# Suppressing shock-induced oscillations of a compliant panel with viscoelastic materials

Carmen Riveiro Moreno, Marie Couliou, Nicolò Fabbiane, Reynald Bur, Olivier Marquet

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Carmen Riveiro Moreno, Marie Couliou, Nicolò Fabbiane, Reynald Bur, Olivier Marquet. Suppressing shock-induced oscillations of a compliant panel with viscoelastic materials. Journal of Fluids and Structures, 2025, 137, pp.104335. 10.1016/j.jfluidstructs.2025.104335. hal-05123343

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# Suppressing shock-induced oscillations of a compliant panel with viscoelastic materials

C. Riveiro Moreno<sup>a</sup>, M. Couliou<sup>a, ID, *</sup>, N. Fabbiane<sup>b, ID</sup>, R. Bur<sup>a, ID</sup>, O. Marquet<sup>a,c, ID</sup>

## ARTICLE INFO

**Keywords:**

Fluid-structure interaction

Shock-waves

Compliant panels

Compressible flows

Viscoelasticity

## ABSTRACT

The influence of structural viscosity on the fluid-structure interaction between a normal shock-wave and a compliant panel is investigated. Two compliant panels are designed to allow significant static deformation while exhibiting different dynamic behaviors. For this purpose, two distinct elastomeric materials are used based on their dynamic responses within the shock oscillation frequency range: polyurethane 40A for the elastic panel and Tango Polyjet 61A for the viscoelastic panel. Dynamic mechanical analyses characterize the materials' dynamic properties, enabling the evaluation of natural vibration modes and frequencies of both structures. The elastic panel exhibits natural vibration frequencies that align with the ones of the natural oscillations of the shock-wave. Wind-tunnel experiments reveal strong dynamic coupling between the elastic panel and the shock-wave, leading to large-amplitude, synchronized oscillations when the shock is centered on the panel. In contrast, the viscoelastic panel is designed to avoid the fluid-structure coupling observed in the elastic panel. The marked viscoelastic properties of the material shift the natural vibration modes to higher frequencies, outside the shock-wave's natural oscillation range. As a result, in its interaction with the shock-wave, the viscoelastic panel exhibits only a large static deformation – greater than that of the elastic panel – without any dynamic fluid-structure coupling, regardless of the shock position. These findings demonstrate that viscoelastic materials hold significant potential for flow control applications, providing structural damping and frequency-dependent stiffening, which effectively decouple static deformation from dynamic interaction. Our results suggest that viscoelastic panels could be optimized as adaptive bumps for shock control, responding to fluid dynamics without inducing unwanted dynamic coupling.

## 1. Introduction

Shock-waves are ubiquitous in compressible flows around high-speed aeronautical vehicles such as commercial aircrafts (transonic regime), missiles and military fighters (supersonic regime) or re-entry vehicles (hypersonic regime). On commercial aircraft, the drag of transonic wings is strongly affected by the presence of shock-waves on their upper surface mainly for three reasons. Firstly, shock-waves entail a production of entropy, which translates into an additional generation of drag commonly known as wave drag (Déléry, 1999). Secondly, as the incoming laminar or turbulent boundary layer is subject to the strong pressure gradients induced by the shock-wave, it might separate and trigger high-frequency fluctuations responsible for additional viscous

<footnote><sup>a</sup> DAAA, ONERA, Institut Polytechnique de Paris, Meudon, 92190, France</footnote>

<footnote><sup>b</sup> DAAA, ONERA, Institut Polytechnique de Paris, Chatillon, 92320, France</footnote>

<footnote><sup>c</sup> DAAA, ONERA, Université Paris Saclay, Meudon, 92190, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: marie.couliou@onera.fr (M. Couliou).</footnote>
drag. Finally, for specific combinations of Mach number and angle of attack, natural low-frequency oscillations of the shock-wave increase rapidly in amplitude, yielding the so-called transonic buffet (Babinsky and Harvey, 2011). Fluctuations of wing loading caused by transonic buffet (Giannelis et al., 2017) may couple with natural vibration modes of the wings, potentially leading to structural failure due to high cycle fatigue. For more details on the shock-wave/boundary-layer interaction, we refer to the reviews by Gaitonde (2015) and Ligrani et al. (2020). Suppressing or mitigating the aforementioned detrimental (steady or unsteady) effects of shock-waves is crucial for ensuring structural integrity and enhancing performance of commercial aircraft.

In the last thirty years, several control methods have been investigated to mitigate the detrimental effects of shock-waves in both internal and external flows (see the reviews Dolling (2001), Huang et al. (2020)) and more specifically to reduce the wave drag (Bushnell, 2004; Rashid et al., 2022). Such control devices can be classified into active or passive, depending on whether they require a source of energy. Among the active control methods, suction through slots or perforated rigid plates (Bur et al., 2002), and arrays of pulsed-plasma jet actuators (Narayanaswamy et al., 2012a) were investigated to control shock-wave induced separation and its unsteadiness (Narayanaswamy et al., 2012b). Arrays of steady micro-jets were investigated by Souverein and Debiève (2010), Verma et al. (2015), Verma and Manisankar (2019), and fluid jet actuators were also used to control of the transonic shock buffet (Molton et al., 2013). In the category of passive control methods, arrays of vortex generators have proved to successfully mitigate shock-induced boundary layer separation in transonic flows (Titchener and Babinsky, 2015; Bur et al., 2009). The present study is predominantly interested by the so-called shock control bump, introduced to decrease the wave drag of transonic wings (see the review Bruce and Colliss (2015)). The shape of a two-dimensional bump is designed in order to produce a nearly isentropic compression in its upstream part, thus acting as a compression ramp. The most effective contour of this ramp reproduced the evolution of the boundary layer displacement thickness in the case without control (Bur et al., 2002). Placed on the upper surface of the airfoil, these bumps are designed to split the near-normal shock-wave into a number of weaker oblique shock-waves or compression waves (Bur et al., 2002). The flow deceleration being more gradual, and hence closer to an isentropic compression than for the near-normal shock-wave, the pressure loss in the wake is reduced, thus lowering the pressure/wave drag of the airfoil. Well-designed two-dimensional shock control bumps allows to increase the lift-to-drag ratio without increasing so much the viscous drag. However, their performance may significantly deteriorate for off-design operating conditions. For instance, changing the Mach number or the angle of attack may induce a translation of the shock-wave along the bump which in turn produces undesirable expansions, secondary shock-wave systems, and flow separation. This is in contrast with the more robust but less optimal performance obtained with arrays of three-dimensional shock control bumps (Eastwood and Jarrett, 2012).

Adaptive shock control bumps were more recently investigated to improve the operational robustness of shock control
bumps. Ashill et al. (1992) proposed the use of a flexible surface that could be deformed to generate the desired bump shape
beneath a shock-wave in an extended range of operation points. The authors explore this principle in a series of papers (Jinks et al.,
2014, 2016, 2018; Gramola et al., 2018; Tan et al., 2019; Gramola et al., 2020) where the plate deforms due to the aerodynamic
loading imposed by the shock-wave. In order to obtain a beneficial panel shape, the authors modify the cavity pressure, use spacers,
and actuators. In the computational study by Jinks et al. (2018), an optimization framework comprising aerodynamic and structural
computational tools were developed and used to assess the performance of various adaptive bump geometries. In the experimental
study by Gramola et al. (2018), the control of a transonic shock-wave at a Mach number of 1.4 with adaptive bumps of various
height was investigated, the latter being obtained by deforming a flexible aluminum alloy plate using spanwise actuation. All the
bumps tested reduce shock-wave unsteadiness, but wave drag saving was only observed when the shock-wave was on the upstream
portion of the flexible plate. Adaptive bumps were also investigated to control shock in other flow configurations, such as oblique
shock wave boundary layer interaction (Tan et al., 2019), with promising results. Overall, adaptive shock control bumps are a
promising solution to off design shock control, and therefore it is worth developing them further.

