cross-spectral density matrix is

(4.7) \begin{align} \boldsymbol{S}_n \equiv \frac {1}{N_{\textit{blk}}}\widehat {{\boldsymbol{Y}}}_n\widehat {{\boldsymbol{Y}}}_n^{\textit{H}}. \end{align}

The eigenvectors, $\widehat {\boldsymbol{\varXi }}_{n}$ , and the eigenvalues, $\boldsymbol{\varLambda }_n$ , of the eigenvalue problem

(4.8) \begin{align} \boldsymbol{S}_n \boldsymbol{W}_{\!y} \widehat {\boldsymbol{\varXi }}_{n} = \boldsymbol{\varLambda }_n \widehat {\boldsymbol{\varXi }}_{n}, \end{align}

give the SPOD modes and their corresponding modal energy. The SPOD modes are the columns of $\widehat {\boldsymbol{\varXi }}_{n}$ , denoted by ${{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n}^{(k)}$ , where $k$ is the column index. The eigenvalues are sorted in decreasing order of energy (by singular value) as

(4.9) \begin{gather} \lambda _n^{(1)}\geq \lambda _n^{(2)}\geq {\cdots} \geq \lambda _n^{(N_{\textit{blk}})}. \end{gather}

At the same frequency, the SPOD modes are mutually orthogonal as

(4.10) \begin{gather} \langle {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k_1)}, {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k_2)}\rangle _{\!y} = \delta _{k_1,k_2}, \end{gather}

and are optimal for modal energy under the norm induced by (4.2). Here, $\delta _{\boldsymbol{\cdot },\boldsymbol{\cdot }}$ is the Kronecker delta function, and the indices $k_1,k_2 =1,2,{\cdots} ,N_{\textit{blk}}$ refer to the modal ranks. We refer the reader to Towne et al. (Reference Towne, Schmidt and Colonius2018) for more details of SPOD. Spectral proper orthogonal decomposition provides an optimal decomposition of each realisation of the flow field into energy-ranked structures as

(4.11) \begin{align} \widehat {\mathsf{{y}}}_n = \sum _{k=1}^{N_{\textit{blk}}} a_n^{(k)} {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{n}^{(k)}, \quad \text{where} \quad a_n^{(k)} \equiv \langle {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{n}^{(k)},\widehat {\mathsf{{y}}}_n\rangle _{\!y}, \end{align}

is the corresponding SPOD expansion coefficient. Across different ranks at a given frequency, the SPOD expansion coefficients are uncorrelated (Nekkanti & Schmidt Reference Nekkanti and Schmidt2021b ; Chu & Schmidt Reference Chu and Schmidt2025), that is

(4.12) \begin{align} \mathrm{E}\left \{ \left ({a_n^{(k_1)}}\right )^{\textit{H}} a_n^{(k_2)} \right \} = \lambda _{n}^{(k_1)} \delta _{k_1,k_2}. \end{align}

4.2. Representative SPOD eigenvalue spectrum and modes

Spectral analysis is performed on the reduced domain $\varOmega$ defined and justified in § 3.2. The SPOD sampling interval, $\Delta t_{\textit{spod}}$ , is set to 2000 simulation time steps for $\textit{St} \le 8 \textit{St}_{0}$ and 1500 time steps for $\textit{St} \ge 10\textit{St}_0$ . The SPOD is performed using 640 snapshots ( $N_t = 640$ ), partitioned into nine blocks with 87.5 % overlap with a block length $N_{\textit{DFT}} = 320$ . A rectangular window is used because the acoustic signal is periodic within each analysis segment. Spectral convergence is confirmed by the large separation between the leading and suboptimal SPOD eigenvalues, indicating accurate identification of dominant coherent structures.

Figure 14.

The SPOD-based spectral energy analysis for $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{St}=4\textit{St}_0$ and $\textit{Re}=\textit{Re}_0$ : (a) eigenvalue spectra, $\lambda ^{(k)}_n$ ; and (b) energy associated with each rank, $\eta ^{(k)}$ , at four harmonic frequencies ( $l=1$ – $4$ defined in (4.13)).

As an example, figure 14(a) shows the SPOD eigenvalue spectrum for the case of 150 dB with a Reynolds number of $\textit{Re}_0$ , where the imposed acoustic frequency $f_s={2}\,\textrm{kHz}$ corresponds to a Strouhal number of $\textit{St}=4\textit{St}_0$ . The SPOD eigenvalues of the leading rank exhibit four tonal peaks of decreasing magnitude at integer multiples of $f_s$ , implying a harmonic response. We introduce the harmonic index

(4.13) \begin{align} l \equiv f_n/f_s, \end{align}

such that the fundamental frequency corresponds to $l=1$ and higher-order harmonics are given by $l=2,3,{\cdots} ,\lfloor (1/(2\Delta t_{\textit{spod}} f_s)\rfloor$ , up to the Nyquist limit. These spectral peaks indicate the presence of coherent vortex shedding triggered by the imposed acoustic excitation (Tam et al. Reference Aulitto, Hirschberg, Arteaga and Buijssen2001, Reference Awasthi, McCreton, Moreau and Doolan2005). The remaining eigenvalues form a broadband spectrum that decreases monotonically. This behaviour is characteristic of mixed broadband–tonal flows, whose complex dynamics combines both stochastic and deterministic features.

Figure 14(b) shows the distribution of energy across the mode ranks for the four tonal peaks. The magnitude of the leading SPOD rank is at least one order of magnitude larger than that of the suboptimal ranks. This pronounced separation reflects the distinct low-rank dynamics of the flow field. As later shown in figure 16, these trends are qualitatively consistent across all parameter combinations listed in table 1. We define the contribution of each rank $k$ , for $k=1,2,{\cdots} ,N_{\textit{blk}}$ , via the energy fraction as

(4.14) \begin{align} \eta ^{(k)}= \sum _n \eta _n^{(k)} = \frac {\sum _n \lambda _n^{(k)}}{\sum _n\sum _k \lambda _n^{(k)}}, \end{align}

which measures the portion of modal energy. Accordingly, we focus on the first three ranks, which represent 95 %, 3.5 % and 1 % of the total energy, respectively. Collectively, these three modes account for 99 % of the energy across the examined parameter combinations. Contributions from higher-order suboptimal modes are neglected in the spectral analysis because of their low energy content.

