# Extremely tortuous sound absorbers with labyrinthine channels in non-porous and microporous solid skeletons

Tomasz G Zieliński, Kamil C Opiela, Nicolas Dauchez, Thomas Boutin, Marie-Annick Galland, Keith Attenborough

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Tomasz G Zieliński, Kamil C Opiela, Nicolas Dauchez, Thomas Boutin, Marie-Annick Galland, et al. Extremely tortuous sound absorbers with labyrinthine channels in non-porous and microporous solid skeletons. Applied Acoustics, 2024, 217, pp.109816. 10.1016/j.apacoust.2023.109816. hal-04363476

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Submitted on 25 Dec 2023

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
This is a preprint version of the article submitted in 2023 to:
*Applied Acoustics*, Vol. ??? (2024), ???.
https://doi.org/10.1016/???

Please, cite this document as:
T. G. ZIELIŃSKI, K. C. OPIELA, N. DAUCHEZ, T. BOUTIN, M.-A. GALLAND, K. ATTENBOROUGH,
"Extremely tortuous sound absorbers with labyrinthine channels in non-porous and microporous solid skeletons."
*Applied Acoustics*, Vol. ??? (2024), ???. DOI: 10.1016/???

# Extremely tortuous sound absorbers with labyrinthine channels in non-porous and microporous solid skeletons

TOMASZ G. ZIELIŃSKI <fnref n="1" />,ª KAMIL C. OPIELAª NICOLAS DAUCHEZᵇ THOMAS BOUTINᵇ
MARIE-ANNICK GALLANDᶜ KEITH ATTENBOROUGHᵈ

## Abstract

An assembly of additively-manufactured modules to form two-dimensional networks of labyrinthine slits results in a sound absorber with extremely high tortuosity and thereby a relatively low-frequency quarter wavelength resonance. Fully analytical modelling is developed for the generic design of such composite acoustic panels, allowing rapid exploration of various specific designs. In addition to single-porosity panels, a double porosity case is also investigated, in which the skeleton of the composite is microporous with a permeability very much lower than that of the tortuous slit network. The analytical modelling is verified by various numerical calculations, as well as sound absorption measurements performed on several 3D printed samples of modular composite panels. The experimental validation required overcoming the non-trivial difficulties related to additive manufacturing and testing samples of extreme tortuosity. On the other hand, due to the substantial two-dimensionality and modularity of the proposed design, such absorbers can be manufactured by traditional methods suitable for mass production of large-surface panels. The experimental results fully confirmed the theoretical predictions that significant sound absorption, almost perfect at the peak, can be achieved at relatively low frequencies using very thin panels, especially those with double porosity.

**Key words:** Sound absorption; Extreme tortuosity; Double porosity; Additive manufacturing; Multiscale modelling; Acoustic panels.

<footnote>ªInstitute of Fundamental Technological Research, Polish Academy of Sciences, ul. Pawińskiego 5B, 02-106 Warsaw, Poland</footnote>

<footnote>ᵇUniversité de Technologie de Compiègne, Alliance Sorbonne Université, Laboratoire Roberval,
Centre de recherche Royallieu, CS 60319, 60203 Compiègne CEDEX, France</footnote>

<footnote>ᶜLaboratoire de Mécanique des Fluides et d'Acoustique, Université de Lyon, École Centrale de Lyon, INSA Lyon,
Université Claude Bernard Lyon I, CNRS, UMR 5509, 36 Avenue Guy de Collongue, F-69134, Ecully, France</footnote>

<footnote>ᵈThe Open University, School of Engineering and Innovation, Milton Keynes MK7 6AA, UK</footnote>

<footnote n="1">¹Corresponding Author: tzielins@ippt.pan.pl</footnote>
# 1 Introduction

To be useful in vehicles and white goods, the sound absorbing layers deployed on surfaces should be as thin and lightweight as possible. This is rather difficult with traditional porous materials such as polymer foams or glass fibres since thick layers are required to offer absorption at low frequencies. Similarly, absorbers based on Helmholtz resonators need to be bulky to offer absorption at low frequencies. Ideally, a useful hard-backed thin absorbing layer will offer a first absorption peak associated with its quarter wavelength resonance at a wavelength that is much larger than the layer thickness. Often this is called sub-wavelength performance. To meet the requirement of sub-wavelength performance, there has been a lot of exploration of what can be achieved through additive manufacturing. Many of the resulting designs are based on metamaterial approaches [1]. A particular structural form uses the notion of ‘coiled’ or ‘coiled up’ space based on relatively long internal channels. This strategy was used by Cai et al. [2], who studied thin acoustic panels that effectively absorb low-frequency sound due to coplanar coiled tubes and/or chambers. A recent example of such a design consists of 3D printed stackable solid plate elements, each of which contains a coplanar spiral rectangular channel open only at one end [3]. Since the channel widths are on the order of millimetres and they are of finite length, the attenuation mechanisms include both the visco-thermal processes associated with propagation in the pores of porous absorbers and resonance associated with the finite lengths. By choosing channel lengths and cross-sections appropriately the quarter wavelength resonances overlap, and a wideband absorption spectrum is obtained. Similar research shows results obtained for sound absorbers composed of a perforated plate covering a coiled coplanar air chamber [4], acoustic metasurfaces composed of coiled channels and embedded apertures [5], or acoustic panels with coiled Helmholtz resonators [6] or multi-layer Helmholtz resonators with extended necks [7]. Several studies have explored composite structures with labyrinthine channels [8–11], or coiled-up cavities that resemble labyrinths, usually covered by perforated or slitted facesheets [12–14].

An alternative way of decreasing the quarter wavelength resonance of a hard-backed layer of porous material is to increase its tortuosity. Increasing tortuosity increases the sound path in the material thereby increasing its effective thickness and reducing the effective sound speed within it. Previously published methods of increasing tortuosity in idealised rigid porous materials include inclining straight pores, non-uniform pore cross-sections, intersecting pores, partitioning and labyrinthine perforations [15, 16]. Of these, the use of labyrinthine perforations is expected to result in the largest values of tortuosity. Twisted or labyrinthine channels or perforations may be regarded as tortuous macropores in a solid or microporous matrix, if they are relatively narrow, i.e. rather not formed of coiled-up cavities or wide passages as in the works cited above. Thus, extreme values of tortuosity can be achieved with proper design. Moreover, an additional attenuation mechanism of pressure diffusion can be introduced if the matrix is microporous and the micropores are much smaller than the width or cross-section of the labyrinth channels, to provide a high permeability contrast between the microporous matrix and the labyrinthine pore network. The high contrast of permeability is necessary not only for pressure diffusion to occur, but also to avoid downgrading the main pore network’s extreme tortuosity by viscous leakages through the micropores. This paper describes the theoretical basis, analytical models and rigorous numerical and experimental validation of such extremely tortuous modular acoustic composite materials involving labyrinthine channels within both non-porous and microporous matrices. Note that this design is fundamentally different from acoustic composite with internal labyrinthine structures formed by thin, impermeable, rigid walls embedded in a porous matrix, usually made of conventional acoustic foams or wools [17–20].

The outline of this paper is as follows. Section 2 presents the resizeable modular geometry of the proposed acoustic composite panels and all necessary formulae for analytical modelling of such structures with extreme tortuosity. The additive manufacturing of samples of a few prototype panels is discussed in Section 3. Numerical verification of the analytical modelling is extensively discussed in Section 4, where the theoretical predictions of sound absorption by extremely tortuous panels are also validated by the results of acoustic measurements on the 3D printed samples. The main findings, conclusions, and research perspectives are summarised in Section 5.

# 2 Theory of modular acoustic panels with high tortuosity

## 2.1 Resizeable modular geometry

Let us consider a composite panel composed of L-shaped modular elements. Two such elements are arranged in a rectangular cell, as depicted in Figure 1(a), forming a tortuous slit of width $w$ between them. The L-shape proportions and cell dimensions are fully defined by $a_L$, $b_L$, $c_L$, and $w$, as shown in the picture, so that one side of L is equal to $a_L + b_L$ and the other side – which also defines the cell width $W_c$ – equals $2(c_L + w)$. Note that a small part (of area $a_L w$) of the top L lies outside of the rectangular cell, but it is replaced from the other side by the same part of the top L of the neighbouring cell, which means that the solid part in the rectangular cell is equal to the double area of L. The cells are lined up in a row, in $N_c$ columns inside a single panel section which in turn contains $N_r$ rows such that for $N_r > 1$ each even row is a mirror copy of the odd row. This is a type A panel. Figure 1(b) shows its top view focusing on one section, where a single cell
<figure><img src="image_2.png" /><figcaption>**Figure 1:** Modular panel geometry.</figcaption></figure>
and the first odd row are highlighted. The alternative composite panel setup – called type B – is shown in Figure 1(c,d). In this case, there is no distinction between odd and even rows, but a vertical slit of width w is added at the bottom of each row. As a consequence, the cell and row height is increased by the slit width, see Figure 1(c). Regardless of the type of panel, its sections are separated by partitions with a width $c_L$. Therefore, the entire width of row or section is the sum of $N_c$ cell widths increased by the slit width and the width of the partition, i.e.

$$W_r = N_c W_c + c_L + w, \quad W_c = 2c_L + 2w. \qquad (1)$$

On the other hand, the panel thickness $H_p$ is simply a multiple of the row height $H_r$, namely

$$H_p = N_r H_r, \quad H_r = \begin{cases} 2a_L + b_L + w & \text{for type A,} \\ 2a_L + b_L + 2w & \text{for type B.} \end{cases} \qquad (2)$$

The proposed geometry is resizeable and fully defined by six independent geometric parameters: four lengths, w, $a_L$, $b_L$, and $c_L$, and two positive integers, $N_c$ and $N_r$. In general, the lengths should be greater than zero and it is also advised that the L-shape dimensions are greater than (or equal to) the slit width w. Then, the cell slits are connected throughout the panel section and form a single, continuous, tortuous and narrow channel, shaped like a labyrinth, see Figure 1(b,d). In an exceptional case when $b_L = 0$, which is shown in Figure 1(e,f) for both panel types, the slit channel, although still winding, has a simpler shape, and with one important correction, all the formulae derived for the panel will work in this case as well.