While designing adaptive shock control bumps, the dynamics interaction of the shock with the flexible panels should be carefully considered (Jinks et al., 2014, 2016) so as to avoid undesirable unsteady effects. In particular, the excitation of natural vibration modes of the panels by low-frequency oscillations of the shock position may lead to large amplitude shock-induced limit cycle oscillations. The latter were characterized experimentally at a Mach number (M) equal to 2 in Brouwer et al. (2021) using high-speed stereoscopic digital image correlation while investigating the motion of a thin panel excited by turbulent flow and shock/boundary-layer interactions. At the same Mach number, the dynamic interaction between a compliant panel in a flow under a compression ramp-induced shock-wave/boundary-layer interaction was also investigated experimentally in Eitner et al. (2021) and Eitner et al. (2023). High-speed stereoscopic digital image correlation and fast-response pressure sensitive paint were used to obtain the panels' simultaneous deformation and surface pressure. Their analysis of the shock foot motion showed that the panel primarily oscillates at its fundamental natural frequency. Strongly coupled oscillations between an impinging oblique shock-wave at M = 2 and a panel were also experimentally found by D'Aguanno et al. (2023). In their study, the motion of the shock wave coupled with the first and third bending modes of the panel. In a more recent experimental study by Riveiro Moreno et al. (2024), the authors investigated the static and dynamic interaction of an elastic compliant wall with normal shock at M = 1.35. Various shock-wave/panel dynamics were observed depending on the relative (time-averaged) position of the shock-wave with the compliant panel. On one hand, small-amplitude oscillations of the panels around a deviated static shape were observed when the shock was positioned near the leading or trailing edges of the compliant panel. On the other hand, large amplitude limit cycle oscillations of the elastic wall synchronized with the shock position were observed when the time-averaged position of the shock lies in the middle of the compliant panel. As shown by these studies, there is a risk of strong dynamic coupling of the shock-wave with flexible elastic structures, as such as those that constitute shock control bumps.

In the present study we will exploit the dynamics properties of viscoelastic materials to suppress the large amplitude oscillations observed for elastic panel while retaining their large static deformation. As their name suggests, viscoelastic materials respond to
<figure><img src="image_5.png" /><figcaption>**Fig. 1.** (a) Sketch of the side view along the xy-plane of the compliant panel in the wind tunnel. The normal shock is sketched in blue (b) Sketch of the top view along the xz-plane The compliant panel (orange area) is clamped to the container on its sides and free to move on its top and bottom panels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

external forces in a way that is intermediate between the behavior of an elastic solid and that of a viscous liquid. Whereas elastic materials deform when stretched and immediately return to their initial state once the stress is removed, viscoelastic materials exhibit time-dependent deformation and return to their initial state with a time delay once the stress is removed. Thanks to their viscous component, they dissipate the energy of propagating waves (Lee and Kanter, 1953) and are therefore naturally used to design shock-absorbing materials (Dura et al., 2002). In the present context, they can therefore be used to design flexible panels whose oscillations are reduced when they interact with normal shock. To completely suppress these oscillations, we will in fact exploit another interesting property of these materials, namely the variation of their stiffness as a function of an external dynamic forcing that results from the time dependence of their deformation. In general terms, the higher the external excitation frequency, in this case the shock oscillations, the stiffer will be the viscoelastic material's reaction (Ward and Sweeney, 2012). Therefore, the dynamic stiffening property of viscoelastic materials will be exploited to change the natural vibration frequencies of the compliant panel, and thus avoid the coupling with natural frequencies of the normal shock-wave. Through wind tunnel experiments we will demonstrate the capability of viscoelastic panels to exhibit large static deformations when interacting with a normal shock, without displaying large amplitude oscillations, as observed otherwise for elastic panels.

The article is organized as follows. The experimental setup defining the study case – namely, the interaction of a normal shock wave with two distinct compliant panels – is described in Section 2, along with the employed measurement techniques. The design principles of the two panels and their structural dynamic characterization are then introduced in Section 3. Finally, Section 4 presents the results of the wind tunnel tests on the coupled interaction between the panels and the shock wave.

## 2. Experimental setup and measurement techniques

The interaction of compliant panels with a normal shock-wave is investigated experimentally in the transonic open-circuit wind tunnel S8Ch located at the ONERA's center in Meudon, France. The test section of this wind tunnel is a transonic channel of height 100 mm and span 120 mm. As sketched in Fig. 1(a), the lower wall of the wind tunnel is rectilinear while the upper wall is curved and designed to obtain a half-nozzle producing a uniform supersonic flow of incoming Mach number equal to 1.35 at the end of the nozzle (Bur et al., 2006). This setup allows to mimic a near-normal shock raising around transonic airfoil. Upstream the test section, the stagnation temperature in the settling chamber is 300 ± 10 K and the stagnation pressure is 96.0 ± 0.3 kPa. The air is going through an air dryer prior to enter the wind tunnel to control the humidity in the flow and, thus, ensure reproducibility of shock behavior (Sartor et al., 2015, for a similar setup). Downstream the test section, a second throat of adjustable cross-section (not visible in the figure) is placed to choke the flow. This allows not only to isolate the test section from downstream perturbations but also to change, in a continuous and precise manner, the time-averaged streamwise position of the normal shock-wave in the test section.

In the following, the origin O of the Cartesian coordinate system (O, x, y, z) is placed on the lower wall and in the mid-span plane of the wind tunnel, at the streamwise position corresponding to the mid-length of the panel. This streamwise location coincides with the end of the curved section of the half nozzle. The second throat is adjusted so that the streamwise position of the shock is close to the x = 0 mm, as sketched in Fig. 1(a). During the experiments, the instantaneous streamwise position $x_s(t)$ of the shock-wave at the mid-height of the test section (y = 50 mm) is tracked using schlieren images. Optical accesses of the wind tunnel on the two lateral doors enable schlieren visualizations. These are conducted with a Z-type schlieren setup using a high speed Phantom camera and a blue LED source. A vertical knife orientation is used, which yields images whose light intensity is proportional to horizontal density gradients. The schlieren videos are acquired at a sampling frequency of 7 kHz for a duration of 3.87s, which uses the maximum camera memory capacity. The spatial resolution is 0.22 mm/pixel. The field of view corresponds to the length of the compliant panel in the streamwise direction and the height of the wind tunnel for the panel-normal direction, i.e. 160 mm × 100 mm. In the present study, the nominal configuration of the normal shock oscillating over a rigid (lower) wall corresponds to the time-averaged streamwise position $\bar{x}_s = -3.6$ mm. As previously investigated in Riveiro Moreno et al. (2024), the unsteady oscillations of the shock around this time-averaged position are characterized by a broadband spectrum in the frequency range $70 \text{ Hz} \le f \le 300 \text{ Hz}$.
To investigate the interaction of the normal shock with compliant panels, the latter is integrated into (and mounted flush to) the lower wall of the test section, as seen in orange in Fig. 1. The center position of the compliant panels in the streamwise and spanwise directions corresponds to the origin O of the coordinate system so that the nominal position of the normal shock-wave is close to panel’s center. The two compliant panels investigated in the present study present a length of L = 160 mm, width W = 85 mm, and a nominal height of H = 20 mm. This geometry is given by the size of the container. Structural numerical studies as well as tests in the wind tunnel with different panel heights showed that this geometry promotes the interaction of the normal shock with the compliant panels (Riveiro Moreno, 2024). The compliant panels are clamped on the lateral walls of the container, while their upper and lower surfaces are free to move. The top surface of the compliant panel is the interface with the transonic flow in the test section. The bottom surface of the compliant panel is in contact with a cavity. Inside this cavity, a constant pressure P<sub>bottom</sub> = 45 kPa is measured with a static pressure tap, corresponding to the average pressure value of the pressure distribution along the test section. The average temperature at the compliant panel surface is estimated to be 293 K, as derived from the isentropic relations for a compressible flow and using a recovery factor of 0.9, used for turbulent boundary layer (for more details see Riveiro Moreno (2024)).

In the same manner as for the shock-wave, the displacement of the compliant panel and its dynamics can be tracked with the schlieren setup. When deflected upwards, the compliant wall casts a shadow onto the schlieren images and the upper surface of the compliant panel can therefore be tracked. However, when deflected downward, the compliant panel is not visible in the schlieren images. This partial information on the position of the upper wall of the panel is used to reconstruct its complete, instantaneous displacement field. The procedure is based on the fitting of the natural vibration modes of the panel and a precomputed static deformation of the wall on the scattered space-time information recovered from the schlieren images. Further details on the reconstruction algorithm can be found in Riveiro Moreno et al. (2024).

The measurement of the compliant panel dynamics is limited by the spatial resolution of the schlieren images. If this is not an issue when large oscillations of the wall are observed, it can become problematic when small dynamic displacements occur. For this reason, the compliant panel’s displacement is also measured through a non-intrusive optical point-wise displacement sensor Keyence LK-G402. The displacement resolution of this sensor is 0.2 µm meaning it provides a panel position 1000 times more precise than the one extracted from schlieren images. The sensor is mounted on a metallic structure outside of the wind tunnel facing the wind tunnel’s upper panel, which is also equipped with a window. Due to restrictions on the optical window of the test section, the sensor is installed so that the laser beam impinges the compliant panel with 4° of inclination. The measured point is located at x = 27 mm and z = 2 mm. Note that this is not exactly the centerline of the test section. The signal is acquired at 20 kHz during 60 s and is filtered in at the acquisition card at 1 kHz. A high pass sixth-order Butterworth filter at 600 kHz is applied to the displacement signal to filter out noise generated by the acquisition equipment. These measurements are available only for the case of the viscoelastic panel.