Figure 15.

Leading SPOD modes, $\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l}^{(1)} )$ , at $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{St}=4\textit{St}_0$ and $\textit{Re}=\textit{Re}_0$ . Cases in (a)–(c) correspond to $l=1$ , $2$ and $3$ . Cases in (i) and (ii) correspond to velocities in the $x$ - and $y$ -coordinate directions, $u$ and $v$ .

Figure 15 shows the leading fundamental, 2nd and 3rd harmonic SPOD modes for the case examined in figure 14. The spatial structures are active both upstream and downstream of the slit, reflecting the periodic nature of the acoustically driven flow field. Across all harmonics, the modes are concentrated near the slit, with pronounced intensities at the four slit corners. These localised structures highlight strong interactions between the flow and the slit geometry. In particular, the induced oscillatory boundary-layer dynamics enables the conversion of incident acoustic energy into coherent vortical motion. As expected, the spatial modes exhibit smaller-scale structures as the harmonic order increases. This trend is consistent with the higher effective excitation frequencies associated with harmonic tones, which induce faster oscillations in the flow field.

The slight asymmetry of the modes about the horizontal centreline, especially in the $y$ -direction velocity component, is consistent with observations from previous analyses based on both numerical simulations (Qiang et al. Reference Qiang, Wang and Liu2022) and experimental measurements (Tang et al. Reference Tang, Wang and Liu2025). Since the configuration is geometrically symmetric about the centreline, it reflects the dominant coherent structures captured by SPOD rather than a strict symmetry breaking of the statistically averaged flow. In the instantaneous vorticity field in figure 5(a), vortex shedding is not symmetric with respect to the centreline, and the leading SPOD modes inherit this pattern. Such asymmetric coherent structures may subsequently be amplified through nonlinear vortex interactions (Mizushima & Hatsuda Reference Mizushima and Hatsuda2014; Frantz et al. Reference Frantz, Mimeau, Salihoglu, Loiseau and Robinet2025). Nevertheless, full statistical convergence is more difficult to achieve in regimes with stronger broadband content (lower $\textit{St}$ and higher $\textit{Re}$ ), and additional data would further sharpen the modes in that range.

4.3. Spectral features across the parameter space

To understand the spectral behaviour, we compare the SPOD eigenvalue spectra across the parameter space at $\textit{ISPL} = {150}\,\textrm{dB}$ , as shown in figure 16. In general, the harmonic peaks become more distinct from the broadband content for smaller $\textit{Re}$ . This trend is consistent with the reduced spectral density at lower $\textit{Re}$ , which lowers the broadband energy level and thereby enhances the spectral contrast between coherent harmonic peaks and incoherent broadband components. In addition, both the leading and suboptimal ranks exhibit more visible harmonic peaks at lower $\textit{Re}$ , without a reduction in the separation between their SPOD eigenvalues. This observation suggests that increased viscosity at smaller $\textit{Re}$ promotes the coherence of fluctuations at the forcing frequency across cycles, thereby suppressing broadband fluctuations.

Figure 16.

The SPOD eigenvalue spectra, $\lambda _n^{(k)}$ , at different ranks $k$ for different $\textit{St}$ – $\textit{Re}$ combinations at $\textit{ISPL} = {150}\,\textrm{dB}$ . Cases in column (a)–(c) represent $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ . Cases in row (i)–(iv) represent $\textit{St}=\textit{St}_0$ , $4\textit{St}_0$ , $8\textit{St}_0$ and $12\textit{St}_0$ . The fundamental frequency ( $l=1$ ) and its higher-order harmonics ( $l \ge 2$ ) manifest as peaks in the spectrum. For example, two tonal peaks arise at $f_n = {4}\,\textrm{kHz}$ ( $l=1$ ) and $f_n = {8}\,{kHz}$ ( $l=2$ ) in case (a, iii).

An increase in acoustic excitation frequency also leads to stronger harmonic peaks. At $\textit{St}=12\textit{St}_0$ , additional peaks appear at non-integer multiples of the forcing frequency and become more pronounced at lower $\textit{Re}$ . These additional peaks suggest more complex nonlinear interactions within the flow field, involving not only vortex-shedding harmonics but also interactions with the slit boundaries. This behaviour can be qualitatively observed in the instantaneous fields shown in figure 5, where the shed vortices at $\textit{St}=12\textit{St}_0$ travel along and interact with the $y$ -direction boundaries of the slit, rather than propagating in phase with the acoustic wave.

Figure 17.

Leading SPOD modes, $\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)} )$ , for different $\textit{St}$ – $\textit{Re}$ combinations at $\textit{ISPL} = {150}\,\textrm{dB}$ , plotted at their respective fundamental frequencies ( $l=1$ ) as their $x$ -directional components, $u$ . Cases in column (a)–(c) represent $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ . Cases in row (i)–(iv) represent $\textit{St}=\textit{St}_0$ , $4\textit{St}_0$ , $8\textit{St}_0$ and $12\textit{St}_0$ .