Below we present the necessary formulae for several geometric quantities that will be used in the analytical modelling of the composite panel. Let us first determine the total length $s_r$ of the continuous slit channel passing through a single row in the panel section, as it will be useful for further derivations. We have

$$s_r = N_c s_c + H_r, \quad s_c = \begin{cases} 2b_L + 2c_L + 2w & \text{for type A,} \\ 2b_L + 4c_L + 4w & \text{for type B,} \end{cases} \qquad (3)$$

where $s_c$ is the total length of the channel inside a single cell. The above definitions describe the length of the channel's central line with sharp bends. This line should indicate the flow path and we will demonstrate that for greater accuracy one should rather use curved paths that better resemble streamlines of viscous and inviscid flows through the channel, evidenced by numerical simulations – see streamlines in Figure 5(a,b) in Section 4.1 for both types of flows. Thus, let us consider the case where all the bends of the central line (but not the channel walls) are rounded with circular arcs and find the length of such a path with rounded bends across the entire row $\check{s}_r$ and in a single cell $\check{s}_c$. The latter is marked with a dashed red line in Figure 1(a,c). To do so, the length of each sharp bend – equal to $2\frac{w}{2}$ – has to be replaced by the smaller length $\frac{\pi w}{2}$ of the corresponding quarter-circle arc. There are four bends in each cell (A or B type), which means that the length of the channel path inside the cell is modified by subtracting $4w$ and adding $\pi w$. Moreover, in the B-type panel, there are two additional bends in each row. Therefore, the respective expressions (for $b_L > 0$) are

$$\check{s}_r = \begin{cases} N_c \check{s}_c + H_r & \text{for type A,} \\ N_c \check{s}_c + H_r - 2w + \pi w/2 & \text{for type B,} \end{cases} \quad \check{s}_c = s_c - 4w + \pi w. \qquad (4)$$

These formulae do not hold when $b_L = 0$, because then there are no bends inside the cell. In that case $\check{s}_c = W_c$ for type A and $\check{s}_c = 2W_c$ for type B, and the formula for $\check{s}_r$ has to be corrected to

$$\check{s}_r = \begin{cases} N_c W_c + H_r - 2w + \pi w/2 & \text{for type A with } b_L = 0, \\ 2N_c W_c + H_r - 4w + \pi w & \text{for type B with } b_L = 0. \end{cases} \qquad (5)$$

We will also use the total length of the channel walls $S_{wr}$ in the plane of a single row of the panel and the corresponding area $A_{rf}$ of the fluid-saturated channel in the single row. These can be calculated from the channel length $s_r$ as

$$S_{wr} = 2s_r, \quad A_{rf} = s_r w. \qquad (6)$$

Finally, the area of the solid part $A_{rs}$ inside a single row and the total area of the row $A_r$ are

$$A_{rs} = A_r - A_{rf}, \quad A_r = W_r H_r. \qquad (7)$$

## 2.2 Analytical formulae for geometry-based modelling

It is assumed that the frame of the composite panel is stiff enough to allow the use of the so-called rigid-frame models that essentially replace the acoustic composite with an equivalent fluid [21]. However, due to the additional property of
the material of the frame, two kinds of panels will be considered: (1) single-porosity panels with the frame made of solid impervious material, and (2) double-porosity panels – with the microporous frame. In the latter case, it is assumed that the micropore network is open with the characteristic sizes much smaller than $w$, ensuring good scale separation and high permeability contrast between this network and the slit channels [22–24].

In order to predict sound absorption by a modular composite panel we need first to determine its effective properties, i.e. the effective density, compressibility, speed of sound. It will be shown below that those are calculated on the basis of the characteristic dynamic functions related to visco-inertial and thermal effects in the tortuous slit channels, as well as to pressure diffusion in the microporous skeleton – in the latter case only for panels with double porosity. One way to calculate the dynamic functions is by using the scaling functions that depend on a set of macro-parameters (viz. tortuosity, volume fractions, characteristic lengths, static permeabilities) determined from the panel’s internal geometry. In general, the macro-parameters can be calculated numerically by solving three or, in the double-porosity case, four boundary value problems defined on the appropriate representative domains [25, 24]. However, in the case of the modular panels, they can be calculated or very accurately estimated using analytical expressions presented below.

Perhaps the most important macro-parameter of the proposed modular composites is tortuosity, because it strongly influences the viscous permeability, and the recommended modular approach allows creation of extremely tortuous panels. The tortuosity is often treated as a geometric parameter and then usually defined as the squared ratio of the average flow path length to the straight-line length across the porous medium [26–28]. Following this definition we will analytically calculate the tortuosity of panel as

$$ \alpha_{\infty} = (N_r \bar{s}_r / H_p)^2 = (\bar{s}_r / H_p)^2. \qquad (8) $$

We will demonstrate that this formula is very accurate when $w < \min(b_L, c_L)$, or if $b_L = 0$ then $w < N_c W_c$. Here, the symbol ‘<’ practically means ‘a few times larger’, however, we can even have $w = b_L < c_L$.

The slit channels of the composite panel are saturated with a fluid, and that fluid is air. The frame is solid, although it can also be microporous. The volume fraction of the fluid is porosity $\phi$ associated with the main pore network, i.e. the slit channels. The solid volume fraction $\phi_d$ will only be needed in the case of double porosity, referring in practice to the microporous domain. These two volume fractions are complementary, viz.

$$ \phi = A_{rf}/A_r, \quad \phi_d = A_{rs}/A_r = 1 - \phi. \qquad (9) $$

The characteristic lengths associated with viscous, thermal and pressure diffusion effects are

$$ \Lambda_v = w, \quad \Lambda_{th} = 2A_{rf}/S_{wr}, \quad \Lambda_d = 2A_{rs}/S_{wr}, \qquad (10) $$

respectively. Note that here, the viscous length $\Lambda_v$ is fairly accurately but nevertheless approximated with the exact result for straight slits and may slightly differ from the numerically determined value due to bends in the channel. On the other hand, the expressions for the thermal length $\Lambda_{th}$ and the pressure diffusion length $\Lambda_d$ are their formal definitions, although in a general three-dimensional case $A_{rf}$ and $A_{rs}$ denote the corresponding volumes and $S_{wr}$ is the surface of the channel walls. Moreover, when using equations (6) one can see at once that the thermal length is in fact equal to the channel width. It is also easy to show that $\Lambda_d/\phi_d = \Lambda_{th}/\phi$. Therefore,

$$ \Lambda_v = \Lambda_{th} = w, \quad \Lambda_d = w\phi_d/\phi. \qquad (11) $$

The next required macro-parameters are the static viscous $K_0$ and thermal $\Theta_0$ permeabilities, and in the case of double porosity also the static pressure diffusion $B_0$. Each of them can be related to the square of the corresponding characteristic length multiplied by the corresponding volume fraction, viz.

$$ K_0 = \frac{\phi \Lambda_v^2}{\zeta_v \alpha_{\infty}}, \quad \Theta_0 = \frac{\phi \Lambda_{th}^2}{\zeta_{th}}, \quad B_0 = \frac{\phi_d \Lambda_d^2}{\zeta_d}. \qquad (12) $$