## 3. Design principles and characterization of the viscoelastic materials and compliant panels

Two compliant panels of identical structure described in the previous section but made of different materials are designed in this section. After explaining in Section 3.1 the underlying principle for using viscoelastic materials to avoid lock-in effect with normal-shock oscillations, the dynamical properties of the two selected viscoelastic materials, i.e. polyurethane 40 A and Tango Polyjet 61 A, are characterized experimentally in Section 3.2 with a Dynamic Mechanical Analysis. The experimental determination of the storage and loss moduli then allows to determine numerically in Section 3.3 the lowest natural vibration frequencies and the related modes of the two compliant panels made by these viscoelastic materials.

### 3.1. Design principle

A material exhibits elastic behavior when it returns to its original state once a load is removed (Meyers and Chawla, 2008). For small deformations, this behavior follows Hooke’s law, which describes a linear relationship between the internal forces in the solid, stress, and its deformation, strain. The behavior of viscoelastic materials, however, is more complex (Shaw and MacKnight, 2018). As their name suggests, these materials exhibit both elastic and viscous properties. This results in a constitutive law – i.e., the relation between stress and strain – that depends not only on the instantaneous strain state but also on its time evolution. In the case of harmonic deformations, a Hooke-like relation can still be recovered (Ward and Sweeney, 2012). The time-dependent strain and stress tensors can be Fourier decomposed as $\hat{e}(t) = \int \hat{e}(\omega) e^{i\omega t} d\omega + \text{c.c.}$ and $\hat{\sigma}(t) = \int \hat{\sigma}(\omega) e^{i\omega t} d\omega + \text{c.c.}$, where $\omega$ is the angular frequency, while $\hat{e}(\omega)$ and $\hat{\sigma}(\omega)$ are the complex Fourier component of the tensors at that frequency. The frequency-dependent linear relation between these two complex components reads

$$ \hat{\sigma}(\omega) \sim \hat{E}(\omega) \hat{e}(\omega), \quad \text{where} \quad \hat{E}(\omega) = E'(\omega) + iE''(\omega) \qquad (1) $$

is the complex, frequency-dependent Young’s modulus. The real part, $E'(\omega)$, is called the storage modulus and represents the material’s ability to store energy, reflecting its elastic behavior. The imaginary part, $E''(\omega)$, is known as the loss modulus, and it quantifies energy dissipation, representing the material’s viscous behavior. The elastic response of the material is recovered in the static case ($\omega = 0$), with the storage modulus $E'(0)$ corresponding to the classical Young’s modulus E and the loss modulus vanishing, i.e. $E''(0) = 0$, since there is no viscous effect in a static case. This classical Young’s modulus will therefore govern the amplitude of static deformation of the panel, whether it is made of elastic or viscoelastic materials. In the design of adaptive shock control bumps (Ashill et al., 1992), a material with a low value of the static Young’s modulus is of interest for obtaining large static displacements. For the dynamic response ($\omega \neq 0$), the viscoelastic behavior gradually onsets when increasing the excitation
frequency. At low excitation frequencies, the storage modulus $E'$ remains nearly constant and equal to its static value. The material then exhibits an elastic behavior that may extend within a large frequency range, depending on the nature of the material. Purely elastic materials, such as metals, are de facto always in this state, with their Young modulus constant with respect to the excitation frequency. In viscoelastic materials, such as elastomers, the viscoelastic effects gradually emerge as the frequency increases, up to the point where a large increase of both $E'(\omega)$ and $E''(\omega)$ is observed (Ward and Sweeney, 2012). The magnitude of these changes and the frequency range in which they occur are properties of the material.

In this work, the design of the two compliant panels aims to leverage this frequency-dependent behavior of viscoelastic materials to show how it can be exploited to pilot the dynamic response of the structure. For an isolated structure, the dynamic response is primarily governed by its natural vibration modes, i.e., the free harmonic oscillations that occur at specific frequencies $\omega_i^n$, known as natural vibration frequencies, with $i$ a positive integer. These frequencies are intrinsically linked to both the material's rigidity and inertia, and they scale according to the relation:

$$ \omega_i^n \sim \left( \frac{|\hat{E}(\omega)|}{\rho} \right)^{\frac{1}{2}}, \qquad (2) $$

where $\rho$ is the material's density, $\hat{E}(\omega)$ is the complex Young's modulus introduced earlier and $\omega$ the frequency of the excitation. For purely-elastic materials, the natural frequencies of the structure scale as $\sqrt{E/\rho}$ and are thus directly linked to its static mechanical properties (since $E'(\omega) = E$ and $E''(\omega) = 0$). For viscoelastic materials, however, the relationship is more complex due to the frequency-dependent nature of $\hat{E}(\omega)$. This frequency dependence introduces an additional degree of freedom in the design of the structure's dynamics—specifically. If the value of $|\hat{E}(\omega)|$ is greater than that of the static Young's modulus $E$ in the frequency range of interest, a statically-flexible but dynamically-rigid structure can be obtained. In the present case, we should select materials so that is $|\hat{E}(\omega)|$ is large in the range $70 \text{ Hz} \le f \le 300 \text{ Hz}$, with $f = \omega/2\pi$.

Following this design principle, we select two different elastomeric materials for our compliant panels: polyurethane 40A and Tango Polyjet 61A. Materials in this family exhibit the high flexibility needed for adaptive bump design (Ashby, 2011), while offering a range of viscoelastic behaviors (Ward and Sweeney, 2012). A Dynamical Mechanical Analysis (DMA) is first performed in Section 3.2 to measure the loss $E'$ and storage $E''$ modulus of each material in a wide range of frequencies. A numerical finite-element analysis is then proposed in Section 3.3 to determine the lowest frequencies $\omega_i''$ of the natural vibration modes for panels made with these two materials. We will see that the polyurethane behaves as an elastic material within the frequency range of the natural shock oscillations ($70 \text{ Hz} \le f \le 300 \text{ Hz}$) whereas the Tango Polyjet behaves as a viscoelastic material within the same frequency range.

## 3.2. Experimental characterization of the viscoelastic materials

In this section, the mechanical properties of the two materials – polyurethane 40 A and Tango Polyjet 61 A – are experimentally characterized. Both materials are elastomers, with a density $\rho$ measured to be approximately $1000 \text{ kg/m}^3$ for each. They also exhibit similar static Young's moduli $E$, estimated at $1.8 \text{ MPa}$ for polyurethane and $0.84 \text{ MPa}$ for Tango Polyjet. As elastomers, both materials behave as nearly incompressible, with a Poisson's ratio of $\nu \approx 0.499$. Despite these similarities, the two materials are made by different polymeric chains and they exhibit significantly different dynamic behaviors, as it will be discussed in the following sections.

Dynamic mechanical analysis (DMA, Dunson, 2017) is employed to characterize the frequency-dependent dynamic response of the two materials. This technique consists on applying a harmonic excitation onto the material at different frequencies. At each frequency, the complex viscoelastic moduli of the material, such as the complex Young modulus, are measured. In order to obtain this information for a wider frequency range than that provided by the machine, one can leverage the time-temperature equivalence (Ward and Sweeney, 2012). In its simplest form, this principle suggests that the material's response, and thus its mechanical properties, at one frequency is equivalent to the one displayed at a higher or lower frequency, if the temperature is decreased or increased respectively. This is a characteristic property of polymers, and thus elastomers. Therefore, commonly during DMA tests, the complex modulus $\hat{E}(\omega)$ is measured for a sweep of frequency at different working temperatures. Then, in a second step, the data is rearranged according to the time-temperature equivalence, for the temperature at which the material will be operating. As a result, one obtains the so called master curve of the material, which displays the measured storage $E'(\omega)$ and loss $E''(\omega)$ moduli, for a wide range of frequencies at a given temperature (Ward and Sweeney, 2012). It is worth noting that temperature dependent behavior of elastomers can be exploited to control the viscoelastic properties of a material. An increase in temperature shifts the master curve towards higher frequencies, while a decrease in temperature shifts it towards lower frequencies. Thus, in principle, a single material could be employed instead of two, with the panel being heated or cooled to attain the desired viscoelastic properties. However, this approach would require a more complex experimental setup, which led to the decision to use two different materials in this study. We note that the influence of environmental humidity on the mechanical properties of the viscoelastic material (e.g., Moreland et al., 1994) is negligible in our setup due to the controlled, low humidity levels maintained within the test section.