Figure 17 presents the leading $x$ -direction SPOD modes at their respective fundamental frequencies, corresponding to the cases shown in figure 16. The spatial axes follow the same convention as in figure 15, but are omitted here for clarity. At the lowest acoustic frequency of $\textit{St} = \textit{St}_{0}$ , a decrease in Reynolds number leads to more flattened large-scale structures along the horizontal centreline, while the smaller-scale structures are diminished. Combined with the corresponding SPOD eigenvalue spectra, where lower- $\textit{Re}$ cases exhibit reduced broadband energy and more pronounced spectral peaks, these observations suggest that stronger viscous effects at this frequency enhance the coherence of vortex shedding. This enhancement occurs while suppressing the remaining flow content.

For larger acoustic frequency, the leading SPOD modes exhibit similarly compact spatial support near the slit, with reduced dependence on $\textit{Re}$ . Specifically, at $\textit{St} = 4\textit{St}_{0}$ , the modes display dumbbell-shaped structures spanning the slit, whereas at $\textit{St} = 8\textit{St}_{0}$ and $12\textit{St}_{0}$ , they exhibit X-shaped patterns localised near the slit. With a virtually doubled slit thickness (i.e. the Womersly number $\textit{Wo}$ doubled), the case with $\textit{St} = 8\textit{St}_0$ and $\textit{Re} = \textit{Re}_0$ exhibits a more confined X-shaped pattern as opposed to the case with $\textit{St} = 4\textit{St}_0$ and $\textit{Re} = \textit{Re}_0/2$ , which appears to be dumbbell shaped. Figure 31 in Appendix C documents the modal structures of the cases not shown in figure 17.

Taken together, these observations indicate that increasing the acoustic frequency or the slit thickness compresses the dominant vortex-shedding structures toward the near-slit region. At the same time, more confinement can introduce complex interactions near and within the slit.

Figure 18.

Profiles of the $x$ -directional components, $u$ , of the leading SPOD modes at the fundamental frequency, $\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)} )/\max \{\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)})\}$ , within the slit for different $\textit{St}$ – $\textit{Re}$ combinations at $\textit{ISPL} = {150}\,\textrm{dB}$ . The maximum mode gradient in the $y$ direction on the top and bottom halves of each profile, $|\partial \textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)} )/\partial y|_{\textit{max}}$ , is marked as a hollow circle, which is connected by arrows across all the profiles. Cases in column (a)–(c) correspond to $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ . Cases in row (i)–(iv) correspond to $\textit{St}=\textit{St}_0$ , $4\textit{St}_0$ , $8\textit{St}_0$ and $12\textit{St}_0$ .

To have a detailed understanding of mode structure inside the slit, we show the normalised velocity profiles at the inlet ( $x=-d/2$ ), centre ( $x=0$ ) and outlet ( $x=d/2$ ) of the slit in figure 18. For most $\textit{St}$ – $\textit{Re}$ combinations, the inlet and outlet profiles are nearly identical, indicating weak $x$ -direction variation of the fundamental mode between the inlet and outlet. We calculated the mode gradient in the $y$ direction using a first-order finite-difference approximation. We highlighted two points on each profile where the maximum mode gradients occur in both the top and bottom halves. These points have implications on the distribution of the VL field, which will be discussed in § 4.5. At the lower $\textit{St} \le 4\textit{St}_0$ and higher $\textit{Re} \ge \textit{Re}_0/2$ , the profiles are asymmetric about the horizontal centreline. In this regime, the oscillatory motion produces a thicker boundary layer, and we observe transient reversal of the wall shear stress and a localised flow detachment at the slit corner, consistent with the larger-scale mode structures seen in figure 17. In contrast, at $\textit{St} \gt 4\textit{St}_0$ and $\textit{Re} \lt \textit{Re}_0/2$ , the profile is similar to the Womersley flow profile with a Womersley number $\textit{Wo} \ge 16$ (Womersley Reference Womersley1955), exhibiting symmetrical structure according to the horizontal centreline. The increased acoustic frequency yields a thinner boundary layer.

4.4. Three-dimensional effects

The present database and SPOD-based energetics analysis are obtained from DNSs. This choice enables a broad sweep in $(\textit{ISPL},\,\textit{St},\,\textit{Re})$ at tractable cost. Nevertheless, acoustic-driven slit flows may, in principle, develop a 3-D dynamics, including spanwise vortex stretching, secondary instabilities of shear layers and eventual breakdown to turbulence. Such mechanisms are excluded by construction in two dimensions and may alter both the near-slit vortex dynamics and the spatial distribution of dissipation under certain operating conditions. To assess the role of three-dimensionality, we compare the 2-D and 3-D simulations at $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{Re} = \textit{Re}_0/3$ and $\textit{St} = 2\textit{St}_0$ . Two 3-D slit configurations are examined, with spanwise extents $L_{z,1}=11d$ and $L_{z,2}=22d$ , as described in figure 3.

We obtain $\alpha = 0.1628$ for the 2-D case and $\alpha = 0.1450$ for both 3-D cases, a difference of roughly 11 % that reflects the additional spanwise degrees of freedom in three dimensions. To enable direct comparison, an approximate spanwise average of the 3-D streamwise velocity field, $\langle u\rangle _z(x,y,t)$ , is constructed by sampling $10$ evenly spaced ( $x$ , $y$ ) planes across the span. The SPOD procedure described in § 4 is subsequently applied to $\langle u\rangle _z$ , using $N_t=320$ time snapshots instead of $640$ in the main database, owing to the shorter 3-D simulation. As shown in figure 19(a), the leading SPOD mode of the 3-D simulation exhibits a near-slit spatial structure that is almost identical to that of the 2-D case, even for the shorter spanwise extent $L_{z,1}=11d$ . Figure 19(b) shows that, for this parameter combination, the leading SPOD eigenvalue spectrum from the 3-D cases is quantitatively consistent with the corresponding 2-D spectrum. Overall, these results suggest that the 2-D model for this configuration represents well the dominant flow structures and statistics.

Figure 19.

