2.1. Conventional and phononic compliant surfaces

To attenuate the amplification of these TSWs in the boundary-layer flow, a compliant structure of fixed length ( $l = 47.1\,{\rm mm}$ ) and depth ( $h = 2.35\,{\rm mm}$ ) is flush-mounted at the wall starting at a distance $x=11.8\,{\rm mm}$ from the inlet of the computation domain, i.e. the origin of our coordinate system. Length, depth and location of the compliant walls are chosen to match to those employed by Pfister et al. (Reference Pfister, Fabbiane and Marquet2022) and they correspond to $100\delta ^*_0$ , $5\delta ^*_0$ and $25\delta ^*_0$ , respectively. Two different structures – schematically illustrated in figure 1 – are then considered to delay the laminar-to-turbulent transition: a conventional compliant surface (figure 1 a) obtained by considering a structure made of a homogeneous soft material and a phononic compliant surface (figure 1 b) obtained by periodically placed, stiffer inclusions in the soft material.

The conventional compliant surface, illustrated in figure 1(a), is obtained here by using a soft elastomer (Ashby Reference Ashby1999). More specifically, the density of this soft elastomer is chosen as $\rho _{s}=1000\,\mathrm{kg\, m^{-3}}$ , i.e. equal to the density of water $\rho _{\!f}$ . Its elastic and viscous properties are defined through the Young’s modulus $E_s = 0.1\,\, {\rm MPa}$ and the material viscosity $D_s = 0.94\,{\rm Pa\,s}$ , respectively. Additionally, elastomers are usually modelled as incompressible materials and, thus, a Poisson coefficient $\nu _s = 0.5$ is considered. This specific choice corresponds to that proposed by Pfister et al. (Reference Pfister, Fabbiane and Marquet2022). In particular, their work showed numerically that the flexibility of the compliant surface allows one to attenuate the amplification of TSWs occurring at low frequency of around $f\sim 200\,{\rm Hz}$ , with a streamwise wavelength $\lambda _{TSW}\approx 30\delta ^*_0 = 14.1\,{\rm mm}$ ; this resulted in a ratio between the length of the compliant wall ( $l$ ) and the TSW wavelength ( $\lambda _{TSW}$ ) of about $3$ . Meanwhile, the viscosity of the compliant surface prevents the destabilisation of natural vibration modes of the structure by the flow at higher frequencies $f \sim 1300\,\mbox{Hz}$ . Still, the structural viscosity is not large enough to avoid the amplification of the so-called TWF instabilities by an external forcing in the flow. These TWF instabilities propagate in the streamwise direction in the vicinity of the fluid–solid interface and are characterised by a streamwise wavelength $\lambda _{x} \sim 4.7\,{\rm mm}$ , for the current choice of dimensional parameters.

The phononic compliant surface, illustrated in figure 1(b), is here introduced to specifically attenuate the amplification of TWF instabilities. The term phononic refers to metamaterials or structures that possess an inherent periodicity in their internal geometry (Hussein et al. Reference Hussein, Leamy and Ruzzene2014). This periodicity, when properly tuned, imparts unique dynamical properties, such as blocking the propagation of waves in specific directions and frequency ranges. In the case of TWF instabilities, we exploit the wave-blocking properties of phononic materials to prevent the streamwise propagation of solid waves in the phononic compliant surface. Thus, we aim at breaking the unstable interaction between natural vibration modes of the surface and flow perturbations. To that aim, we add circular inclusions (red circles) made of a stiffer material ( $E_{I}=1\,\, {\rm MPa}$ ) into a homogeneous matrix (orange) made from the same material ( $E_{s}=0.1\,\, {\rm MPa}$ ) as that used for the conventional homogeneous compliant surface. The two materials (matrix and inclusions) share the same density ( $\rho _{I}=1000\,\mathrm{kg\, m^{-3}}$ ); note that the stiffer material corresponds to common elastomers such as natural rubber or polydimethylsiloxane. These circular inclusions have a diameter $d_I= 0.94\,{\rm mm}$ and they are periodically placed in the streamwise direction with a constant spacing of $l_I = 2.35\,{\rm mm}$ , so as to achieve the band-gap properties of the phononic surface for TWF of wavelength $\lambda _{x} = 4.7\,{\rm mm}$ . The details of the design process that led to the considered spacing of the circular inclusions are addressed in § 3. The phononic compliant surface is finally made of $20$ inclusions that are periodically arranged in the same direction as the flow. This is different from the phononic surface first proposed by Hussein et al. (Reference Hussein, Biringen, Bilal and Kucala2015) and later by Michelis et al. (Reference Michelis, Putranto and Kotsonis2023) who introduced a periodic arrangement of material in the wall-normal direction to attenuate TSWs. With our phononic compliant surface, the natural compliance of the matrix materials should still be exploited to attenuate those waves, as in the case of the conventional compliant surface.