The DMA of a sample of polyurethane 40 A was performed using a Modular Compact Rheometer (MCR 502) at the Solid Mechanics Laboratory (LMS) at École Polytechnique in Palaiseau, France (Riveiro Moreno, 2024). Fig. 2(a) shows the measured storage ($E'$, red dots) and loss ($E''$, orange dots) moduli as a function of the excitation frequency $f$, at the estimated working temperature of the compliant panel in the wind tunnel ($T = 293 \text{ K}$). The typical behavior of viscoelastic materials is observed. At low frequencies, the storage modulus $E'$ approaches the static Young's modulus (represented by the black horizontal solid line), while the loss modulus $E''$ is approximately an order of magnitude lower. This characterizes the rubbery regime, where the viscoelastic material exhibits an almost elastic behavior. At higher frequencies, the material transitions to the so-called glassy state, which
<figure><img src="image_6.png" /><figcaption>**Fig. 2.** Experimental characterization of the viscoelastic behavior of polyurethane 40 A (a) and Tango Polyjet 61 A (b). The complex Young modulus $\hat{E}(f) = E'(f) + iE''(f)$ ($E'$ red dots, $E''$ orange dots) is reported at a temperature $T = 293$ K. The horizontal solid line reports the value of the static Young's modulus $E = \hat{E}(0)$ for reference. The black dots show the dissipation phase $\delta = \arctan(E''/E')$. The shaded areas indicate the range of frequencies for which shock oscillations are observed in the rigid wall configuration ($70 \text{ Hz} \le f \le 300 \text{ Hz}$). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

fully onsets for $f > 10^7$ Hz, with the storage modulus stabilizing around 1 GPa, i.e., 1000 times greater than the measured static Young's modulus $E = 1.8$ MPa. Due to the significant variation in the magnitude of the storage and loss moduli, the loss phase $\delta(\omega) = \arctan(E''/E')$ (black dots in the figure) is introduced to quantify the relative contributions of elasticity and viscosity to the material's behavior. The typical bell-shaped curve is observed, with a dissipation peak occurring during the rubbery-to-glassy transition (Dunson, 2017).

A similar behavior, though shifted towards lower frequencies, is observed for the Tango Polyjet 61 A. Fig. 2(b) presents the results of the DMA conducted on this material using a Metravib DMA+1000 at the Department of Materials and Structures (DMAS) at ONERA in Châtillon, France (Riveiro Moreno, 2024). In this case, the rubbery state occurs at much lower frequencies ($f < 10^{-1}$ Hz), while the glassy state is reached at frequencies higher than $f > 10^3$ Hz with a storage modulus around 1 GPa, three orders of magnitude higher than its Young modulus $E = 0.84$ MPa. In the same manner as for polyurethane, the loss phase $\delta$ attains its maximum value of approximately 60° within the transition between the two states, highlighting the importance of viscosity in this phase.

When comparing the materials' master curves shown in Fig. 2 at the frequency range of the natural oscillation of the shock (gray shaded area), it is possible to see that polyurethane is in the rubbery state, while Tango Polyjet has almost completed the transition to the glassy state. The rubbery state of polyurethane implies that its static and dynamic behavior are determined by the static Young's modulus of the material, as it would be the case for a purely elastic material. For this reason, in the following, we will refer to the polyurethane made panel as the *elastic* one. Within the same range of frequencies, Tango Polyjet's storage ($E'$, red dots) and loss ($E''$, orange dots) moduli exhibit values two orders of magnitude higher than the static Young's modulus $E$ (black horizontal solid line). At this point, the viscous component of the material – quantified by the loss modulus $E''$ – cannot be neglected, and thus the material presents a viscoelastic behavior. Because of this, we will refer to the Tango Polyjet made panel as the *viscoelastic* one. Therefore, under the excitation of the shock oscillations, we would expect one panel to display an elastic behavior (polyurethane made panel), and the other a viscoelastic (Tango Polyjet made panel).

The shock oscillations can also excite vibration modes of the structure, leading to a strong dynamic coupling. As previously mentioned in Section 3.1 in Eq. (2), the natural frequencies of a structure, and thus those of the panel are linked to the complex Young's modulus $\hat{E}(\omega) = E'(\omega) + iE''(\omega)$. As the master curve of Tango Polyjet is displaced towards the lower frequencies with respect to the polyurethane, overall, for a given frequency, Tango Polyjet displays higher storage and loss modulus than polyurethane. Therefore, we anticipate that the panel fabricated with Tango Polyjet will exhibit natural vibration modes at higher frequencies than those of the polyurethane made panel. This indicates that the Tango Polyjet has the dynamic stiffening properties we aimed for, potentially preventing a coupling between the shock-wave dynamics and the panel's natural oscillations. In the following sections, this expectation is validated through numerical simulations that take into account the characterized viscoelastic behavior of both materials.

### 3.3. Numerical characterization of the dynamic properties for the two compliant panels

The previous results provide a quantitative characterization of the viscoelastic behavior of the two materials used in the fabrication of the compliant panels. The results indicate that the natural frequencies of the compliant panels should be of different order of magnitude. In order to verify this prediction, the natural vibration modes of a homogeneous viscoelastic structure are computed, and the natural vibration frequencies of the two panels are numerically evaluated.
Because of the viscoelastic nature of the materials, the natural vibration modes are not purely harmonic and, therefore, a time decay of their amplitude has to be taken into account. To this end, the following form for the time-dependent displacement field $\xi(x,t)$ in the solid is considered

$$ \xi(x,t) = \hat{\xi}(x) e^{(i\omega - \gamma)t} + \text{c.c.}, \quad (3) $$

with $\omega$ being the angular frequency of the mode and $\gamma$ the damping. A positive value of $\gamma$ indicates exponential decay of the motion amplitude, corresponding to a stable solution, whereas a negative one implies exponential growth, signifying an unstable behavior. The complex displacement field $\xi(x)$ indicates the spatial structure of the natural vibration mode.

The local equation of motion is then introduced (Altenbach and Eremeyev, 2014):

$$ -(\omega + i\gamma)^2 \rho \hat{\xi} = \nabla \cdot \hat{\sigma}(\omega, \hat{\epsilon}(\xi)), \quad (4) $$

This equation is defined within the solid domain; with the respective clamp and free-stress boundary conditions on the sides of the plate, it describes the dynamic of the structure. The three dimensional (3D) local stress state of the material is given by the stress tensor $\hat{\sigma}$, which is a function of the frequency $\omega$ via the complex Young's modulus $\hat{E}(\omega)$, and the 3D local strain tensor $\hat{\epsilon}$. The detailed relation linking the stress tensor and the strain tensor reads,

$$ \hat{\sigma}(\omega, \hat{\epsilon}) = \hat{E}(\omega) \underbrace{\frac{1}{(1 + \nu)} \left( \frac{\nu}{(1 - 2\nu)} \operatorname{tr}(\hat{\epsilon}) + \hat{\epsilon} \right)}_{\mathcal{A}(\nu)\epsilon} = \hat{E}(\omega) \mathcal{A}(\nu) \hat{\epsilon}, \quad (5) $$

where $\nu$ is the Poisson ratio, introduced in the previous section. This equation completes via the operator $\mathcal{A}(\nu)$ the linearity relation between stress and strain already introduced in Eq. (1). For small displacements, the strain tensor is defined as $\hat{\epsilon} = \frac{1}{2}(\nabla\xi + (\nabla\xi)^T) = \mathcal{E}\hat{\xi}$, where $\mathcal{E}$ is a linear operator. Substituting this relation and Eq. (5) into Eq. (4), we obtain,

$$ -(\omega + i\gamma)^2 \rho \hat{\xi} = \hat{E}(\omega) \underbrace{\nabla \cdot (\mathcal{A}(\nu) \mathcal{E} \hat{\xi})}_{\mathcal{K} \hat{\xi}} \Rightarrow ((\omega + i\gamma)^2 \rho \mathcal{I} + \hat{E}(\omega) \mathcal{K}) \hat{\xi} = \mathbf{0}, \quad (6) $$

where the complex Young's modulus $\hat{E}(\omega)$ can be factored out of the divergence operator, since the structure is composed of a homogeneous material. Moreover, since the term $\nabla \cdot (\mathcal{A}\mathcal{E}\hat{\xi})$ is linear with respect to the displacement field $\hat{\xi}$, it can be rewritten as $\mathcal{K}\hat{\xi}$, where $\mathcal{K}$ is a linear operator that depends only on the geometry of the structure and the material's Poisson ratio. The natural vibration modes of the structure are then given by the eigenvalue problem in Eq. (6), which is non-linear because of the dependency of $\hat{E}$ by the natural frequency $\omega$.