Comparison of the 2-D and 3-D simulation on (a) leading SPOD modes, $\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)} )$ and (b) eigenvalue spectrum $\lambda _n^{(1)}$ at $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{Re} = \textit{Re}_0/3$ and $\textit{St} = 2\textit{St}_0$ plotted at their respective fundamental frequencies ( $l=1$ ) as their $x$ -directional components, $u$ .

Figure 20.

Spanwise Fourier decomposition of the fluctuating KE in the 3-D DNS. The energy is concentrated at the spanwise-uniform component ( $k_z=0$ ), with negligible contribution from non-zero spanwise wavenumbers, indicating an effectively 2-D response over the analysed interval.

A spanwise-wavenumber analysis further supports this interpretation. We quantify three-dimensionality by decomposing the $x$ -component fluctuating velocity field into spanwise Fourier modes and quantifying the KE contained in each spanwise wavenumber. Figure 20 shows that the energy in non-zero spanwise wavenumbers remains negligible compared with the $k_z=0$ component, indicating an effectively 2-D response for these cases. Together with the SPOD eigenvalue spectra and modes shown in figure 19, these results suggest that the 2-D DNSs can capture the dominant coherent structures and energetics observed in the corresponding 3-D simulations for the specific parameter combination examined, even at the high acoustic amplitude of $\textit{ISPL}={150}\,\textrm{dB}$ . The subsequent flow analysis can therefore be effectively carried out in the two-dimensional domain at substantially reduced cost, without loss of the essential physics.

4.5. Dissipation mechanism

The two principal contributions to the energy budget associated with the consumption of the input acoustic energy are the KE, dominated by vortical dynamics at the shedding frequency, and the VL, spanning all frequency components (Tam & Kurbatskii Reference Tam and Kurbatskii2000; Tam et al. Reference Tam, Kurbatskii, Ahuja and Gaeta2001). Here, KE quantifies the energy stored in the flow over the analysis window, whereas VL quantifies the portion of energy irreversibly dissipated by viscosity. Accordingly, KE and VL are distinct quantities in the energy budget rather than overlapping dissipation channels: the KE of shed vortices will eventually be viscously dissipated, but this subsequent dissipation occurs outside the analysis window and is not double counted in VL. By decomposing the flow field into optimally energy-ranked SPOD modes, we analyse the respective KE and VL contributions of each spectral component. For each frequency $f_n$ , we express the velocity and stress tensor as linear combinations of the SPOD modes, ${\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{n}^{(k)}$ , and corresponding expansion coefficients, $a_n$ , as

(4.15) \begin{align} \widehat {\mathsf{{y}}}_n(\mathsf{{x}}) \approx \sum _{k=1}^{3} a_n^{(k)}{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{n}^{(k)}(\mathsf{{x}}), \quad \text{and} \quad \widehat {\mathsf{{T}}}_n(\mathsf{{x}}) \approx \sum _{k=1}^{3} a_n^{(k)}{\widehat {\boldsymbol{\boldsymbol{{T}}}}}_{n}^{(k)}(\mathsf{{x}}), \end{align}

where

(4.16) \begin{gather} {\widehat {{\boldsymbol{{T}}}}}_{n}^{(k)}(\mathsf{{x}}) \equiv \frac {1}{\overline {\beta } \textit{Re}} \left [\boldsymbol{\nabla }{\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)}+\left ( \boldsymbol{\nabla }{\widehat {{\boldsymbol{{\xi }}}}}_{n}^{(k)}\right )^{\textit{H}} -\frac {2}{3}\left (\boldsymbol{\nabla } \boldsymbol{\cdot }{\widehat {{\boldsymbol{{\xi }}}}}_{n}^{(k)}\right )\mathsf{{I}}\right ](\mathsf{{x}}) \end{gather}

is the rank- $k$ stress-tensor mode.

To enable a direct, dimensionally consistent comparison between the spectral KE and VL components, we weight the KE spectrum by the corresponding angular frequency, $\omega _n=2\pi f_n$ . This process converts the spectral KE density (energy per frequency) into an energy-rate scale (power per frequency) with units consistent with the spectral VL density. Frequency weighting is introduced to maintain dimensional consistency and facilitate comparison, whereas a budget interpretation requires applying the same scaling consistently to all terms of the spectral KE equation. The frequency-weighted spectral KE field can then be expressed as

(4.17) \begin{align} \widehat {\boldsymbol{{K}}}_n(\mathsf{{x}}) = \frac {1}{2}\mathrm{E}\left \{\left [\omega _n {\widehat {\mathsf{{y}}}_n}^{\textit{H}}{\widehat {\mathsf{{y}}}_n} \right ](\mathsf{{x}})\right \} & \approx \pi f_n \mathrm{E}\left \{\left [ {\left (\sum _{k=1}^{3}a_n^{(k)}{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k)}\right )}^{\textit{H}}\left (\sum _{k=1}^{3}a_n^{(k)}{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k)}\right ) \right ](\mathsf{{x}})\right \} \\[-12pt]\nonumber \end{align}

(4.18) \begin{align} &= \pi f_n \mathrm{E}\left \{ \sum _{k_1=1}^{3} \sum _{k_2=1}^{3}\left [ \left ({a_n^{(k_1)}}\right )^{\textit{H}} a_n^{(k_2)} \left ({{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k_1)}}\right )^{\textit{H}} {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k_2)} \right ](\mathsf{{x}})\right \} \\[-12pt]\nonumber \end{align}

(4.19) \begin{align} & = \pi f_n \sum _{k=1}^{3} \left [ \lambda _{n}^{(k)} \left ({{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k)}}\right )^{\textit{H}} {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_n^{(k)} \right ](\mathsf{{x}}) = \sum _{k=1}^{3} {\widehat {\boldsymbol{\boldsymbol{K}}}}_n^{(k)} (\mathsf{{x}}). \end{align}