The values of $\zeta_v$, $\zeta_{th}$, and $\zeta_d$ can be found analytically for simple pore and solid domain geometries. In particular, $\zeta_v = \zeta_{th} = 8$ for straight cylindrical channels, and similarly, $\zeta_d = 8$ for straight cylindrical solids, while $\zeta_v = \zeta_{th} = \zeta_d = 12$ for straight, infinite slits and solid partitions separating them. In [29] the values of the dividing factor $\zeta$ were calculated numerically for straight rectangular perforations (in that case $\zeta_{th} = \zeta_v = \zeta$ and $\alpha_{\infty} = 1$) by solving dedicated Poisson problems defined on rectangular domains with successively changing proportions. The proportions can be described by the parameter $\xi$ defined as the ratio of the side lengths of the rectangular domain. Here, we propose the analytical function

$$ \zeta(\xi) = \zeta_{||} - (\zeta_{||} - \zeta_{\square}) \frac{(2 + \beta)\xi}{1 + \beta\xi + \xi^2} = 12 - \frac{15.98\xi}{1 + 1.27\xi + \xi^2} \qquad (13) $$

which very accurately reproduces the numerical results presented in [29] for the fitting parameter $\beta = 1.27$. As shown in Figure 2, the function $\zeta(\xi)$ reaches its maximum value $\zeta_{||} = 12$ for straight, infinite slits, i.e. when $\xi = 0$ or $\xi \to \infty$,
<figure><img src="image_3.png" /><figcaption>Figure 2: The dividing factor versus the rectangle proportions: numerical calculations [29] vs. analytical formula (13).</figcaption></figure>

while the minimum value $\zeta_{\square} = 7.114$ is obtained for square domains, i.e. when $\xi = 1$. Formula (13) will be used to estimate $\zeta_d$. In practice, $\zeta_d \approx \zeta((a_L + b_L)/c_L)$ can be a good estimate when $b_L$ is larger than $c_L$ and $a_L$. In the special cases shown in Figure 1(e,f), reasonable estimates are: $\zeta_d \approx \zeta(N_c W_c / a_L)$ for Panel A with $b_L = 0$ and $N_r > 1$, and $\zeta_d \approx \zeta(N_c W_c / (2a_L))$ for Panel B with $b_L = 0$ as well as Panel A with $b_L = 0$ and $N_r = 1$. As for the viscous and thermal effects we will simply assume the upper limit, i.e. $\zeta_v = \zeta_{th} = 12$, although a slightly lower value (nearer 11) should be a better estimate as clearly seen from Figure 2.

The macro-parameters introduced above are used to calculate characteristic frequencies and shape factors of the porous medium. The angular characteristic frequencies associated with the viscous, thermal, and pressure diffusion effects are defined as

$$ \omega_v = \frac{\phi \nu_a}{K_0 \alpha_\infty}, \quad \omega_{th} = \frac{\phi \tau_a}{\Theta_0}, \quad \omega_d = \frac{\phi_d D_{0m}}{B_0}, \qquad (14) $$

respectively. Here: $\nu_a = \eta_a / \rho_a$ is the kinematic viscosity of air, i.e. the ratio of the dynamic viscosity $\eta_a$ to the density $\rho_a$; $\tau_a = \kappa_a / (\rho_a C_{pa})$ is the thermal diffusivity of air, i.e. the thermal conductivity $\kappa_a$ divided by the density $\rho_a$ and the specific heat capacity at constant pressure $C_{pa}$; $D_{0m} = K_{0m} P_0 / (\phi_m \eta_a)$ is the static pressure diffusivity of the microporous material of skeleton related to its static viscous permeability $K_{0m}$ and microporosity $\phi_m$, as well as the ambient mean pressure $P_0$ and dynamic viscosity of air saturating the pores and micropores. Note that the term $\phi_m / P_0$ means the static effective compressibility of air in micropores.

The shape factors related to viscous, thermal, and pressure diffusion effects are defined as

$$ M_v = \frac{8K_0\alpha_\infty}{\phi\Lambda_v^2}, \quad M_{th} = \frac{8\Theta_0}{\phi\Lambda_{th}^2}, \quad M_d = \frac{8B_0}{\phi_d\Lambda_d^2}, \qquad (15) $$

respectively. Finally, if the static viscous $\alpha_{0v}$, thermal $\alpha_{0th}$, and pressure diffusion $\alpha_{0d}$ tortuosities are specified, e.g. by numerical calculations, the corresponding low-frequency correction factors

$$ P_v = \frac{M_v}{4(\alpha_{0v} / \alpha_\infty - 1)}, \quad P_{th} = \frac{M_{th}}{4(\alpha_{0th} - 1)}, \quad P_d = \frac{M_d}{4(\alpha_{0d} - 1)}, \qquad (16) $$

can be calculated. These corrections will be omitted (i.e. $P_v = P_{th} = P_d = 1$) in fully analytical modelling.

The dynamic functions mentioned at the beginning of this section are the dynamic viscous $K(\omega)$ and thermal permeability $\Theta(\omega)$, and the dynamic pressure diffusion function $B(\omega)$. They can be determined as

$$ K(\omega) = X_{\omega}(K_0, M_v, P_v, \omega_v), \quad \Theta(\omega) = X_{\omega}(\Theta_0, M_{th}, P_{th}, \omega_{th}), \quad B(\omega) = X_{\omega}(B_0, M_d, P_d, \omega_d), \qquad (17) $$

respectively, where the scaling function

$$ X_{\omega}(X_0, M, P, \omega_c) = X_0 \left( \frac{i\omega}{\omega_c} + 1 - P + \sqrt{P^2 + \frac{M i\omega}{2 \omega_c}} \right)^{-1} \qquad (18) $$

is the key expression of the so-called Johnson-Champoux-Allard-Lafarge-Pride (JCALP) model [30–33].
We can simplify the calculation procedure in fully analytical modelling. Namely, when the static permeabilities $K_0$ and $\Theta_0$, and static pressure diffusion $B_0$ are determined using equations (12), the formulae for the corresponding characteristic frequencies (14) reduce to

$$ \omega_v = \frac{\zeta_v \nu_a}{\Lambda_v^2}, \quad \omega_{th} = \frac{\zeta_{th} \tau_a}{\Lambda_{th}^2}, \quad \omega_d = \frac{\zeta_d D_{0m}}{\Lambda_d^2}, \qquad (19) $$

while those for shape factors (15) simplify to

$$ M_v = 8/\zeta_v, \quad M_{th} = 8/\zeta_{th}, \quad M_d = 8/\zeta_d. \qquad (20) $$

It is now easy to see that for cylindrical pores $M_v = M_{th} = 1$, and similarly for cylindrical rods $M_d = 1$, which means that non-unit values of shape factors describe the deviation from the cylindrical shape. For example, all shape factors equal $2/3$ for straight slits. Proceeding further in this way, formulae (16) can be reduced to $P_v = (2/\zeta_v)/(\alpha_{0v}/\alpha_\infty - 1)$, $P_{th} = (2/\zeta_{th})/(\alpha_{0th} - 1)$, and $P_d = (2/\zeta_d)/(\alpha_{0d} - 1)$. Recall, however, that in analytical calculations we will neglect the low-frequency corrections and assume that $P_v = P_{th} = P_d = P = 1$. Now, the dynamic functions (17) can be expressed in terms of porosity, tortuosity, dividing factors, and characteristic lengths as follows

$$ K(\omega) = Y_{\omega}(\phi/\alpha_{\infty}, \zeta_v, \Lambda_v, \nu_a), \quad \Theta(\omega) = Y_{\omega}(\phi, \zeta_{th}, \Lambda_{th}, \tau_a), \quad B(\omega) = Y_{\omega}(1 - \phi, \zeta_d, \Lambda_d, D_{0m}), \qquad (21) $$

where the scaling function

$$ Y_{\omega}(\psi, \zeta, \Lambda, \mathcal{N}) = \psi \left( \frac{i\omega}{\mathcal{N}} + \frac{\zeta}{\Lambda^2} \sqrt{1 + \frac{4\Lambda^2 i\omega}{\zeta^2 \mathcal{N}}} \right)^{-1} \qquad (22) $$

is a reshaped expression of the so-called Johnson-Champoux-Allard-Lafarge (JCAL) model which is obtained from the JCALP formula (18) for $P = 1$. If we recall that $\Lambda_v = \Lambda_{th} = w$ and the dividing factors $\zeta_v$ and $\zeta_{th}$ can be fixed to 12 (or a slightly lower value), then the only required geometric parameters that have to be calculated in the analytical design process are $\phi$ and $\alpha_\infty$ in the case of modular composite panels with single porosity, and additionally $\zeta_d$ and $\Lambda_d = w(1 - \phi)/\phi$ when the composite frame is microporous. Moreover, these parameters are determined almost at once from the analytical formulae given above, and then only equations (21) and a few analytical expressions given below have to be computed for quick assessment of sound absorption by the proposed composite panels, which allow for rapid analytical optimisation of such composites.

Alternatively, a model based on the exact analytical solution obtained for straight slits [21] can be used for viscous and thermal dynamic permeabilities. In the case of visco-inertial effects, the solution is modified by tortuosity $\alpha_\infty$, which is very important if the slits are oblique or tortuous. In this modelling, the dynamic functions $K(\omega)$ and $\Theta(\omega)$ can be expressed in terms of porosity, tortuosity, and characteristic lengths as follows

$$ K(\omega) = T_{\omega}(\phi/\alpha_{\infty}, \Lambda_v, \nu_a), \quad \Theta(\omega) = T_{\omega}(\phi, \zeta_{th}, \Lambda_{th}, \tau_a), \qquad (23) $$

where the function based on the exact analytical solution is

$$ T_{\omega}(\psi, \Lambda, \mathcal{N}) = \frac{\psi \mathcal{N}}{i \omega} \left( 1 - \frac{\tanh\left(\frac{1}{2} \Lambda \sqrt{i \omega / \mathcal{N}}\right)}{\frac{1}{2} \Lambda \sqrt{i \omega / \mathcal{N}}} \right). \qquad (24) $$

We have verified that $T_\omega(\psi, \Lambda, \mathcal{N}) \approx Y_\omega(\psi, 12, \Lambda, \mathcal{N})$, i.e. this model gives practically the same results as the JCAL model (22) with $\zeta_v = \zeta_{th} = 12$ which is exact for straight, parallel, infinite slits.