2.2. Linear fluid–structure perturbations: global stability and resolvent analyses

The interaction of these compliant surfaces with flow perturbations developing in the boundary layer are investigated numerically based on the framework detailed in Appendix A. It was first developed by Pfister et al. (Reference Pfister, Marquet and Carini2019) for the study of vortex-induced vibrations of a thin plate and later applied to the control of boundary-layer flow instabilities with a conventional homogeneous compliant surface (Pfister et al. Reference Pfister, Fabbiane and Marquet2022). Similar theoretical and numerical tools have also been used in the literature for the study of the dynamics of gas bubbles in straining flows (Sierra-Ausin et al. Reference Sierra-Ausin, Bonnefis, Tirri, Fabre and Magnaudet2022). The equations governing the interaction between an incompressible flow and a hyperelastic structure are written in the arbitrary Lagrangian–Eulerian (ALE) framework. To investigate the linear dynamics of flow perturbations interacting with solid perturbations, these equations are linearised around steady solutions of the coupled fluid–structural equations. In the present case, the static deformation $\boldsymbol{\varXi }_{s}(\boldsymbol{x})$ of both compliant surfaces induced by the zero-pressure-gradient boundary-layer flow remains small with respect to the boundary-layer thickness ( $|\boldsymbol{\varXi }_{s}| \lt 10^{-3}\,\delta ^*_0$ ), thus confirming the assumption made in Pfister et al. (Reference Pfister, Fabbiane and Marquet2022) of an non-deformed compliant wall. The linear dynamics of coupled fluid–structure perturbations is then investigated based on (i) global stability analysis to determine the onset of self-sustained fluid–structure oscillations and (ii) resolvent analysis to investigate the amplification of flow perturbations sustained by an external forcing, in the case of globally stable steady solution.

The equations governing infinitesimal flow perturbations $\boldsymbol{{q}}_{\!f}^\prime$ and solid perturbations $\boldsymbol{{q}}_{s}^\prime$ of the steady flow state $\boldsymbol{{Q}}_{\!f}$ may be written as the coupled linear equations

(2.1) \begin{align} \left [\begin{array}{cc} \boldsymbol{{B}}_{\!f} & 0\\[4pt] 0 & \boldsymbol{{B}}_s \end{array}\right ] \frac {\partial }{\partial t} \left [\begin{array}{c} \boldsymbol{{q}}_{\!f}^\prime \\[4pt] \boldsymbol{{q}}_s^\prime \end{array}\right ] - \left [\begin{array}{cc} \boldsymbol{{A}}_{\!f}(\boldsymbol{{Q}}_{\!f}) & \boldsymbol{{C}}_{\!\textit{fs}}\\[4pt] \boldsymbol{{C}}_{\!\textit{sf}} & \boldsymbol{{A}}_s \end{array}\right ] &\left [\begin{array}{c} \boldsymbol{{q}}_{\!f}^\prime \\[4pt] \boldsymbol{{q}}_s^\prime \end{array}\right ] = \left [\begin{array}{c} \boldsymbol{{P}}_{\!f}\\[4pt] 0 \end{array}\right ] \boldsymbol{f}^\prime , \end{align}

where the first line corresponds to the linearised ALE incompressible Navier–Stokes equations while the second line corresponds to the linear viscoelastic equations (Kelvin–Voigt model). The exact definitions of the operators are detailed in Appendix A. The right-hand side forcing term $\boldsymbol{f}^\prime$ corresponds to an external forcing in the momentum flow equations that is introduced for the resolvent analysis but is dropped in the global stability analysis.