This can be simplified by introducing the eigenvalue $\Lambda$, defined as,

$$ \Lambda^2 \frac{\hat{E}(\omega)}{\rho} = (\omega + i\gamma)^2. \quad (7) $$

The eigenvalue problem thus becomes

$$ (\Lambda^2 \mathcal{I} + \mathcal{K}) \hat{\xi} = \mathbf{0}, \quad (8) $$

which is linear and does not depend on the mechanical properties of the material. Thanks to this formulation, we decouple the geometric characteristics of the structure from the viscoelastic properties of the material, allowing a simpler two-steps computation of the natural vibration modes of the viscoelastic plate. Once the eigensolutions $(\Lambda_i, \hat{\xi}_i)$ are computed from Eq. (8), the corresponding natural frequencies $\omega_i^n$ and damping factors $\gamma_i^n$ can be calculated by solving the scalar non-linear equation in Eq. (7), which incorporates the viscoelastic constitutive law $\hat{E}(\omega)$ of the material, previously determined experimentally in Section 3.2. The different notation between $\omega$ and $\omega_i^n$ indicates that the latter are solutions of the eigenvalue problem and, thus, natural frequencies of the structure.

From a numerical point of view, the linear eigenvalue problem in Eq. (8) is addressed using the finite-element method (FEM), where the displacement field $\hat{\xi}(x)$ is discretized over the solid volume using a 3D mesh. The operator $\mathcal{K}$ is then constructed accordingly using the FreeFem++ toolbox (Hecht, 2012), as detailed in Riveiro Moreno et al. (2024). Once the eigenvalue problem is formulated, the eigenvalues $\Lambda_i$ and corresponding mode shapes $\hat{\xi}_i(x)$ are computed. From these, the natural frequencies $\omega_i^n$ and damping ratios $\gamma_i^n$ are determined for the two materials by solving Eq. (7) using the secant method. To regularize the solution of this non-linear problem, a fitting of the experimentally characterized $\hat{E}(\omega)$, based on the generalized Maxwell model, is applied [for more details on the fitting, see Riveiro Moreno (2024)].

Thanks to Eq. (7), we also recover the scaling between the natural frequencies $\omega_i^n$ and the frequency-dependent modulus $\hat{E}(\omega)$, introduced by Eq. (2) in Section 3.1. This confirms that $\omega_i^n$ is proportional to the square root of $\hat{E}(\omega_i^n)$, with the proportionality factor given by the eigenvalue $\Lambda_i$.

The squares in Fig. 3(a) indicate the computed frequency and damping of the first twelve natural vibration modes of the elastic panel, fabricated with the polyurethane 40 A. The natural frequencies mostly sit within the frequency range in which the polyurethane exhibits a rubbery behavior, i.e. the loss modulus $E'(\omega)$ is approximately equal to the static Young's modulus $E = E'(0)$, see Fig. 2(a). The first vibration modes are within the frequency range of the natural shock oscillation observed in the rigid wall case (gray shaded area); therefore, we expect that the elastic panel would show a lock-in behavior when interacting with the flow. The
<figure><img src="image_7.png" /><figcaption>Fig. 3. Natural frequencies, damping ratio, and modes for the two designed compliant panels. (a) reports the frequency $f_1^n = \omega_1^n / (2\pi)$ and damping ratio $\gamma_1^n / \omega_1^n$ of the first 12 natural vibration modes for the two panels made by polyurethane (squares) and Tango Polyjet (circles). The shaded areas indicate the range of frequencies for which shock oscillations are observed in the rigid wall configuration. The shape of the first three natural vibration modes (filled markers in a) are reported in (b) for the polyurethane and in (c) for the Tango Polyjet. The colormap (negative in blue, positive in red, and zero in white) shows the panel-normal displacement of the compliant panel in the 2D xz-plane at the interface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

modes are all stable, with a damping ratio $\gamma_i^n / \omega_i^n > 0$: this non-dimensional quantity is often used instead of the natural damping $\gamma_i^n$, since it is related to the attenuation of the motion amplitude at each oscillation cycle.

The circles in Fig. 3(a) report instead the natural frequencies and damping factors of the viscoelastic panel, fabricated with the Tango Polyjet. They occur at a frequency one order of magnitude higher than the ones of the elastic wall, therefore confirming the final remarks in Section 3.2. At these frequencies the Tango Polyjet is in its glassy behavior (see Fig. 2(b)) and, therefore, the natural vibration frequencies are driven by the storage modulus $E'$, which in the glassy state reaches approximately 100 times the value of static Young's modulus $E$. In this regime, the viscoelastic nature of the material effectively decouples the static response of the panel determined by the $E$ from its dynamic response determined by $E'$. The natural frequencies sit outside the frequency range of the natural shock oscillation observed in the rigid wall case and, therefore, we do not expect any resonance occurring when tested in the wind-tunnel.

The modal shapes $\hat{\xi}_i(x)$ corresponding to the first three natural vibration modes are shown in Fig. 3(b) for the elastic panel, and in Fig. 3(c) for the viscoelastic panel. The colormap depicts the vertical displacement on the xz-plane at the interface with the flow, i.e., the upper surface of the panel. Both panels exhibit identical modal shapes, as expected, since they are governed by the eigensolutions $(\Lambda_i, \hat{\xi}_i)$ of Eq. (8). However, we observe that the modes occur in the same sequence for both panels. The first mode is characterized by a single-lobe upward/downward motion, the second mode by two counter-phase lobes along the streamwise direction, and the third mode by three lobes, further subdividing the motion.

Based on this analysis, which highlights the distinct dynamic behavior of the two panels, we anticipate that their responses when interacting with the shock-wave will differ significantly. The elastic panel, made of polyurethane, is expected to resonate with the shock-wave, while the viscoelastic panel, made of Tango Polyjet, will likely exhibit only a static deformation.

## 4. Wind-tunnel experimental results of the normal shock interacting with compliant panels

In the following, the fluid-structure interaction of a normal shock with both compliant panels is experimentally characterized, illustrating the two completely diverse behaviors (Section 4.1): a strong dynamic coupling between the elastic structure and the
<caption>**Table 1**</caption>

Properties of the two fabricated compliant panels. The Young's moduli at 0 Hz – static – and 100 Hz are reported for both materials. The last columns list the three lowest computed natural frequencies of the panels.

<table><thead><tr><th>Panel</th><th>Material</th><th><i>E</i>'(0 Hz)</th><th><i>E</i>'(100 Hz)</th><th><i>E</i>''(100 Hz)</th><th><i>f</i><sub>1</sub><sup>n</sup></th><th><i>f</i><sub>2</sub><sup>n</sup></th><th><i>f</i><sub>3</sub><sup>n</sup></th></tr></thead><tbody><tr><td>Elastic</td><td>Polyurethane 40 A</td><td>1.8 MPa</td><td>2.15 MPa</td><td>0.25 MPa</td><td>118 Hz</td><td>152 Hz</td><td>207 Hz</td></tr><tr><td>Viscoelastic</td><td>Tango Polyjet 61 A</td><td>0.84 MPa</td><td>195 MPa</td><td>145 MPa</td><td>1936 Hz</td><td>2551 Hz</td><td>3542 Hz</td></tr></tbody></table>

shock-wave, and no dynamic interaction between the viscoelastic panel and the shock-wave. Lastly, the position of the shock-wave along the viscoelastic compliant wall is modified, in order to understand the impact of the shock position on the fluid-structure interaction (Section 4.2).

The elastic and viscoelastic panels are thus fabricated and tested in the wind tunnel. The elastic panel is produced by mixing the two components in their liquid state and pouring them directly into the container (black region in Fig. 1(b)). During the curing process, the material chemically bonds to the container's lateral walls. In contrast, the viscoelastic panel is 3D-printed and subsequently attached to the container using silicone-based adhesive. As a result, the compliant panels can be considered clamped to the lateral walls of the container, with their upper and lower surfaces free to move. The thickness of both panels is then measured to ensure compliance with the design specifications. During this inspection, we found that the fabricated elastic panel has a thickness of 19 mm, slightly less than the nominal 20 mm. No dimensional inconsistencies were observed in the viscoelastic panel.

Table 1 summarizes the characteristics of the fabricated elastic and viscoelastic compliant panels. The reported natural frequencies of the elastic panel have been recomputed to take into account its real thickness: because of this reason, they are slightly lower (less than 5 Hz) than the ones computed within the design process and reported in Fig. 3.

The elastic panel is fabricated in polyurethane 40 A; this material has a static Young's modulus *E* of 1.8 MPa, thus allowing for considerable static deformation. At 100 Hz, within the frequency range of the natural oscillations of the shock-wave (70 Hz ≤ *f* ≤ 300 Hz), the dynamic storage modulus *E*′ is comparable with the static one, with a low viscous dissipation, as indicated by the loss modulus *E*′′ ≪ *E*′. This indicates that the panel will behave almost in an elastic manner when interacting with the shock. Moreover, the natural frequencies fall within the frequency range of the natural oscillations of the shock-wave, suggesting a potential resonance effect that could amplify the panel's vibrations during the interaction with the shock-wave.