The simplification from (4.18) to (4.19) follows from the mutual uncorrelatedness of the SPOD coefficients shown in (4.12). Following the canonical treatment of VL (Landau & Lifshitz Reference Landau and Lifshitz1987), the total spectral VL can be decomposed into shear (deviatoric) and dilatational (compressible) contributions by separating the terms of the viscous stress tensor in (2.4) into its symmetric and divergence-related parts. In particular, we focus on the shear-driven contribution associated with the symmetric strain-rate tensor

(4.20) \begin{align} \widehat {\boldsymbol{{D}}}_n(\mathsf{{x}}) \approx \mathrm{E}\left \{\left [\widehat {\mathsf{{T}}}_n:\boldsymbol{\nabla }{\widehat {\mathsf{{y}}}_n} \right ](\mathsf{{x}})\right \} &\approx \mathrm{E}\left \{\left [ \left ( \sum _{k=1}^{3} a_n^{(k)}{\widehat {{\boldsymbol{{T}}}}}_n^{(k)} \right ): \boldsymbol{\nabla }{\left (\sum _{k=1}^{3}a_n^{(k)}{\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)} \right )} \right ](\mathsf{{x}})\right \} \\[-12pt]\nonumber \end{align}

(4.21) \begin{align} & = \sum _{k=1}^{3} \lambda _n^{(k)}\left [ {\widehat {{\boldsymbol{{T}}}}}_n^{(k)}: \boldsymbol{\nabla }{\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)} \right ](\mathsf{{x}}) = \sum _{k=1}^{3} {\widehat {\boldsymbol{D}}}_n^{(k)} (\mathsf{{x}}). \end{align}

which represents dissipation associated with strain rates. Here, ${{\widehat {\boldsymbol{D}}}}_n^{(k)}$ and ${\widehat {\boldsymbol{D}}}_n$ are effectively real with negligible imaginary components. The compressible term depends on the dilatation, $\boldsymbol{\nabla }\boldsymbol{\cdot }\hat {\mathsf{{u}}}$ , and associated density gradient, $\boldsymbol{\nabla }\beta$ , which together represent volumetric and pressure-related damping. As shown in figure 36 in Appendix E, the compressible term contributes less than 1 % to the total, and can be reasonably neglected in the present analysis.

The rank- $k$ spectral KE and VL fields take the forms of

(4.22) \begin{gather} {\widehat {\boldsymbol{K}}}_n^{(k)} (\mathsf{{x}}) = \pi f_n \lambda _{n}^{(k)} \left [ \left ({{\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)}}\right )^{\textit{H}} {\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)} \right ](\mathsf{{x}}) \quad \text{and} \quad {\widehat {\boldsymbol{D}}}_n^{(k)}(\mathsf{{x}}) = \lambda _{n}^{(k)} \left [ {\widehat {{\boldsymbol{{T}}}}}_n^{(k)}:\boldsymbol{\nabla }{\widehat {{\boldsymbol{{\xi }}}}}_n^{(k)} \right ](\mathsf{{x}}), \end{gather}

respectively. Their respective integral measures within the computational domain are

(4.23) \begin{gather} \widehat {{{K}}}_n^{(k)} = \sum _{\varOmega }{\widehat {\boldsymbol{K}}}_n^{(k)}(\mathsf{{x}}) \varDelta _x\varDelta _{\!y} = \pi f_n \lambda _n^{(k)}, \quad \text{and} \quad \widehat {{{D}}}_n^{(k)} = \sum _{\varOmega }{\widehat {\boldsymbol{D}}}_n^{(k)}(\mathsf{{x}}) \varDelta _x\varDelta _{\!y}, \end{gather}

where the normality of SPOD modes in (4.10) is used. Summing the contributions from the first three ranks yields

(4.24) \begin{gather} \widehat {{{K}}}_n = \sum _{\varOmega }\widehat {\boldsymbol{{K}}}_n(\mathsf{{x}}) \varDelta _x\varDelta _{\!y} = \pi f_n \left (\lambda _n^{(1)}+\lambda _n^{(2)}+\lambda _n^{(3)}\right )\!, \quad \text{and} \quad \widehat {{{D}}}_n = \sum _{\varOmega } \widehat {\boldsymbol{{D}}}_n(\mathsf{{x}}) \varDelta _x\varDelta _{\!y}. \end{gather}

We approximate the integration over the discrete frequency $f_n$ up to the Nyquist frequency using the summation operator

(4.25) \begin{gather} \mathcal{S}( \, \boldsymbol{\cdot }\, )=\sum _{f_n\in \left (0, 1/(2\Delta t_{\textit{spod}})\right ]}( \, \boldsymbol{\cdot }\, )\,\Delta f_n, \end{gather}

where $\Delta f_n=1/(N_{\textit{DFT}}\Delta t_{\textit{spod}})$ is the frequency-bin width. In practice, the sum is evaluated using the trapezoidal rule. The rank- $k$ contributions, obtained by integrating over the frequency domain, are defined as

(4.26) \begin{align} \widehat {\mathcal{K}}^{(k)} = \mathcal{S} (\widehat {{K}}_n^{(k)}), \quad \text{and} \quad \widehat {\mathcal{D}}^{(k)} = \mathcal{S} (\widehat {{D}}_n^{(k)}), \end{align}

respectively. Likewise, the corresponding total (all-rank) contributions are

(4.27) \begin{align} {\mathcal{K}} = \mathcal{S} (\widehat {{K}}_n), \quad \text{and} \quad {\mathcal{D}} = \mathcal{S} (\widehat {{D}}_n). \end{align}

From Parseval’s theorem, the spectral integral of the viscous-loss density, $\mathcal{D}$ , recovers the mean irreversible dissipation power, $\overline {D}$ . The integrated frequency-weighted KE is a power-like spectral moment satisfying $\mathcal{K}=\overline {K}\overline {\omega }_k$ , where $\overline {K}$ is the mean KE and $\overline {\omega }_k$ is the energy-weighted characteristic angular frequency. In general, $\mathcal{K}$ provides a convenient rate scale for comparison with $\mathcal{D}$ but does not, by itself, imply closure of the spectral KE budget. In the present study, the KE spectrum is strongly concentrated around the vortex-shedding frequency such that $\overline {\omega }_k$ is dominated by that peak, making $\mathcal{K}$ an effective single-time-scale measure of the KE dynamics. This interpretation is consistent with the metric used by Tam et al. (Reference Tam, Kurbatskii, Ahuja and Gaeta2001), which quantifies cycle-based energy conversion by averaging the KE of large-scale shedding vortices over an acoustic period.