In the global stability analysis, the long-term temporal amplification or decay of linear fluid–structural perturbations is investigated by assuming an exponential temporal growth of perturbations as

(2.2) \begin{equation} \left [\begin{array}{c} \boldsymbol{{q}}_{\!f}^\prime (\boldsymbol{x},t)\\[4pt] \boldsymbol{{q}}_s^\prime (\boldsymbol{x},t) \end{array}\right ] = \left [\begin{array}{c} \boldsymbol{{\tilde {q}}}_{\!f}(\boldsymbol{x})\\[4pt] \boldsymbol{{\tilde {q}}}_s(\boldsymbol{x}) \end{array}\right ] \, {\rm e}^{(\gamma + \mathrm{i} \omega )\,t} + {\underbrace {\left [\begin{array}{c} \boldsymbol{{\tilde {q}}}^*_{\!f}(\boldsymbol{x})\\[4pt] \boldsymbol{{\tilde {q}}}^*_s(\boldsymbol{x}) \end{array}\right ] \, {\rm e}^{(\gamma - \mathrm{i} \omega )\,t}}_{\mbox{complex conjugate (c.c.)}}}, \end{equation}

where $\omega$ ( $\gamma$ ) is the frequency (growth rate) associated with the complex spatial structures denoted $\boldsymbol{{\tilde {q}}}_{\!f}$ in the flow and $\boldsymbol{{\tilde {q}}}_s$ in the solid. In the following, terms like the under-braced expression will be abbreviated as ‘c.c.’, indicating that they represent the complex conjugate of their preceding term. Injecting the above decomposition into (2.1) yields the following coupled fluid–structural eigenvalue problem:

(2.3) \begin{align} (\gamma + \mathrm{i} \omega ) \left [\begin{array}{cc} \boldsymbol{{B}}_{\!f} & 0\\[4pt] 0 & \boldsymbol{{B}}_s \end{array}\right ] \left [\begin{array}{c} \boldsymbol{{\tilde {q}}}_{\!f}\\[4pt] \boldsymbol{{\tilde {q}}}_s \end{array}\right ] - \left [\begin{array}{cc} \boldsymbol{{A}}_{\!f}(\boldsymbol{{Q}}_{\!f}) & \boldsymbol{{C}}_{\!\textit{fs}}\\[4pt] \boldsymbol{{C}}_{\!\textit{sf}} & \boldsymbol{{A}}_s \end{array}\right ] &\left [\begin{array}{c} \boldsymbol{{\tilde {q}}}_{\!f}\\[4pt] \boldsymbol{{\tilde {q}}}_s \end{array}\right ] = \left [\begin{array}{c} 0\\[4pt] 0 \end{array}\right ] . \end{align}

In the resolvent analysis, we investigate the linear response of the fluid and solid perturbations to external forcing, that is here restricted to the momentum flow equations. The forcing and response may be decomposed into temporal harmonics as

(2.4) \begin{equation} \boldsymbol{f}^\prime (\boldsymbol{x},t)= \boldsymbol{\hat {f}}(\boldsymbol{x})\,{\rm e}^{ \mathrm{i} \omega t} + \mbox{c.c.} \quad \mbox{and}\quad \left [\begin{array}{c} \boldsymbol{{q}}_{\!f}^\prime (\boldsymbol{x},t)\\[4pt] \boldsymbol{{q}}_s^\prime (\boldsymbol{x},t) \end{array}\right ] = \left [\begin{array}{c} \boldsymbol{{\hat {q}}}_{\!f}(\boldsymbol{x})\\[4pt] \boldsymbol{{\hat {q}}}_s(\boldsymbol{x}) \end{array}\right ] \, {\rm e}^{ \mathrm{i} \omega t} + \mbox{c.c.}, \end{equation}

where the forcing frequency, also denoted $\omega$ , is now a parameter, while $\boldsymbol{\hat {f}}$ , $\boldsymbol{{\hat {q}}}_{\!f}$ and $\boldsymbol{{\hat {q}}}_s$ are spatial structures. The latter are related by the following input–output relation that is obtained after injecting the above harmonic decomposition into (2.1):