On the other hand, Tango Polyjet 61 A is considered for the fabrication of the viscoelastic panel; this material exhibits a lower static Young's modulus of 0.84 MPa, but a much higher storage and loss moduli at 100 Hz. This would allow for large static deformation, but a viscoelastic dynamic interaction with the shock. This significant increase in storage modulus results also in higher natural frequencies, with the first one at 1936 Hz, which are well above the frequency range of the natural oscillations of the shock-wave. As a result, the Tango Polyjet panel is expected to avoid coupling and thus minimize adverse dynamic effects during the interaction with the shock-wave.

## 4.1. Comparison of the coupled response of the two compliant panels with the normal shock

The first case investigated of shock-induced fluid-structure interaction corresponds to the interaction of a normal shock with the elastic panel made in polyurethane. Fig. 4(a) displays an instantaneous schlieren image of the fluid-structure interaction between the normal shock and the elastic panel. We observe that the shock displays a large λ shock foot. The front leg is located slightly downstream of the position of the maximal deformation. Large structures are shed on the mixing layer downstream the shock's rear leg. The presence of the large λ-shaped shock foot is an indicator of the existence of a large separation bubble under the shock foot (Babinsky and Harvey, 2011). As a supplementary material of the present article, a schlieren movie of several cycles is provided, which shows large periodic synchronized oscillations of the shock and the compliant panel occur, as detailed in Riveiro Moreno et al. (2024).

Schlieren images are analyzed to study the deformation of the elastic panel under the normal shock. As the elastic panel deforms upwards, it blocks the passage of light and thus casts a shadow on the image — e.g. the black area at the bottom left of schlieren image in Fig. 4(a). Downwards deformations cannot be captured with the present setup, because of lack of optical access; when moving downwards the elastic panel is hidden by the container and thus, not visible in the schlieren — e.g. the black area at the bottom right of schlieren image in Fig. 4(a). A reconstruction method based on the Schlieren data and the 3D linear equation that describes the material behavior (see Riveiro Moreno et al. (2024)) is used to reconstruct the instantaneous 3D motion of the elastic panel. The orange dash line in Fig. 4(a) corresponds to the instantaneous reconstructed panel displacement at the centerline ($z = 0$). The time-averaged deformation of the elastic panel is reported in Fig. 4(b) by the black solid, as well as the plus and minus of the standard deviation in dashed orange. The time-averaged deformation consists of an upward-downward displacement at the center line of the panel, as the result of the pressure difference across the elastic panel. Upstream the shock wave, the cavity pressure is higher than that of the supersonic region and thus it generates a positive deformation of the compliant panel. The opposite takes place downstream of the shock wave, where the pressure in the subsonic region is lower than that of the cavity.

The space-time diagram of the elastic panel deformation over the center line is presented in Fig. 4(c) where the horizontal axis is the streamwise coordinate, the vertical axis is the time and the colormap is reconstructed panel normal displacement of the elastic panel at the centerline. To assess the phase difference between the shock and the panel oscillations, the shock streamwise position was also extracted from the schlieren images. The evolution of the shock position retrieved at $y = 50$ mm (blue dot in Fig. 4(a)) is indicated by the dash-dotted in Fig. 4(c). We observe that an upward movement of the panel upstream of the shock coincides with
<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Interaction between the elastic panel and the normal shock-wave. (a) displays an instantaneous schlieren image for the elastic material with the shock at $\bar{x}_s = -3.6$ mm. The orange dash line corresponds to the instantaneous reconstructed panel displacement at the center line $z=0$. The blue dot refers to the shock position and the orange rectangle to the panel position where the panel dynamic is analyzed. (b) shows the reconstructed displacement of the panel: the black solid line reports the average displacement of the panel, while the dashed orange lines indicate plus and minus the standard deviation. The time-space evolution of the panel displacement at the center-line ($z=0$) is reported by the contour plot in (c); the dash-dotted line indicates the evolution of the shock position. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

a downstream displacement of the shock and vice-versa. The motion of the structure consists of an alternating periodic positive-negative displacement. In Riveiro Moreno et al. (2024) it was shown that it is mainly the second bending mode of the structure (see Fig. 3(b) middle) the one responsible for this oscillation. Interestingly, the duration and amplitude of the negative displacement for $x > 0$ are longer and larger than for the positive displacement. The opposite pattern is found for $x < 0$. This results from the existence of a static deformation of the compliant panel caused by the pressure gradient imposed by the shock.

The interaction between the viscoelastic panel and the normal shock is now analyzed and compared to one of the elastic one. Fig. 5 reports the equivalent data as in Fig. 4 for the viscoelastic panel. Fig. 5(a) displays an instantaneous schlieren image of the flow of the viscoelastic panel. The flow features a large $\lambda$ shock foot, and large structures shed on the mixing layer, that resemble those obtained with the elastic compliant panel in Fig. 4(a). A schlieren movie is also made available as supplementary material of the present article. Contrary to the elastic panel configuration, where both the shock and the panel oscillate with large amplitude, the shock barely oscillates and the viscoelastic panel presents only a static deformation.

The time-averaged reconstructed panel displacement at the center line $z=0$ is reported as a black solid line in Fig. 5(b). As for the elastic panel deformation (Fig. 4(b)), the time-average deformation presents an upward-downward displacement, although of higher magnitude. The maximum upward displacement upstream the shock is around +2.5 mm for the viscoelastic panel (+1 mm, for the elastic one), while the reconstructed downward displacement downstream the shock sets around -1.5 mm (-0.6 mm, for the elastic one). Both the maximum and minimum displacement of the viscoelastic panel (dashed orange lines in Fig. 5(b)) are coincident with the averaged deformation, meaning that no motion of the viscoelastic panel is observed from the schlieren. This is even clearer from the space-time diagram of the panel deformation over the center line presented in Fig. 5(c), where the horizontal axis is the streamwise coordinate, the vertical axis is the time and the colormap is the panel-normal displacement of the viscoelastic panel. As expected, there is no dynamic motion of the structure in time. To assess the phase difference between the shock and panel oscillations, the evolution of the shock position (blue dot in Fig. 5(a)) is indicated by the dash-dotted line in Fig. 5(c). We observe that the shock remains located in the middle of the viscoelastic panel. The peak-to-peak amplitude of shock wave oscillation is of few millimeters and much lower than the one observed for the elastic panel, which sets around 8 mm and is synchronized with the panel motion.

The dynamic of the panels is analyzed by extracting the panel displacement signal. For the viscoelastic panel, the displacement is recorded using the point laser displacement, located at the position $x = 27$ mm and close to the centerline ($z = 2$ mm), see Section 2 and orange rectangle in Fig. 5(a). For the elastic panel configuration, however, this measurement is unavailable due to the oscillating shock intercepting the laser path. Instead, the displacement signal is extracted along the centerline ($z=0$) at $x = 27$ mm (orange rectangle in Fig. 4(a)) from the reconstructed signal, which still allows for a direct comparison with the dynamics of the viscoelastic
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 5.** Interaction between the viscoelastic panel and the normal shock-wave. (a) displays an instantaneous schlieren image for the viscoelastic panel with the shock at $\bar{x}_s = -3.6$ mm. The orange dash line corresponds to the instantaneous reconstructed panel displacement at the center line $y=0$. The blue dot refers to the shock position and the orange rectangle to the panel position measured by the point sensor where the panel dynamic is analyzed (b) shows the reconstructed displacement of the panel: the dashed orange line reports the average displacement of the panel, while the solid black lines indicate plus and minus the standard deviation. The time-space evolution of the panel displacement at the center-line ($z=0$) is reported by the contour plot in (c); the dash-dotted line indicates the evolution of the shock position. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 6.** Power Spectrum Density (PSD) of the panel (a) and shock (b) motion, for the cases introduced in Fig. 4 (elastic panel) and Fig. 5 (viscoelastic panel). The data for the elastic and viscoelastic panels are reported by the solid and dashed lines, respectively. The vertical solid lines indicate the first three natural vibration frequencies of the elastic panel, see Table 1.</figcaption></figure>

panel. For both cases, the Power Spectrum Density (PSD) of the panel signal at this position is calculated. The Welch method with a Hanning window is used with 50% of overlap between blocks. The number of samples per block is set to 4096 samples per block. The resulting spectra are displayed in Fig. 6(a) where the orange solid line refers to the elastic panel and the dashed orange line to the viscoelastic one. On the elastic panel spectrum, a dominant peak $f_2$ at 153 Hz is observed as well as two other minor peaks $f_1$ and $f_3$, located at 119 Hz and 205 Hz. These three frequencies ($f_1$, $f_2$ and $f_3$) coincide with the computed natural frequencies of the elastic structure i.e. $f_1^n = 118$ Hz, $f_2^n = 152$ Hz, $f_3^n = 207$ Hz. (vertical black lines in Fig. 6(a), Table 1). When compared to the elastic panel, the viscoelastic panel displacement spectrum presents no dominant peak implying that the viscoelastic compliant panel does not present any low or medium frequency dynamics. Unfortunately, due to limitations of the acquisition system (electric noise and amplifier filter), frequencies higher than 600 Hz cannot be detected. Therefore, it is not possible to assess whether the flow excites the natural frequencies of the viscoelastic structure which are calculated at $f_1^n = 1936$ Hz, $f_2^n = 2551$ Hz, $f_3^n = 3542$ Hz (Table 1). In any case, since no viscoelastic panel motion is observed on schlieren images, if higher frequency motion would exist, their associated motion would be below 0.22 mm, i.e., the spatial resolution of the image.
<caption>**Table 2**</caption>

Investigated shock positions for the viscoelastic panel (Tango Polyjet). The average position of the shock ($\bar{x}_s$) and its standard deviation ($\sigma_{x_s}$) are reported in the first two rows. Maximal and minimal values of the reconstructed position of the panel along the streamwise direction ($\bar{y}_{w,min}$ and $\bar{y}_{w,max}$) are also reported, along the standard deviation of the point-wise displacement measured by the Keyence probe $\sigma_{y_u}|_{x=27 \text{ mm}}$. The values between rounded parentheses rely only on the numerical model, since the panel is not visible in the schlieren images. The last row indicates the figures in which further data are presented. The nominal case is highlighted in bold.