As already mentioned above, to determine the sound absorption of a modular panel, one first needs to calculate its effective properties. The effective compressibility of the single-porosity panel is

$$ C_{\text{sg}}(\omega) = \frac{\phi}{P_0} \left( 1 - \frac{\gamma_a - 1}{\gamma_a} \frac{\Theta(\omega)}{\phi \tau_a} i \omega \right), \qquad (25) $$

where $\gamma_a$ is the heat capacity ratio for air. In the case of double-porosity panels, the effective compressibility is calculated as

$$ C_{\text{db}}(\omega) = C_{\text{sg}}(\omega) + \phi_d \mathcal{F}_d(\omega) \frac{\phi_m}{P_0}, \quad \mathcal{F}_d(\omega) = 1 - \frac{B(\omega)}{\phi_d D_{0m}} i \omega, \qquad (26) $$

where the function $\mathcal{F}_d(\omega)$ describes the ratio of the averaged pressure locally fluctuating in the microporous skeleton of the composite panel to the pressure in the slit channel which is locally constant. Let us also recall that $\phi_m/P_0$ is the effective compressibility of air in micropores. Now, the effective density $\rho_e$ and speed of sound $c_e$ are

$$ \rho_e(\omega) = \frac{\eta_a}{i\omega K(\omega)}, \quad c_e(\omega) = \sqrt{\frac{i\omega K(\omega)}{\eta_a C(\omega)}}, \qquad (27) $$
where $C(\omega) = C_{sg}(\omega)$ for the case of single porosity, and $C(\omega) = C_{db}(\omega)$ for the case of double porosity with high permeability contrast.

The effective properties (27) can be used to solve the Helmholtz problem of acoustic wave propagation in the composite modelled as an equivalent fluid. Since the composite panel has a constant thickness $H_P$, analytical solutions are available for the propagation of plane acoustic waves. These can be used to calculate the surface acoustic impedance $Z_S(\omega)$ at the front face of the panel [21]. In particular, for the composite panel backed by a rigid wall and subject to plane acoustic waves at normal incidence

$$Z_S(\omega) = Z_e(\omega) \coth(H_P i\omega/c_e(\omega)), \quad (28)$$

where $Z_e(\omega) = \rho_e(\omega)c_e(\omega)$ is the effective characteristic impedance of the equivalent fluid replacing the composite. When the composite panel is backed by an air gap of thickness $H_G$ between the back face of the composite and the rigid wall, the surface acoustic impedance of such a two-layer configuration [21] is

$$Z_S(\omega) = Z_e(\omega) \frac{Z_e(\omega) + Z_G(\omega) \coth(H_P i\omega/c_e(\omega))}{Z_G(\omega) + Z_e(\omega) \coth(H_P i\omega/c_e(\omega))}. \quad (29)$$

Here, $Z_G(\omega) = Z_a \coth(H_G i\omega/c_a)$ is the impedance of the air gap, which in particular depends on the characteristic impedance of air $Z_a = \rho_a c_a$, which in turn is the product of the air density $\rho_a$ and speed of sound $c_a$. Note that the total thickness of two-layer configuration is $H_P + H_G$. For both configurations, i.e. with or without an air gap, the sound absorption coefficient [21] is determined as

$$\mathcal{A}(\omega) = 1 - \left| \frac{Z_S(\omega) - Z_a}{Z_S(\omega) + Z_a} \right|^2. \quad (30)$$

## 3 Additively manufactured samples of prototype panels

Four panel designs were manufactured in order to validate the theory and demonstrate the sound-absorbing properties of the proposed modular acoustic materials with extreme tortuosity. The first design is Panel A with $N_c = 3$ and $N_r = 4$, as shown in Figure 1(b), but with slightly different proportions, viz.: the designed slit width is $w = 1$ mm, while the L-shaped module dimensions are $a_L = 1$ mm, $b_L = 5$ mm, and $c_L = 1.30$ mm. Consequently, the row (or section) width is $W_r = 16.10$ mm, the row height is $H_r = 8$ mm, and the composite panel thickness is $H_P = 32$ mm. This panel design will be referred to as **T38-H32**, where the first number in the label corresponds to the tortuosity of the labyrinthine slits (for the exact calculated value see Section 4.1) and the second is the thickness of the composite in millimetres. The second panel design is labelled as **T38-H16**. Here, the slit width and L-shaped module are the same as in the previous design, but the number of rows is $N_r = 2$, so this composite is two times thinner, i.e. $H_P = 16$ mm. These two designs are locally identical (note that the two **T38-H16** panels set side by side form the **T38-H32** panel) and together they will be referred to as **T38**. The third design, labelled as **T21-H16** (abbreviated as **T21**), also has two rows and thickness 16 mm. Its labyrinthine slit network has lower tortuosity (see Section 4.2), because of the lower number of module columns $N_c = 2$, although $c_L = 1.55$ mm, and consequently $W_r = 12.75$ mm. The slit width and other dimensions are the same as in the previous designs. Finally, due to some technological issues explained below, the fourth panel was designed with much wider slits, i.e. $w = 1.50$ mm, and also thicker labyrinth walls, i.e. $a_L = c_L = 1.50$ mm, while $N_c = 3$ and $N_r = 2$ are the same as in the **T38-H16** panel. The row (or section) width in the new design is $W_r = 21$ mm. The remaining dimension $b_L = 8$ mm was chosen so that the tortuosity is about 36, i.e. intentionally very close to the value designed in the **T38** panels. Consequently, the row height is $H_r = 12.50$ mm, and the composite panel thickness is $H_P = 25$ mm. Therefore, this panel design is labelled as **T36-H25**.

In spite of many limitations, additive manufacturing technologies [34] are widely used in recent times to develop and prototype new acoustic materials [35–40]. For the specific purposes of this study, the panel samples were additively manufactured from two different raw materials in two technologies: (i) Stereolithography (SLA), a.k.a. vat photopolymerisation [41], using a low-viscosity photopolymer resin and the Formlabs Form 3B device; (ii) Binder Jetting 3D Printing (BJP) [42], using a coarse-grain gypsum powder and the 3D Systems ProJet 160 device. The samples of the panels with labyrinthine slits 1 mm wide, i.e. **T38** and **T21**, were produced from resin in SLA technology. We also tried to 3D print these composite panels from gypsum powder in BJP technology, but with this geometry it was extremely difficult to remove the powder residue from the slits without breaking some of the walls. Therefore, the **T36-H25** panel with wider (i.e. 1.5 mm) slits and thicker walls was designed to be manufactured in BJP technology to achieve the expected double porosity. To avoid various limitations [43], all samples were made for testing in an impedance tube with a square cross-section instead of a circular one.

The resin samples manufactured in the SLA technology are of single porosity. Two of them, namely one for each of the 16 mm panels are depicted in Figure 3(a,b), where their labels are suffixed with **-SLA**. The bordered versions of
<figure><img src="image_4.png" /><figcaption>**Figure 3:** Square samples of: (a,b) two 16 mm panels 3D printed in SLA technology from a transparent resin, and (c) a double-porosity 25 mm panel 3D printed in BJP technology from a gypsum powder (this material is not transparent, so hidden parts of the labyrinthine slits are drawn in the middle photo showing the side of this sample).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4:** Three square samples with clamping borders 3D printed from transparent resin in SLA technology.</figcaption></figure>
these samples are shown in Figure 4 along with the bordered sample of the 32 mm panel. Three views of each sample are presented in a column, one below the other. There, we also show the sample dimensions (in millimeters), a single modular cell, the number of entry slits in the insonified surface of the sample, and even the plane wave incidence direction. The samples shown in Figure 4 were manufactured with clamping borders – which is denoted by adding the acronym (w.c.b.) to each of the labels – in order to clamp them firmly to the end of a square impedance tube with cross-sectional dimensions 66 mm × 66 mm, or between two parts of the square tube in the case of measurements with air gaps. The bordered samples with a thickness of 16 mm are otherwise virtually the same as their borderless counterparts shown in Figure 3(a,b). These latter samples can be directly inserted into the 66 mm × 66 mm square impedance tube. As expected from previous reports [44], the quality of all resin samples – in terms of the shape mapping and surface smoothness – is very high, which can be verified, because the resin used for their 3D printing is transparent which makes the labyrinthine network visible, see Figures 3(a,b) and 4. The viscosity of this photopolymer resin is relatively low, which facilitated the removal of the residual (i.e. non-solidified) resin from the labyrinthine networks, although in the case of bordered samples, the borders blocked the flow through the side walls. Therefore, the manufacturing of the bordered samples required greater care in removing residual resin, especially in the case of the 32 mm sample, but they were easier to test and the reproducibility of the measurement results was excellent. In fact, this was the main reason these samples were eventually fabricated (see Section 4 for more information).