Figure 21.

Acoustic energy dissipation at $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{St} = 4\textit{St}_0$ and $\textit{Re} = \textit{Re}_0$ : (a) energy dissipation spectra; and (b) spectral VL and frequency-weighted KE fields at harmonic indices $l=1$ (i) and $l=2$ (ii), together with the field as $\mathcal{S}({\widehat {{\boldsymbol{D}}}_n})(\mathsf{{x}})$ for VL and $\mathcal{S}({\widehat {{\boldsymbol{K}}}_n})(\mathsf{{x}})$ for KE contributions of all spectral components (iii).

Consistent with this representative case of $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{St}=4\textit{St}_0$ analysed in figure 14, figure 21 presents the corresponding spectral KE and VL spectra, computed from the contributions of the first three leading SPOD modes. Within the analysis domain $\varOmega$ , incident acoustic energy is primarily damped by conversion into KE of the induced flow fluctuations and irreversible viscous attenuation (Tam et al. Reference Tam, Kurbatskii, Ahuja and Gaeta2001). We quantify these contributions using the spectral KE and VL components and define the total mode-frequency-resolved contribution as

(4.28) \begin{align} {\widehat {\boldsymbol{P}}}_n^{(k)}(\mathsf{{x}})= {\widehat {\boldsymbol{K}}}_n^{(k)}(\mathsf{{x}})+{\widehat {\boldsymbol{D}}}_n^{(k)}(\mathsf{{x}}), \quad \widehat {P}^{(k)}_n \equiv \widehat {K}^{(k)}_n + \widehat {D}^{(k)}_n \quad \text{and} \quad {\mathcal{P}} \equiv {\mathcal{K}} + {\mathcal{D}}. \end{align}

Here, ${\widehat {{\boldsymbol{P}}}}_n^{(k)}$ provides an estimate of the total spectral density of acoustic energy dissipation for mode $k$ , combining energy stored in coherent vortical motion and energy irreversibly converted to heat through viscosity. In this representative case, the $\widehat {D}_n$ is roughly 20 % of $\widehat {K}_n$ , indicating that most of the incident acoustic energy loss results from conversion into KE. In contrast, VLs still make a non-negligible contribution.

Figure 21(b) shows the spectral KE and VL fields associated with the fundamental frequency, 2nd harmonic frequency and spectral integration, respectively. At the fundamental frequency, both the spectral VL and KE fields show dominant structures, with the VL field concentrated within the slit and the KE field extending slightly outside the slit opening. At the second harmonic, the spectral VL field is localised near the slit corners. Due to the higher frequency, the associated vorticity exhibits reduced coherence. The vorticity cannot be convected as easily into the central region of the slit, resulting in concentrated dissipation at the slit periphery. This localisation is consistent with the reduced propagation distance of high-frequency vortical structures. The spectral KE field at the second harmonic is primarily located outside the slit, with its magnitude approximately 25 % that at the fundamental frequency. The limited spatial extent and energy content of this mode suggest that the KE is dissipated before forming large-scale coherent structures, which aligns with the localised dissipation observed in the spectral VL field. The overall VL field, $\mathcal{S} ({\widehat {{\boldsymbol{D}}}_n} )(\mathsf{{x}})$ , remains confined to the slit region, reinforcing the role of boundary-layer interactions as the dominant mechanism of viscous energy losses. In contrast, the overall KE field, $\mathcal{S} ({\widehat {{\boldsymbol{K}}}_n} )(\mathsf{{x}})$ , is more broadly distributed across the domain, with significant upstream and downstream structures.

Figure 22.

Profiles of the $x$ -directional components, $u$ , of the real component of the leading SPOD modes at the fundamental frequency, $ \textrm{Re} ({{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)}/\max \{{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)}\}})$ , within the slit for $\textit{ISPL} = {150}\,\textrm{dB}$ , $\textit{St}=4\textit{St}_0$ and $\textit{Re}=\textit{Re}_0$ . The maximum mode gradient in $y$ direction on the top and bottom halves of each profile, $|\partial \textrm{Re} ({{\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)}})/\partial y|_{\textit{max}}$ , is marked as a hollow circle. The maximum mode gradient aligns the maximum VL at the fundamental frequency observed in figure 21, using ${\widehat {{\boldsymbol{D}}}_{l=1}}(\mathsf{{x}})$ .

To confirm the role of the boundary layer in viscous dissipation, figure 22 shows the normalised $x$ -direction velocity profiles of the fundamental mode within the slit. These profiles are used to assess the spatial alignment between regions of maximum velocity gradients and maximum VL. The profiles exhibit a Poiseuille-like core in the central region of the slit, primarily driven by the pressure gradient between the upstream and downstream sides of the slit. This pressure difference also reflects the energy losses of the acoustic waves as they propagate through the slit. Near the walls, the profiles show velocity sign reversal, indicative of a pressure gradient reversal and consistent with the strong boundary-layer dissipation observed in the spectral VL field. Near the walls, and in particular at the slit corners, the profiles deviate from the Stokes-layer solution for channel flow, demonstrating that corner detachment is distinct from the simple phase reversal of an oscillating Stokes layer driven by the alternating acoustic pressure gradient. It is instead associated with a transient reversal of the wall shear stress and a localised flow detachment at the slit corner. The steepest velocity gradients occur within these boundary-layer zones, where the interaction between the oscillatory motion and the no-slip boundary condition generates intense shear. This large boundary-layer zone results in a thicker and more asymmetric dissipation field, consistent with previously discussed modal asymmetries. The spectral VL field is concentrated within the oscillatory boundary layer, aligning with regions where the $y$ -direction gradient of the $x$ -direction velocity, $\partial u/\partial y$ , is maximum. This alignment confirms that the boundary-layer dynamics is responsible for viscous energy losses.