<table><thead><tr><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th><th></th></tr></thead><tbody><tr><td>$\bar{x}_s$ (mm)</td><td>-61.5</td><td>-50.8</td><td>-26.0</td><td><strong>-3.6</strong></td><td>+4.8</td><td>+29.1</td><td>+52.6</td><td>+76.1</td></tr><tr><td>$\sigma_{x_s}$ (mm)</td><td>0.52</td><td>0.61</td><td>0.68</td><td><strong>0.74</strong></td><td>0.74</td><td>0.86</td><td>0.69</td><td>0.49</td></tr><tr><td>$\bar{y}_{w,min}$ (mm)</td><td>(-2.81)</td><td>(-2.63)</td><td>-2.05</td><td><strong>-1.54</strong></td><td>-0.96</td><td>-0.26</td><td>+0.00</td><td>+0.00</td></tr><tr><td>$\bar{y}_{w,max}$ (mm)</td><td>(+0.00)</td><td>(+0.19)</td><td>+1.10</td><td><strong>+2.45</strong></td><td>+2.66</td><td>+3.24</td><td>+3.69</td><td>+3.93</td></tr><tr><td>$\sigma_{y_u}|_{x=27 \text{ mm}}$ (mm)</td><td>0.07</td><td>0.07</td><td>0.14</td><td><strong>0.02</strong></td><td>0.06</td><td>-</td><td>0.01</td><td>0.03</td></tr></tbody></table>

The dynamic of the shock over the compliant panels is now characterized by computing the PSD of the shock position at the middle of the test section's height (blue dot in Fig. 6(b)). Welch method with Hanning window is used with 50% of overlap between blocks with 1024 samples per block. This number of samples per block is the best compromise between frequency resolution and reduction of noise. The spectra of the shock displacement with the elastic and viscoelastic panels are presented in Fig. 6(b). Note that frequencies below 20 Hz and above 1 kHz are not displayed. In the former case, low frequencies peaks are observed due to fluctuations of the stagnation pressure in the settling chamber. In the latter case, the high-frequency peaks are of low-amplitude and related to the tracking method used to retrieve the shock location. The shock displacement spectrum for the elastic case (blue solid line in Fig. 6(b)) is dominated by a peak at $f_2 = 153$ Hz accompanied by its first harmonic $2f_2$. It should be noted that the $f_2$ is close to the $f_2^n = 152$ Hz i.e. the second natural frequencies of the elastic structure. Therefore, the shock locks-in to the second vibration frequency of the elastic panel. Regarding the first harmonic $2f_2$, it was found in Riveiro Moreno et al. (2024) that it is linked to an asymmetric motion of the shock wave, where the shock displays a plateau on its upstream excursion. The shock displacement spectrum for the viscoelastic case is plotted in a blue dashed line in Fig. 6(b). Since for the viscoelastic panel, the equivalent mode occurs at a much higher frequency ($f_2^n = 2551$ Hz), no lock-in is observed and no dominant frequency emerges from the shock displacement spectrum. The shock motion is low frequency and broadband in a comparable way to what can be observed over a rigid panel (Riveiro Moreno et al., 2024).

From this spectral analysis, we conclude that when using the elastic panel, both the shock and the elastic panel present large synchronized periodic oscillations at $f_2 = 153$ Hz, a frequency close to the second natural vibration frequency of the structure $f_2^n = 152$ Hz. The large amplitude associated with this motion suggests a lock-in of the shock oscillation with the second vibration mode of the structure. On the other hand, when using the viscoelastic panel, no strong dynamic exists between the shock wave and the structure. This is the result of the structure's viscoelastic behavior, and its dynamic stiffening. Being the natural frequencies of the viscoelastic panel higher than the oscillations of the shock wave, these cannot be excited by the shock wave. Moreover, it has been highlighted that both structures present static displacements of the same order of magnitude, being those of the viscoelastic panel larger. Such a result is consistent with the fact that both materials have a low static Young modulus, which is comparable in value as shown in Table 1. The aforementioned results imply that the viscoelastic panel is capable of decoupling its static deformation from its dynamic behavior. Therefore, by using the viscoelastic compliant panel, the dynamic fluid-structure interaction observed in the elastic panel configuration is suppressed.

## 4.2. Effect of the normal shock location on the fluid-structure interaction

In the previous subsection, we demonstrated that the viscoelastic compliant panel effectively suppresses the large dynamic oscillations induced by shock-panel interaction observed in the elastic panel configuration. While those results were based on a shock positioned at the center of the compliant panel, a parametric study of various shock-panel positions has been conducted to assess the robustness of this behavior. Note that the same investigation was also conducted on the elastic case, and the reader is referred to Riveiro Moreno et al. (2024) for further details. The results of the parametric study on the relative position between the shock and the viscoelastic structure (Tango Polyjet) are first discussed. The investigated shock positions are listed in Table 2. The time evolution of the shock position at the channel's mid-height ($y = 50$ mm) is extracted from the schlieren images. The first row shows the average shock position, identifying the shock/viscoelastic-panel configuration. The nominal case presented in Section 4.1 ($\bar{x}_s = -3.6$ mm) is highlighted in bold; lower values correspond to a shock located upstream of the nominal case, while higher values indicate a downstream location. The standard deviation of the shock position signal, $\sigma_{x_s}$, is also reported to assess the amplitude of the shock movement. No significant variation in magnitude is observed across the investigated positions, suggesting that the shock dynamics exhibit similar behavior to that of the nominal case. To complete the characterization of the fluid-structural system as a function of the shock position, the panel position is reconstructed, and the maximum and minimum vertical panel displacements are reported in the third and fourth rows, respectively. Due to the limited optical access of the camera, the entire compliant panel is not visible on the schlieren images for the two most upstream shock positions ($\bar{x}_s = -61.5$ mm and $\bar{x}_s = -50.8$ mm); in these cases, the reported values are based solely on the numerical model and are, therefore, presented in rounded parentheses in the table. The upward deformation of the panel increases as the shock moves downstream, while the downward deformation decreases, with the panel becoming visible at $\bar{x}_s = -26.0$ mm and fully deformed upward at $\bar{x}_s = 52.6$ mm.
<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 7.** Schlieren snapshots and PSD of the position of the shock ($x_s$) and the panel ($y_w$) for some selected cases from Table 2. The orange dashed line in the schlieren image denotes the reconstructed panel deformation. The blue dot and the orange square indicate respectively the position for which shock- and panel- displacement data are measured. The black dashed line in the PSD plots report the behavior of the nominal configuration ($\bar{x}_s = -3.6$ mm, see Fig. 6). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>**Fig. 8.** Maximum amplitude of the PSD of the vertical motion of the interface, as a function of the average shock location $\bar{x}_s$. The results for the elastic panel are reported in (a) and the ones for the viscoelastic panel in (b). The black-edged markers indicate the two nominal location of the shock wave presented in Section 4.1, while the filled ones indicate the cases reported in Fig. 7. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

This behavior is clearly illustrated in the instantaneous schlieren images shown in Fig. 7, where the reconstructed panel displacement is indicated by the orange dashed line for various shock positions. The panel motion is quantified using the standard deviation of the signal from the Keyence sensor, $\sigma_{y_w}$, positioned at $x = 27$ mm and $z = 2$ mm. Across all tested shock positions, panel oscillations remain minimal and below the spatial resolution of the schlieren images. Due to the shock intercepting the laser path, this measurement is unavailable for $\bar{x}_s = +29.1$ mm; however, no oscillations are observed in the schlieren data. To further confirm the absence of coupled dynamics in the tested cases, the PSD of the shock and panel displacement is computed. The solid lines in the plots on the right in Fig. 7 display the results for the cases indicated in Table 2, while the dashed lines the ones for the nominal case ($\bar{x}_s = -3.6$ mm, see Fig. 6). The spectra for both the shock wave and viscoelastic panel resemble those obtained within the nominal case. All spectra indeed exhibit similar behavior, with no common peak between the solid and the fluid data.