Due to technical problems mentioned above, only one gypsum sample of the T36-H25 panel with wider slits was successfully manufactured in the BJP technology. This sample is shown in Figure 3(c) and labelled with the suffix -BJP. Unlike the resin samples, the BJP gypsum sample is not transparent, so hidden parts of the labyrinthine slits are drawn in the picture of the side wall of this sample – see the middle photo of the column of pictures shown in Figure 3(c). Thanks to previous experience with this technology and gypsum material [24, 45], we expected this sample to have double porosity due to the open microporosity of its skeleton made of gypsum powder grains bound with a butyrolactam binder. The sample thickness is 25 mm and it was 3D printed borderless to fit inside the 66 mm × 66 mm square impedance tube. The powder residue was removed from the labyrinthine network of slits manually using a thin metal wire and by blowing it out through the openings in two opposite side walls rather than through the slits in both faces of the composite panel. As expected, the quality of the gypsum sample is inferior to that of the resin samples, but in fact, as already shown in numerous studies [44, 46], some minor imperfections in 3D printing, in particular surface roughness, often slightly increase the soundproofing properties of additively manufactured acoustic materials. More importantly, as already mentioned, the BJP sample has double porosity as its skeletons is microporous due to the use of relatively coarse-grained gypsum powder and proper dosing of the binder [24, 45]. The resulting microporosity of the skeleton, and in particular its contrastingly low permeability as opposed to that of the main network of labyrinthine slits, has a great influence on the acoustic properties of the composite panels as demonstrated in Section 4.

To determine the necessary properties of the microporous material that forms the skeleton of the double-porosity sample, dedicated disc-shape samples (all 15 mm thick and 100 mm in diameter) were 3D printed from the same gypsum powder as the square BJP sample, using the same 3D printer, binder, and process parameters. To ensure this, they were 3D printed in the same powder bed, i.e. during the same manufacturing process as the square sample. These disc samples were used to directly measure the microporosity $\phi_m$, as well as the static airflow resistivity of the microporous material from which the corresponding static viscous permeability $K_{0m}$ was determined. It appeared that the latter parameter was strongly dependent on the orientation of the 3D printed discs in the powder bed during the manufacturing process, and it varied from $0.70 \cdot 10^{-12} \text{ m}^2$ (or even $0.57 \cdot 10^{-12} \text{ m}^2$, if we consider the value determined previously for such a material, see [24]) found in the case of horizontal orientation, up to $1.64 \cdot 10^{-12} \text{ m}^2$ determined for the case of vertical orientation. More than a twofold difference between the permeability measured for two mutually perpendicular directions (vertical and horizontal) proves the anisotropy of the material. One may expect at least a transverse isotropy of the material (if not some other form of anisotropy) related to the vertical direction of the layer-by-layer additive manufacturing process. On the other hand, for all gypsum discs, we obtained quite similar microporosity values between 40% and 44%, which is also in line with the value of 43%, previously measured for such a material [24]. For the purposes of this work, we assumed the lowest measured values of microporosity, i.e. $\phi_m = 0.40$, and permeability, i.e. $K_{0m} = 7 \cdot 10^{-13} \text{ m}^2$, which means that these two microporous material properties were used in double-porosity calculations. We have checked, however, that sound absorption by double-porosity panels does not change much (at least at frequencies below 2 kHz) even if any other measured value of $K_{0m}$ is used. Only the second absorption peak is slightly modified. In order to get a clear difference in the magnitude of the first absorption peak, one needs to assume much lower permeability, e.g. $K_{0m} = 2 \cdot 10^{-13} \text{ m}^2$ (for comparison, in Section 4.2 we will also show the absorption curves predicted for this value).

Note that the specific dimensions in the T21, T38, and T36 designs, in particular $c_L$, were chosen so that the effective area of each panel sample was practically equal to the cross-sectional area of the square impedance tube used in the experiments. This resulted in: 5 entry slits in both faces of the T21 panel samples, 4 slits in the faces of the T38 panel samples, and 3 slits in the faces of the T38 panel sample (see Figures 3 and 4).

Many studies have shown that the actual dimensions of the essential details of the pore networks in additively manufactured acoustic materials deviate from the nominal values set in the designs used for 3D printing [46, 37, 24]. Therefore,
design adjustments are required to improve the accuracy of predictions based on the designed geometry of the pore network. The most essential dimension in the slit networks of modular composite panels is the slit width $w$. It has to be updated using an estimated slit width correction $\Delta w$, which can be negative and depends on the material and technology used for manufacturing of the samples. In the case of photopolymer resin and SLA technology, the slits are slightly narrower than in the design used for 3D printing. Based on microscopic examination of the samples and the long-term experience in using the Formlabs Form 3B 3D printer, we can accurately estimate the slit correction in this case as $\Delta w = -0.1$ mm. The situation is not so obvious in the BJP sample, where the width of the slits varies: in most places it is narrower than the nominal design value, but it is also slightly wider in some other places, mainly due to the rather aggressive removal of powder residues from the slits. However, after careful microscopic examination, we decided that in the gypsum sample, the average slit correction could also be roughly estimated as $\Delta w = -0.1$ mm. In any case, the correction is made by the following replacement

$$w \leftarrow w + \Delta w, \quad a_L \leftarrow a_L - \Delta w/2, \quad c_L \leftarrow c_L - \Delta w. \qquad (31)$$

Note that $b_L$, $W_c$, $W_r$, $H_r$, and $H_p$ are not affected by this replacement, see equations (1) and (2). This correction and all formulae derived in the previous sections are valid provided that $-w < \Delta w < c_L$ and $\Delta w < 2a_L$. Table 1 shows the nominal and corrected dimensions of the modular geometry. The latter are related to the specific technology – SLA or BJP – used to manufacture the panel samples. These corrected values were used in the relevant calculations of the acoustic properties of these panels.

<table><caption>**Table 1:** Dimensions (in millimetres) of the modular geometry: (a) the nominal values in composite panel designs, and (b) the actual values (i.e. corrected by $\Delta w$) in the samples 3D printed using the specified technologies</caption><thead><tr><th>Panel type & technology</th><th></th><th>Δw</th><th>w</th><th>a<sub>L</sub></th><th>b<sub>L</sub></th><th>c<sub>L</sub></th><th>W<sub>r</sub></th><th>H<sub>r</sub></th></tr></thead><tbody><tr><td rowspan="2"><b>T21</b><br>(N<sub>c</sub> = 2)</td><td>(a) nominal</td><td>—</td><td>1.00</td><td>1.00</td><td>5.00</td><td>1.55</td><td>12.75</td><td>8.00</td></tr><tr><td>(b) SLA</td><td>-0.10</td><td>0.90</td><td>1.05</td><td>5.00</td><td>1.65</td><td>12.75</td><td>8.00</td></tr><tr><td rowspan="2"><b>T38</b><br>(N<sub>c</sub> = 3)</td><td>(a) nominal</td><td>—</td><td>1.00</td><td>1.00</td><td>5.00</td><td>1.30</td><td>16.10</td><td>8.00</td></tr><tr><td>(b) SLA</td><td>-0.10</td><td>0.90</td><td>1.05</td><td>5.00</td><td>1.40</td><td>16.10</td><td>8.00</td></tr><tr><td rowspan="2"><b>T36</b><br>(N<sub>c</sub> = 3)</td><td>(a) nominal</td><td>—</td><td>1.50</td><td>1.50</td><td>8.00</td><td>1.50</td><td>21.00</td><td>12.50</td></tr><tr><td>(b) BJP</td><td>-0.10</td><td>1.40</td><td>1.55</td><td>8.00</td><td>1.60</td><td>21.00</td><td>12.50</td></tr></tbody></table>

# 4 Numerical and experimental validation

## 4.1 Results for the 32 mm composite panels

First, consider the thicker 32 mm panel discussed in Section 3 and labelled **T38-H32**, see Figure 4(c). Sound absorption by this panel backed by a rigid wall will be calculated using four approaches: (i) fully analytical modelling, (ii) hybrid multiscale calculations, (iii) direct numerical simulations (DNS) for single-porosity panel, and (iv) DNS for double-porosity composite panel. To accurately show the difference between single and double porosity cases – and since only a single-porosity sample of this panel was manufactured – all numerical and analytical analyses are performed for the actual dimensions of the modular geometry of the **T38-H32-SLA** sample made of resin, see values in the fourth row of numbers in Table 1.