Figure 23.

Frequency-weighted KE and VL spectra for different $\textit{St}$ – $\textit{Re}$ combinations at $\textit{ISPL} = {150}\,\textrm{dB}$ . The fundamental frequencies $(l=1)$ and their higher-order harmonics $(l\geq 2)$ are manifested as distinct peaks in the spectra. Cases in column (a)–(c) represent $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ . Cases in row (i)–(iv) represent $\textit{St}=\textit{St}_0$ , $4\textit{St}_0$ , $8\textit{St}_0$ and $12\textit{St}_0$ .

Figure 23 displays the KE and VL spectra, quantified via spatial integration, across the parameter space, illustrating the dependence of energy production and viscous dissipation on $\textit{St}$ , and $\textit{Re}$ at a fixed input sound pressure level of $\textit{ISPL} = {150}\,\textrm{dB}$ . As $\textit{Re}$ decreases, the VL in the broadband region becomes comparable to the KE rate, and the difference in peak magnitude between KE and VL also diminishes. At higher $\textit{St}$ , the VL spectrum even exceeds the KE spectrum at certain frequencies. This trend is consistent with the inverse proportional relation $\widehat {D}_n \propto 1/\textit{Re}$ , indicating that lower Reynolds numbers suppress irregular activity and reduce the relative contribution of broadband energy dissipation.

At higher $\textit{St}$ , the spectra are dominated by tonal peaks. When $\textit{St}/\textit{St}_0\gt 4$ , the peak amplitudes in both KE and VL spectra exceed the broadband levels by at least two orders of magnitude, suggesting minimal presence of irregular fluctuations. In the extreme case of $\textit{Re}/\textit{Re}_0 = 1/3$ and $\textit{St}/\textit{St}_0 = 12$ , the tonal peak surpasses the broadband level by over six orders of magnitude. Conversely, the opposite extreme case of $\textit{Re}/\textit{Re}_0 = 1$ and $\textit{St}/\textit{St}_0 = 1$ shows a tonal-to-broadband contrast of less than one order of magnitude, indicating stronger irregular activity. These observations are consistent with the SPOD eigenvalue spectra, further confirming that increasing the slit thickness enhances viscous dissipation and leads to more pronounced tonal features. Additional KE and VL spectra at $\textit{ISPL} = {150}\,\textrm{dB}$ have similar features and are included in figure 33 in Appendix C.

Figure 24.

Frequency-weighted KE and VL spectra for different $\textit{Re}$ at $\textit{St} = \textit{St}_0$ and $\textit{ISPL} = {120}\,\textrm{dB}$ . The fundamental frequencies $(l=1)$ and their higher-order harmonics $(l\geq 2)$ are manifested as distinct peaks in the spectra. Cases in column (a)–(c) represent $\textit{Re}=\textit{Re}_0$ , $\textit{Re}_0/2$ and $\textit{Re}_0/3$ .

For comparison, figure 24 displays the KE and VL spectra at $\textit{St}/\textit{St}_0 = 1$ and a lower input sound pressure level of $\textit{ISPL} = {120}\,\textrm{dB}$ , across varying $\textit{Re}$ . At this reduced acoustic forcing, the tonal peaks remain prominent, exceeding broadband levels by at least 4 orders of magnitude in both the KE and VL spectra. However, the overall energy levels are significantly diminished. The entire spectrum is approximately six orders of magnitude lower than that observed at $\textit{ISPL}={150}\,\textrm{dB}$ , reflecting the strong dependence of both KE and viscous dissipation on the ISPL. Dissipation spectra for additional cases are provided in figure 35 in Appendix C.

Figure 25.

Overall KE, $\mathcal{K}$ , and VL, $\mathcal{D}$ , components, obtained by integrating over all frequency components via $\mathcal{S}( \, \boldsymbol{\cdot }\, )$ , across $\textit{St}$ – $\textit{Re}$ combinations for (a) $\textit{ISPL} = {150}\,\textrm{dB}$ and (b) $\textit{ISPL} = {120}\,\textrm{dB}$ .

Figure 25 shows the overall KE and VL obtained by summing the contributions from all frequency components and retaining only the first three leading SPOD modes, across the entire parameter space. At a higher sound pressure level of $\textit{ISPL} = {150}\,\textrm{dB}$ , the total VL accounts for approximately 20 %–60 % of the total KE. The combined total energy, ${\mathcal{P}}={\mathcal{K}}+{\mathcal{D}}$ , exhibits a three-stage trend that aligns with the absorption coefficient shown in figure 8. At low $\textit{St}$ , the total energy shows little sensitivity to $\textit{Re}$ . For intermediate frequencies, $4\leq \textit{St}/\textit{St}_0 \lt 10$ , $\mathcal{P}$ increases monotonically with $\textit{Re}$ . At higher frequencies, $10 \leq \textit{St}/\textit{St}_0 \leq 12$ , the total spectral energy is nearly independent of $\textit{Re}$ , indicating that Reynolds number effects become negligible in this regime because of the absence of significant broadband components. Among the cases, $\textit{Re}=\textit{Re}_0$ consistently yields the highest total energy conversion across the full range of acoustic frequencies, indicating the strongest energy dissipation mechanism. At a lower sound pressure level $\textit{ISPL} = {120}\,\textrm{dB}$ , VL plays a more significant role in the overall dissipation mechanism, ranging from approximately 20 %–180 % of the KE contribution. The $\textit{St}$ - and $\textit{Re}$ -dependence of the spectral KE is weaker than in the higher-amplitude cases, indicating reduced conversion of incident acoustic energy into vortical motion at lower forcing levels. In contrast, for smaller $\textit{Re}$ , VL is larger, highlighting the growing importance of direct viscous attenuation at low ISPL.