The effect of the shock location on the fluid-structure interaction is now compared between the two tested configurations i.e the shock-elastic panel and shock-viscoelastic panel interaction. To compare the two cases, Fig. 8 displays the maximum amplitude of the panel displacement spectrum as a function of the shock wave location. In Fig. 8(a), the maximum amplitude of the elastic panel displacement spectrum is presented. Here it can be seen that the shock position has an impact on the elastic panel dynamics, displaying maximum oscillations when the shock is at $\bar{x}_s \approx 0$ mm. In Riveiro Moreno et al. (2024) it was observed that the shock location plays a crucial role on the elastic panel dynamics. Depending on its position, the shock wave coupled with the first, second, or third vibration frequency of the structure. Fig. 8(b) displays the maximum amplitude of the viscoelastic panel displacement spectrum as a function of the shock wave location. In that case, for all the shock positions, we observe the maximum amplitude of the shock oscillations spectrum remains constant and lower than for the elastic case.

Given the results above, it can be concluded that the viscoelastic panel prevents any dynamic interaction regardless of the shock location. Therefore, it is validated that the viscoelastic panel maintains its capability of decoupling its dynamic behavior from its static one independently of the shock-wave position. This property is of interest when designing adaptive shock control bumps, used for wave drag reduction and delay of transonic buffet onset, as explained in Section 1. The adaptive control bump structure needs to be flexible enough to ensure the optimal bump shape, especially its crest location with respect to the shock position. In the present configuration, for the shock-wave position close to the center of the panel as for $x_s = -3.6$ mm (Fig. 5(a)) and for $x_s = 4.8$ mm (Fig. 7(c)), the triple point is located the furthest from the wall, meaning the spreading of the compression is the largest. Flow is decelerated through isentropic compression waves and a rear shock leg, corresponding to the $\lambda$ structure observed for shock control bumps (Bruce and Colliss, 2015), which leads to a weaker wave drag compared to the rigid case. However, it seems that a boundary layer separation appears in the shock foot, caused by the curvature of the panel, generating potential viscous drag penalties. Nevertheless, a viscoelastic adaptive shock control bumps would still provide the required flexibility, i.e. low static Young modulus, and shift the natural frequencies of the structure out of the range of the shock oscillations. In that manner, strong dynamic oscillations that can destroy the panel after a few seconds are prevented.

## 5. Conclusion

The influence of structural viscosity on the fluid-structure dynamic interaction between a normal shock wave (upstream Mach number of 1.35) and a compliant panel is investigated (Section 2). To emphasize the role of the panel material's viscoelastic properties in coupling with shock dynamics, two compliant panels – with the same geometry but made from distinct materials – are specifically designed and tested experimentally in a transonic wind-tunnel. As an additional design constraint, the two panels are also required to exhibit low rigidity to allow for large static deformations, as needed for adaptive shock-control techniques detailed in Section 1.
The design principle of the two panels is based on the dynamic response of the materials from which they are fabricated (Section 3). This led to the identification of two materials, both elastomers, with similar static flexibility but characterized by a very different dynamic behavior (Section 3.1). The first material, polyurethane 40 A, remains in its rubbery – i.e., elastic – state within the frequency range of the natural shock oscillations (Riveiro Moreno et al., 2024), where it exhibits nearly elastic behavior with minimal energy dissipation from the viscous component. In contrast, the second material, Tango Polyjet 61 A, operates close to its glassy – i.e., viscoelastic – state within the same frequency range, resulting in increased (dynamic) rigidity and high energy dissipation. The identification of these differing dynamic behaviors is enabled by experimental characterization of the frequency-dependent mechanical properties of the two materials using dynamic mechanical analysis (DMA, Section 3.2). It is important to note that understanding the frequency-dependent behavior of the employed material is crucial for this type of application. Although this data is not commonly available from manufacturers and ad-hoc experimental investigations are required, characterization of the most common viscoelastic materials is beginning to appear in the literature (e.g., Abayazid and Ghajari (2020), Kotrocz and Bakonyi (2023)). By upscaling the experimental results of the DMA from the material level to the structural level, we can also numerically predict the natural dynamic behavior of each panel, specifically its natural vibration modes and frequencies (Section 3.3). The natural frequencies of the elastic panel lie within the frequency range of the natural shock oscillations, potentially leading to strong dynamic interaction. Meanwhile, the dynamic stiffening of Polyjet Tango shifts the natural frequencies of the viscoelastic panel to higher values, thereby preventing the possibility of a dynamic coupling with the shock-wave.

These predictions are confirmed by the wind-tunnel experimental results (Section 4). They show a strong dynamic coupling between the elastic panel and the shock-wave, while no dynamic fluid-structure interaction is observed for the viscoelastic structure (Section 4.1). Only static deformations of the viscoelastic panel are found, which are larger than those exhibited by the elastic compliant panel. This behavior is consistent regardless of the shock location, demonstrating the robustness of the design (Section 4.2). We explain this as a consequence of the shift of the natural vibration frequencies of the viscoelastic panel, as quantified by eigenvalue analysis of the isolated panel within the design process. Considering the potential application of viscoelastic panels for shock control bumps, it is interesting to understand whether the present viscoelastic panel in the setup could be a potential candidate.

Although the present results might not have a net positive impact on the total drag – i.e., reduction of wave drag but increase of viscous drag –, a positive effect on the total drag could be attained by modifying the viscoelastic compliant wall deformation, by changing the pressure in the cavity (Gramola et al., 2020), or using actuators (Jinks et al., 2016; Gramola et al., 2018). Consequently, it could be of interest to modify the present setup along these directions to better evaluate this possibility. For future test campaigns, it is recommended to quantify the wave drag benefit from wall pressure measurements – e.g. by sensors or pressure-sensitive painting – and laser doppler velocimetry (LDV) probing. Indeed, to have an estimation of the benefit of the wave drag thanks to the viscoelastic panel, the total drag force could be determined by calculating the momentum balance for a control volume with its entry section located upstream of the interaction region and its exit section downstream of it (Bur et al., 2002).

The main message of this work remains that viscoelasticity can be exploited to shift the natural vibration frequencies of a highly
compliant wall in order to avoid the dynamic coupling with the shock wave. This principle could also be applied to different
configurations and shapes of the plate other than the investigated one, e.g., clamped at the leading and trailing edge (D'Aguanno
et al., 2023), and only at the leading edge [cantilever plate in Currao et al. (2016)]. More generally, these results open to new
applications for viscoelastic materials in flow control: we highlight that the viscoelastic behavior of the material not only provides
structural damping, but also frequency-dependent stiffening. The latter can be thus leveraged to effectively decouple the static and
dynamic behavior of the panel, i.e., providing the large static deformation of a soft material, while presenting the high natural
frequencies usually related to a rigid one. If this effect can be used to avoid undesired coupling with the flow as in the present
study, it can more broadly be exploited as an additional degree of freedom to pilot the coupled interaction between a fluid and a
flexible surface.

CRediT authorship contribution statement

C. **Riveiro Moreno**: Writing – review & editing, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. **M. Couliou**: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **N. Fabbiane**: Writing – review & editing, Visualization, Supervision, Software, Methodology, Funding acquisition, Formal analysis, Conceptualization. **R. Bur**: Writing – review & editing, Supervision, Project administration, Funding acquisition. **O. Marquet**: Writing – review & editing, Supervision, Software, Resources, Project administration, Methodology, Funding acquisition, Formal analysis, Conceptualization.

**Declaration of competing interest**

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Carmen Riveiro Moreno reports financial support was provided by General Directorate of Armaments. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments

This work is co-funded by the Direction Générale de l'Armement, France (AID Convention N° 2020 65 0022) and by an internal research project FLUTE, funded by the Scientific Direction of ONERA. The authors would like to express their gratitude to Anne Marvel (ONERA, DMAS), Florence Saffar (ONERA, DMAS), and Julie Diani (Laboratoire de Mécanique des Solides (LMS), École Polytechnique) for their assistance in the characterization of the materials. Special thanks are extended to the ONERA DAAA measurement team: Cédric Illoul, Pascal Audo, Jean-Pierre Tobeli, and Philippe Geffroy. The authors also thank Thibault Leudière (ONERA, DAAA) for his technical support in the preparation of the materials and the execution of the experimental tests.

Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.jfluidstructs.2025.104335.

Data availability

Data will be made available on request.

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