In the fully analytical modelling, the dynamic functions (21) are calculated using the JCAL model (22) where the porosity $\phi$, tortuosity $\alpha_\infty$, dividing factors $\zeta_v = \zeta_{th} = 12$ and $\zeta_d$, and characteristic lengths $\Lambda_v = \Lambda_{th} = w$ and $\Lambda_d$ are determined analytically or known a priori. The remaining macro-parameters (12), characteristic frequencies (14), and shape factors (15) are calculated from the relevant analytical formulae and listed in Table 2 for comparison with the corresponding numerical results obtained by the hybrid multiscale approach, as discussed below. Note that $f_v = \omega_v/(2\pi)$, $f_{th} = \omega_{th}/(2\pi)$, and $f_d = \omega_d/(2\pi)$.

In the hybrid approach, first, the macro-parameters, viz. the volume fractions $\phi$ and $\phi_d$, kinematic $\alpha_\infty$ and static tortuositities $\alpha_{0v}$, $\alpha_{0th}$, $\alpha_{0d}$, characteristic lengths $\Lambda_v$, $\Lambda_{th}$, $\Lambda_d$, and static permeabilities $K_0$, $\Theta_0$, $B_0$ are calculated numerically by solving relevant boundary value problems as discussed, e.g. in [47, 48, 25, 24], although $\phi$, $\phi_d$, $\Lambda_{th}$ and $\Lambda_d$ are determined directly from the geometry. In particular, we solve the scaled Stokes' flow, Laplace's and Poisson's problems defined in the fluid subdomain, i.e. in the channel inside a single row of the panel. For the double-porosity case, an additional Poisson's problem is solved in the solid subdomain, i.e. in the microporous skeleton domain. We define and solve these problems using the finite element method, strictly adhering to all numerical requirements as discussed, e.g. in [25]. For example, the finite element mesh for the Stokes' flow has boundary layers to ensure correct determination of the high velocity gradient close to the channel walls where the no-slip boundary condition is applied. For the Laplace's problem, all
<table><caption>**Table 2:** Macro-parameters (upper table) and other related parameters (lower table) determined analytically and numerically for the T38 panel.</caption><thead><tr><th rowspan="2">type of analysis</th><th rowspan="2">φ<br>%</th><th rowspan="2">φ<sub>d</sub><br>%</th><th colspan="4">α<sub>∞</sub></th><th rowspan="2">Λ<sub>v</sub><br>mm</th><th rowspan="2">Λ<sub>th</sub><br>mm</th><th rowspan="2">Λ<sub>d</sub><br>mm</th><th colspan="3">K<sub>0</sub></th></tr><tr><th>—</th><th>—</th><th>—</th><th>—</th><th>10<sup>-10</sup>m<sup>2</sup></th><th>10<sup>-8</sup>m<sup>2</sup></th><th>10<sup>-7</sup>m<sup>2</sup></th></tr></thead><tbody><tr><td>analytical</td><td>36.20</td><td>63.80</td><td>38.26</td><td>—</td><td>—</td><td>—</td><td>0.900</td><td>0.900</td><td>1.586</td><td>6.386</td><td>2.443</td><td>1.735</td></tr><tr><td>numerical</td><td>36.20</td><td>63.80</td><td>37.94</td><td>46.33</td><td>1.217</td><td>1.489</td><td>0.839</td><td>0.900</td><td>1.587</td><td>6.186</td><td>2.553</td><td>1.696</td></tr></tbody></table>

<table><thead><tr><th rowspan="2">type of analysis</th><th rowspan="2">ζ<sub>v</sub><br>—</th><th rowspan="2">ζ<sub>th</sub><br>—</th><th rowspan="2">ζ<sub>d</sub><br>—</th><th colspan="3">M</th><th colspan="3">P</th><th colspan="3">f</th></tr><tr><th>—</th><th>—</th><th>—</th><th>—</th><th>—</th><th>—</th><th>—</th><th>—</th><th>—</th></tr></thead><tbody><tr><td>analytical</td><td>12</td><td>12</td><td>9.26</td><td>0.667</td><td>0.667</td><td>0.864</td><td>1</td><td>1</td><td>1</td><td>37.6</td><td>53.4</td><td>5528</td></tr><tr><td>numerical</td><td>10.84</td><td>11.48</td><td>9.47</td><td>0.738</td><td>0.697</td><td>0.845</td><td>0.834</td><td>0.804</td><td>0.432</td><td>39.1</td><td>51.1</td><td>5657</td></tr></tbody></table>

sharp corners of the channel walls are filleted (with the radius of w/20) to avoid singularity when determining the viscous length. The antisymmetric boundary conditions are applied on the upper and lower outlet/inlet boundaries of the channel (inside the single row of the panel) in the case of Stokes' flow and Laplace's problem, while the symmetric boundary conditions in the case of the Poisson's problems. Figure 5 shows the respective solution results, i.e. the scaled fields of: (a) viscous flow velocity projected onto the 'vertical' direction, (b) potential flow gradient projected onto the 'vertical' direction, (c) temperature, and (d) pressure diffusion. The 'vertical' direction is the direction of the external pressure gradient driving the viscous flow and consistent with the wave propagation direction. This is also the direction of the external unit vector field that evokes the potential i.e. inviscid flow. Streamlines for both flows are shown in Figure 5(a,b). The scaled potential flow gradient is dimensionless. The velocity, temperature, and pressure diffusion fields are scaled to the permeability unit (m²), so after averaging over the considered panel fragment, they give the static viscous K₀ and thermal Θ₀ permeabilities, and the static pressure diffusion B₀, respectively. The tortuosity α∞ as well as the viscous length Λ_v are calculated using the potential flow gradient. For all necessary formulae and more discussions on the subject see [25]. The numerically computed values of macro-parameters are used to analytically (hence this approach is hybrid) calculate the characteristic frequencies (14), shape factors (15), and low-frequency corrections (16), which in turn are used to determine the dynamic functions (17) using the JCALP model (18).

<figure><img src="image_6.png" /><figcaption>**Figure 5:** Results of FE analyses for hybrid multiscale calculations: (a) Stokes flow, (b) Laplace problem for inviscid flow, (c) Poisson problem for static thermal diffusion, (d) Poisson problem for static pressure diffusion.</figcaption></figure>

Table 2 compares the analytically determined parameters with the corresponding values found on the basis of numerical analyses. In general, the agreement is very good. In particular, the numerical and analytical values of volume fractions φ and φ_d, and two characteristic lengths Λ_th and Λ_d are identical as they all represent the exact values. The analytical approximation of tortuosity (8) proves to be very accurate due to the proposed correction of quarter-circle bends, which is indicated by the streamlines of the viscous and potential flows shown in Figure 5(a,b). The alternative tortuosity value (s_f/H_f)² = 41.93 calculated for sharp bends is evidently less accurate. The numerical value of Λ_v is almost 7% smaller than the slit channel width w adopted a priori for Λ_v in analytical calculations and exact only for straight slits. This
is due the bends of the channel, which affect the viscous characteristic length. Finally, note that the numerical values of $ζ_v$, $ζ_{th}$, and $ζ_d$ were found by inverting equations (12) and taking numerical values for all other parameters present in these formulae. Furthermore, only numerically calculated values are given for $α_{0v}$, $α_{0th}$ and $α_{0d}$, since the static tortuosities are not determined by analytical calculations and are not required by the JCAL scaling function; also recall that the JCALP model reduces to the JCAL model for $P_v = P_{th} = P_d = 1$.

Once the dynamic functions are determined using the analytical or hybrid approach, the effective compressibility can be calculated for the single-, Eq. (25), and double-porosity case, Eq. (26), and then the corresponding effective density and speed of sound (27). In the case of double porosity, the microporosity $φ_m$ and static viscous permeability $K_{0m}$ of the microporous skeleton (see Section 3 for their directly measured values) were first used to determine the required static pressure diffusivity $D_{0m} = 9.44 \cdot 10^{-3} \text{ m}^2/\text{s}$. All dynamic and effective properties as well as the characteristic frequencies and static pressure diffusivity are computed for the relevant air properties listed in Table 3. Their values were determined for the ambient mean pressure and temperature conditions found during acoustic testing of four of the 3D printed samples of prototype panels. The remaining two samples were tested in very similar conditions.

<table><caption>Table 3: Air properties at ambient conditions of pressure $P_0$ and temperature $T_0$.</caption><thead><tr><th>P<sub>0</sub><br>hPa</th><th>T<sub>0</sub><br>°C</th><th>ρ<sub>a</sub><br>kg/m<sup>3</sup></th><th>c<sub>a</sub><br>m/s</th><th>γ<sub>a</sub><br>—</th><th>η<sub>a</sub><br>Pa·s</th><th>κ<sub>a</sub><br>W/(m·K)</th><th>C<sub>pa</sub><br>J/(kg·K)</th><th>ν<sub>a</sub><br>m<sup>2</sup>/s</th><th>τ<sub>a</sub><br>m<sup>2</sup>/s</th></tr></thead><tbody><tr><td>993.6</td><td>25.9</td><td>1.155</td><td>347.0</td><td>1.40</td><td>1.841·10<sup>-5</sup></td><td>2.631·10<sup>-2</sup></td><td>1006</td><td>1.594·10<sup>-5</sup></td><td>2.264·10<sup>-5</sup></td></tr></tbody></table>

Figures 6 and 7 compare the dynamic functions and the corresponding effective compressibilities calculated using both approaches, showing very good agreement between them. Note that the peaks in the negative imaginary parts of all these functions are located close to the respective characteristic frequencies. Pay attention in particular to the second peak on the curve of the negative imaginary part of the normalised effective compressibility for the double-porosity structure (Figure 7). This peak is close to the characteristic frequency $f_d = ω_d/(2π) = 5528 \text{ Hz}$, but it drives the curve up far below this frequency, so that the values at frequencies already above 200 or 300 Hz are much greater than the values of the corresponding curve determined for the single porosity case. This means that the pressure diffusion phenomenon in the double-porosity structure should be significant already at such low frequencies.