The trend of the total energy conversion is consistent with the absorption coefficient shown in figure 8. A fully closed comparison between the SPOD-based acoustic dissipation $\mathcal{P}=\mathcal{D}+\mathcal{K}$ and the far-field acoustic power absorption inferred from the three-microphone method would require evaluating the SPOD over a control volume that includes all boundary flux terms. Performing this full control-volume closure is left for future work.

Figure 26.

Overall KE, $\mathcal{K}$ , and VL, $\mathcal{D}$ , as in figure 25 but plotted against the in-slit Strouhal number $\textit{St}_{\textit{slit,pk}}$ , for (a) $\textit{ISPL} = {150}\,\textrm{dB}$ and (b) ${120}\,\textrm{dB}$ .

Figure 26 recasts the integrated KE and VL against the in-slit Strouhal number. At 150 dB, $\mathcal{K}$ and the total $\mathcal{K}+\mathcal{D}$ are largest at small $\textit{St}_{\textit{slit}}\lesssim 0.2$ , where vortex roll-up is strongest, and decline toward $\textit{St}_{\textit{slit}}=1$ , while $\mathcal{D}$ decreases monotonically; $\textit{Re}=\textit{Re}_0$ gives the largest conversion throughout. At 120 dB, the energies are several orders of magnitude smaller, $\mathcal{D}$ carries a larger share, and the $\textit{Re}$ ordering is clear across the $\textit{St}_{\textit{slit}}$ range. The mode-resolved dissipation is thus described in the same in-slit variables as the impedance and absorption.

Together, these results provide a unified picture for interpreting dissipation mechanisms in acoustic slit flows, specifically those associated with slit-resonator mouths. At low $\textit{St}$ and low ISPL, the incident acoustic energy loss receives comparable contributions from the KE of the induced flow field and spatially distributed VL. For larger $\textit{St}$ and $\textit{ISPL}$ , the dissipation becomes more localised near the slit mouth, and the KE associated with shed vortices comes to dominate the acoustic damping mechanism. This evolving balance between viscous and vortex-driven dissipation mechanisms underlies the nonlinear acoustic damping observed in slit openings at high ISPL, governing energy-transfer efficiency and damping performance.

4.6. The ISPL dependence

In addition to the two baseline forcing levels, $\textit{ISPL}=120$ and ${150}\,\textrm{dB}$ , two intermediate levels, $\textit{ISPL}=140$ and ${145}\,\textrm{dB}$ , are considered to assess the dependence on acoustic amplitude. Figure 27 shows the leading SPOD mode in the four ISPL cases considered, for a representative acoustic excitation with $\textit{St}=2\textit{St}_0$ and $\textit{Re}=\textit{Re}_0/3$ . The corresponding slit Reynolds numbers are $\textit{Re}_{\textit{slit}}=33$ , $325$ , $588$ and $1008$ at $\textit{ISPL}=120$ , $140$ , $145$ and ${150}\,\textrm{dB}$ , respectively. At $\textit{ISPL}={120}\,\textrm{dB}$ , the leading mode remains spatially confined to the vicinity of the slit. As ISPL increases beyond ${140}\,\textrm{dB}$ , the leading mode indicates the onset of vortex shedding, with an increasing characteristic length scale and a progressively larger spatial extent in the near-slit region. These results suggest a transition in the in- and near-slit flow dynamics towards progressively stronger inertial effects and more spatially extended coherent shedding structures.

Figure 27.

Leading SPOD modes, $\textrm{Re} ( {\widehat {\boldsymbol{\boldsymbol{{\xi }}}}}_{l=1}^{(1)} )$ , for different $\textit{ISPL}$ at $\textit{St} = 2 \textit{St}_0$ and $\textit{Re} = \textit{Re}_0/3$ . Cases in (a)–(d) represent $\textit{ISPL} = {120}\,\textrm{}$ , ${140}\,\textrm{}$ , ${145}\,\textrm{}$ and ${150}\,\textrm{dB}$ , respectively.

Figure 28.

Frequency-weighted VL (a) and KE (b) spectra for different $\textit{ISPL}$ at $\textit{Re} = \textit{Re}_0/3$ at $\textit{St} = 2\textit{St}_0$ .

Figure 28 compares the frequency-weighted VL and KE spectra for different ISPL values for $\textit{Re} = \textit{Re}_0/3$ at $\textit{St} = 2\textit{St}_0$ . For this particular case, KE dominates acoustic dissipation across all ISPL values. In contrast, the relative contributions of these two dissipation pathways vary under different acoustic forcing excitations, as shown in figure 25. The tonal peaks in both VL and KE increase monotonically with ISPL, indicating stronger acoustic absorption at higher forcing amplitudes. These trends are consistent with the expectation that larger pressure gradients and higher local inertia relative to viscosity promote a stronger shear-layer dynamics over an acoustic cycle (Shuster & Smith Reference Shuster and Smith2007). The separation of SPOD eigenvalues across ISPL values across the spectrum further suggests that higher ISPL is associated with an increase in the broadband contribution. In addition, the relative contrast between tonal peaks and broadband components is markedly reduced at the highest forcing level, ${150}\,\textrm{dB}$ . This observation indicates that, in addition to the coherent vortical motion and its higher-order harmonics, the irregular nonlinear dynamics makes an increasingly important contribution at high acoustic amplitude.