The effective properties calculated for both 32 mm composite panels, i.e. with single- and double-porosity, allowed the prediction of sound absorption for these composite backed by a rigid wall, as discussed at the end of Section 2.2, using the fully analytic and hybrid approaches. These predictions are shown in Figure 8 and, consequently with the above findings, the differences between the ‘Analytic’ and ‘Hybrid’ results are barely visible in both porosity cases. On the other hand, as expected, there is a significant difference between the sound absorption by the single-porosity panel and the double-porosity composite of the same meso-geometry of tortuous channels due to pressure diffusion in the microporous skeleton. In particular, the pronounced first absorption peak around 0.8 at 400 Hz (single-porosity panel) is advantageously shifted to 290 Hz and even slightly increased to 0.87 for the double-porosity panel, although even the first of these frequencies is already quite low (see the comment above to Figure 7).

<figure><img src="image_7.png" /><figcaption>Figure 6: Dynamic (a) viscous and (b) thermal permeabilities, and (c) pressure diffusion function, calculated analytically (continuous lines) and numerically (dashed lines).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 7:** Normalised effective compressibility for (a) single-porosity and (b) double-porosity absorber, calculated analytically (continuous lines) and numerically (dashed lines).</figcaption></figure>

Figure 8 also presents sound absorption found using computationally more expensive DNS performed for both types of the T38-H32 panel. DNS for the single-porosity panel were performed for each computational frequency (i.e. every 20 Hz) using linear, fully coupled Navier-Stokes-Fourier equations with four degrees of freedom at each node of the finite element mesh, i.e. two components of velocity, pressure and temperature, and with no-slip boundary conditions applied on the tortuous channel walls and on the front face of the composite, which required dense boundary layers in the finite element mesh along these boundaries. These numerical calculations perfectly confirmed the fully analytical and hybrid multiscale modelling. In the case of double-porosity composite panel, Helmholtz analyses – with acoustic pressure as a single degree of freedom at each node of the finite element mesh – were solved in both coupled domains, i.e. in the air and in the fluid corresponding to the microporous skeleton. Therefore, these calculations were less computationally demanding than single-porosity DNS, but ignored the viscosity effects on the channel walls, which resulted in discrepancies from the analytical and hybrid multiscale methods, although the very specific nature of the predicted absorption curves is the same. Note that the sound wave propagation in the air above the composite can be considered lossless, in particular in the adjacent air layer modelled in FE simulations, which means that the sound absorption of the rigidly-backed panel is not changed when an air layer is added at the front. In both DNS, a uniform sound pressure was applied to the upper boundary of the adjacent air layer to simulate the effect of a plane acoustic wave impinging at normal incidence, and the sound absorption was determined using formula (30) for which the surface acoustic impedance $Z_S$ was calculated at the upper boundary as the ratio of the acoustic pressure to the normal velocity determined as the average over the entire

<figure><img src="image_9.png" /><figcaption>**Figure 8:** Sound absorption by the T38-H32 panels of single or double porosity.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Figure 9:** Results of DNS (the real part of the acoustic pressure distribution) for the 32 mm panel of: (a,b) single porosity (Navier-Stokes-Fourier analysis), and (c,d) double porosity (Helmholtz analysis).</figcaption></figure>

boundary, although practically homogeneous along it. Figure 9 shows exemplary results of DNS, i.e. sound pressure distribution computed at two specified frequencies for both porosity cases.

Due to the difficulty of removing powder residues from slits in BJP printouts, only a single-porosity SLA sample of the 32 mm panel was manufactured – see **T38-H32-SLA** sample in Figure 4(c) – and tested in the impedance tube. The results of these measurements confirm the theoretical predictions for this single-porosity panel very well, especially in the target low-frequency range (see Figure 8): some clear yet still acceptable discrepancies between the single-porosity absorption curves occur only at higher frequencies above 1.2 kHz. The double-porosity predictions shown in Figure 8 have not been experimentally verified for this panel, or – for that matter – for other panels with the same slit width, but they illustrate the potential improvement, which has been fully confirmed by experimental tests on the double-porosity composite with wider slits as shown below.

## 4.2 Sound absorption by the 16 mm and 25 mm composite panels

Figures 10 and 11 show sound absorption curves measured and calculated for 16 mm panels with designed tortuosities of approximately 21 and 38, respectively. Each figure shows three separate graphs: (a) for a composite panel directly backed by a rigid wall, and for two configurations with an air gap of (b) 5 mm or (c) 10 mm between the composite and the rigid wall. Both figures present the results of the calculations for the two main cases, namely single and double porosity. All calculations were performed for the actual geometry of the SLA samples (see the respective SLA rows in Table 1). The values of macroparameters calculated for 3D printed samples are listed in Table 4. They were determined analytically – note that the values for **T38** are actually copied for convenience from the first, i.e. ‘analytical’ row of Table 2. The characteristic frequencies for the viscous $f_v$, thermal $f_{th}$, and pressure diffusion $f_d$ effects are given in the last three columns of Table 4 and they were calculated by dividing by $2\pi$ the respective formulae (14), and assuming relevant air properties determined during the acoustic testing (see Table 3). To calculate the values of $f_d$, and consequently for all predictions for double-porosity composites, we used $\phi_m = 0.40$ and $K_{0m} = 7 \cdot 10^{-13} \text{ m}^2$ (so $D_{0m} = 9.44 \cdot 10^{-3} \text{ m}^2/\text{s}$) as determined for the microporus material made from gypsum powder in the BJP technology. When reading Table 4, recall that the pressure diffusion parameters and characteristic frequency $f_d$ are only relevant for the case of double porosity.

<table><caption>**Table 4:** Macro-parameters and characteristic frequencies determined for the designed panels with tortuosities equal to (approximately) 21, 38, and 36. Pressure diffusion parameters and characteristic frequency $f_d$ are relevant only for the case of double porosity.</caption><thead><tr><th rowspan="2">Panel</th><th colspan="5">visco-thermal effects</th><th colspan="3">pressure diffusion</th><th colspan="3">charact. frequencies</th></tr><tr><th>φ<br>%</th><th>α<sub>∞</sub><br>—</th><th>Λ<sub>v</sub> = Λ<sub>th</sub><br>mm</th><th>K<sub>0</sub><br>10<sup>-10</sup>m<sup>2</sup></th><th>Θ<sub>0</sub><br>10<sup>-8</sup>m<sup>2</sup></th><th>φ<sub>d</sub><br>%</th><th>ζ<sub>d</sub><br>—</th><th>Λ<sub>d</sub><br>mm</th><th>B<sub>0</sub><br>10<sup>-7</sup>m<sup>2</sup></th><th>f<sub>v</sub><br>Hz</th><th>f<sub>th</sub><br>Hz</th><th>f<sub>d</sub><br>Hz</th></tr></thead><tbody><tr><td>T21</td><td>33.71</td><td>20.99</td><td>0.900</td><td>10.84</td><td>2.275</td><td>66.29</td><td>8.93</td><td>1.770</td><td>2.326</td><td>37.6</td><td>53.4</td><td>4285</td></tr><tr><td>T38</td><td>36.20</td><td>38.26</td><td>0.900</td><td>6.386</td><td>2.443</td><td>63.80</td><td>9.26</td><td>1.586</td><td>1.735</td><td>37.6</td><td>53.4</td><td>5528</td></tr><tr><td>T36</td><td>41.87</td><td>35.90</td><td>1.400</td><td>19.05</td><td>6.838</td><td>58.13</td><td>9.84</td><td>1.944</td><td>2.232</td><td>15.5</td><td>22.1</td><td>3915</td></tr></tbody></table>

Sound absorption curves predicted for such thin 16 mm panels with networks of extremely tortuous slits have rather specific nature. They are characterised by a peak of perfect or near-perfect absorption at relatively low frequency. For the hard-backed panels with single porosity, this frequency is close to 1100 Hz for $\alpha_{\infty} = 21$, see Figure 10(a), but it
<figure><img src="image_11.png" /><figcaption>**Figure 10:** Sound absorption calculated and measured for the single-porosity T21-H16-SLA samples: (a) with a rigid backing, or (b) backed by an gap of 5 mm or (c) 10 mm. In each graph, the absorption curves measured for the clamped and inserted samples are compared with the results predicted for single and double porosity.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 11:** Sound absorption calculated and measured for the single-porosity T38-H16-SLA samples: (a) with a rigid backing, or (b) backed by an gap of 5 mm or (c) 10 mm. In each graph, the absorption curves measured for the clamped and inserted samples are compared with the results predicted for single and double porosity.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 12:** Sound absorption calculated and measured for the double-porosity T36-H25-BJP sample: (a) with a rigid backing, or (b) backed by an gap of 5 mm or (c) 10 mm. In each graph, two predictions calculated for different values of $K_{0m}$ are shown, and also the results predicted for the case of single porosity.</figcaption></figure>
approaches 800 Hz for $\alpha_\infty = 38$, see Figure 11(a). This is consistent with the well-known fact that by increasing the
tortuosity, the sound absorption efficiency can be shifted to a lower frequency range. Achieving near-perfect absorption
at low frequencies using such thin layers can be considered an achievement. On the other hand, the sound absorption is
not broadband but rather localised in the frequency range around the peak. This is mainly due to the low mesoporosity
of the panels that is less than 40% (see values of $\phi$ for the T21 and T38 panels in Table 4). To a large extent, however,
this situation can be easily improved by the use of a microporous material for the skeleton of the composite panels to
obtain double-porosity structures with well-contrasted permeabilities ensuring pressure diffusion. First of all, in the case
of double porosity, the absorption peaks are still above 0.9 and shifted to much lower frequencies, namely to 780 Hz
in the case of the 16 mm panel with $\alpha_\infty = 21$, and to only 590 Hz for the similar panel with $\alpha_\infty = 38$. Significant
sound absorption is still located around the peak, but the frequency range of relatively good absorption can even be
widened, cf. absorption curves predicted for single- and double-porosity panels in Figure 10(a). Here, we should note that
a frequency range (span) of 300 Hz or even only 200 Hz can be considered as locally wide, i.e. not strictly broadband, but
not narrowband either, when it is located around a lower frequency, i.e. well below 1 kHz, and when the aim is to absorb
mainly low-frequency sound. In this regard, it should be observed that the sound absorption coefficient of the designed
16 mm panels with double porosity is larger than 0.4 in relatively wide low-frequency ranges, namely: (i) from about
640 Hz up to 960 Hz (i.e. 320 Hz span) with the maximum of 0.96 at 780 Hz – in the case of the T21 panel with the main
pore network tortuosity of 21, see Figure 10(a); (ii) from about 500 Hz up to 700 Hz (i.e. 200 Hz span) with the maximum
of 0.93 at 590 Hz – in the case of the T38 panel with the main pore network tortuosity of 38, see Figure 11(a). Panels with
such good low-frequency sound absorption properties can be very useful as this is not achievable with the most efficient
of conventional acoustic foams, when we recall that the thickness of the layer (or composite panel) is only 16 mm.

Experimental validation of the proposed concept turned out to be more difficult than expected due to the extremely
tortuous, labyrinthine pore networks, in particular due to problems with removing powder and resin residues from the
3D printed samples. The samples have to be measured in a square impedance tube and initially the absorption curves
measured for the samples inserted into the tube differed significantly from the predictions, in particular the absorption
peak appeared at a higher frequency. We realised that this was due to sound leakages around the sample edges. This is
a well-known effect that should be avoided even when measuring conventional porous materials in a circular impedance
tube, which is achieved by properly fitting and, if necessary, sealing the sample [49, 50]. However, in the case of the
tested samples, the sealing had to be particularly tight, because even a very small leak around an extremely tortuous
sample can be critical to the accuracy of the measurements, as the edge gaps are in the form of rough but straight slits that
have a tortuosity close to unity, which spoils the designed high tortuosity of the labyrinthine slits. We confirmed this by
successively improving the fit and sealing, and re-measuring the absorption, which eventually agreed with the prediction.
Finally, as already discussed in Section 3, we also manufactured the samples with clamping edges and fastened them
tightly to the square impedance tube. With these samples, the leakage problem was gone and the absorption curves
matched their predictions very well. On the graphs in Figures 10 and 11 we show the absorption curves measured for
the inserted and clamped single-porosity samples manufactured from a photopolymer resin in the SLA technology. The
results for the clamped samples were easy to reproduce and in general agree very well with the predictions. The sound
absorption curves measured for the inserted samples differ slightly but clearly from these results, however, we checked
that the discrepancies decreased after improving the sealing.

As already reported in Section 3, we were unable to remove completely powder residue from 1 mm wide slits of
the labyrinthine networks, so only modelling predictions of the sound absorption by such composite panels with double
porosity are presented (see dash-dotted curves in Figures 10 and 11). Alternatively, we manufactured and tested the T36-
H25-BJP sample, see Figure 3(c), with not optimal but wider 1.5 mm slits and thicker walls. We kept the extremely high
tortuosity in this new design, although this resulted in an undesirable increase in thickness to 25 mm. The sound absorption
curves measured for this sample are confronted with the modelling predictions in Figure 12, in a low-frequency range up
to 1200 Hz. The comparison confirmed the expectations regarding the double porosity effect: sound absorption calculated
for the 25 mm panel with assumed single porosity has the first peak at approximately 550 Hz, see Figure 12(a), and only
when the double porosity effect is taken into account in the modelling, this peak is shifted to about 420 Hz, which is in
perfect agreement with the first peak of the measured absorption. Although we present two alternative double-porosity
results computed for different values of $K_{0m}$, the discrepancies between any of these predictions and the measurement
results are evident. They can be attributed to larger shape and surface imperfections of this 3D printed sample, partly
also to the highly probable inhomogeneities and anisotropy of the microporous skeleton, but mainly to small sound
leaks around the BJP sample as it was inserted into the impedance tube rather than clamped to it. In fact, the nature of
these discrepancies is evidently the same as in the case of the inserted SLA samples. Another reason for the observed
discrepancies – in the case of all samples – can be large distances between the entries to labyrinthine slits, which are
12.75 mm in the case of T21 panels, 16.10 mm in the case of T38 panels, and 21.00 mm in the case of T36 panels. This
may have some effect on wave propagation and therefore measurements at higher frequencies, as these values are for T21
5×, for T38 4×, and for T36 only 3× smaller than 66 mm, which is the size of the square impedance tube used for testing
the composite panel samples.
The measurement and modelling results obtained for the samples backed by an air gap are also affirmative. As expected, the sound absorption peak is shifted to a lower frequency and another absorption peak may appear at higher but still observed frequencies. Note that for a larger backing gap, the first absorption peak for the single-porosity panel becomes very similar to that of its double-porosity counterpart since the pressure diffusion effect is weaker at lower frequencies, cf. the respective results in Figures 10(c), 11(c) and 12(c).

# 5 Conclusions

The proposed composite sound absorbers have relatively simple modular, resizeable and essentially two-dimensional meso-geometry, which provides extreme tortuosity of the main pore network. All multiscale calculations for such sound-absorbing composite panels can be performed analytically, which allows for quick optimisation and examination of various specific designs. Moreover, the fundamental two-dimensionality and modularity of the proposed generic design can be manufactured using more conventional methods instead of 3D printing technologies used for this study, which should allow mass production of large-surface panels.

The correctness of the applied analytical modelling was verified by numerical calculations as well as sound absorption measurements performed on several 3D printed samples of modular panels. The experimental validation required overcoming the non-trivial difficulties related to manufacturing and testing samples of extreme tortuosity: laborious removal of powder and resin residues and extremely tight sealing of sound leaks around the samples to avoid reduction of tortuosity. Experimental results confirmed the predicted acoustic performance of the proposed high-tortuosity structures, proving that significant sound absorption, almost perfect at the peak, can be achieved at relatively low frequencies, using very thin panels. The peak frequency can even be easily lowered by making the skeleton microporous and permeable with a permeability suitably contrasted with that of the labyrinthine pore network. On the other hand, the acoustic efficiency is not broadband but rather localised around the peak frequency, mainly due to the relatively low mesoporosity, i.e. the porosity of the main pore network, which results from the two-dimensional nature of the proposed modular labyrinth designs. This can be improved, however, by designing materials with essentially three-dimensional networks of main pores (winding channels) with similar extreme tortuosity but at the same time much higher mesoporosity. For more complex designs, the matrix can be heterogeneous, i.e. assembled from several parts made of different materials with different microporosity and permeability, which should result in local non-equilibrium pressure states [51]. The channels can also be filled with a highly permeable porous or fibrous materials to form a multi-porous composite [52], but this type of design would require more theoretical development. More straightforward solutions to investigate can be double-layer configurations composed of a thin extremely tortuous composite panel backed with a thin layer of highly-porous, conventional acoustic foam. Note that the effect of the backing air cavity has already been demonstrated in this paper.

# Acknowledgements

This research was financially supported by the National Science Centre (NCN), Poland, as part of the project "Sound-absorbing composites: coupled acoustic energy dissipation mechanisms, multiscale modelling and prototyping", under Grant Agreement No. 2021/41/B/ST8/04492. For the purpose of Open Access, the authors have applied a CC-BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission.

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