(2.5) \begin{equation} \left [\begin{array}{c} \boldsymbol{{\hat {q}}}_{\!f}\\[4pt] \boldsymbol{{\hat {q}}}_s \end{array}\right ] = \boldsymbol{{R}}_{\!\textit{fs}}(\omega )\,\boldsymbol{{P}}\! \boldsymbol{{\hat {f}}}, \quad \mbox{where}\quad \boldsymbol{{R}}_{\!\textit{fs}}(\omega ) =\left ( \mathrm{i} \omega \left [\begin{array}{cc} \boldsymbol{{B}}_{\!f} & 0\\[4pt] 0 & \boldsymbol{{B}}_s \end{array}\right ] - \left [\begin{array}{cc} \boldsymbol{{A}}_{\!f}(\boldsymbol{{Q}}_{\!f}) & \boldsymbol{{C}}_{\!\textit{fs}}\\[4pt] \boldsymbol{{C}}_{\!\textit{sf}} & \boldsymbol{{A}}_s \end{array}\right ] \; \right )^{-1} \end{equation}

is the aeroelastic resolvent operator and $\boldsymbol{{P}}=[\boldsymbol{{P}}_{\!f},0]^T$ is a prolongation operator allowing one to impose the forcing solely on the fluid momentum equations. The resolvent analysis (Sipp & Marquet Reference Sipp and Marquet2013) consists here of identifying, for each forcing frequency $\omega = 2\pi f$ , the spatial structure $\boldsymbol{\hat {f}}$ of the forcing that provides the largest energy amplification $\sigma ^2$ of the flow velocity perturbation $\boldsymbol{\hat {u}}_{\!f}$ , a component of $\boldsymbol{{\hat {q}}}_{\!f}$ . The energy gain is defined as

(2.6) \begin{align} \sigma ^2(\omega ) = \max _{\boldsymbol{\hat {f}}} \left ( \frac {\|\boldsymbol{\hat {u}}_{\!f}\|^2}{\|\boldsymbol{\hat {f}}\|^2} \right ) \quad {\rm with}\quad \|\boldsymbol{\hat {u}}_{\!f}\|^2 &= \int _{\varOmega _{\!f}} \boldsymbol{\hat {u}}_{\!f}^* \boldsymbol{\cdot }\boldsymbol{\hat {u}}_{\!f}\,{\rm d}\varOmega = \boldsymbol{{\hat {q}}}_{\!f}^*\,\boldsymbol{{M}}_{\boldsymbol{u}}\,\boldsymbol{{\hat {q}}}_{\!f},\notag\\ \|\boldsymbol{\hat {f}}\|^2 &= \int _{\hat {\varOmega }_{\!f}} \boldsymbol{\hat {f}}^* \boldsymbol{\cdot }\boldsymbol{\hat {f}} \,{\rm d}\varOmega = \boldsymbol{{\hat {f}}}^*\,\boldsymbol{{M}}_{\!\boldsymbol{f}}\,\boldsymbol{{\hat {f}}}. \end{align}

Here $\|\boldsymbol{\hat {f}}\|^2$ is the norm of the forcing field and $\|\boldsymbol{\hat {u}}_{\!f}\|^2$ is the norm of the harmonic fluid velocity perturbation $\boldsymbol{{\hat {u}}}_{\!f}$ . In order to limit the influence of the outflow boundary condition on the amplification factor $\sigma$ , the fluid domain $\hat {\varOmega }_{\!f}$ utilised for computing the two norms is truncated at $x = 70.6\,{\rm mm}$ . As shown by Brandt et al. (Reference Brandt, Sipp, Pralits and Marquet2011), the optimal flow forcing and response can be determined by computing the largest eigenvalue $\sigma ^2$ of the Hermitian eigenvalue problem

(2.7) \begin{equation} \boldsymbol{{P}}^{T}\,\boldsymbol{{R}}_{\!\textit{fs}}^{H}(\omega )\,\boldsymbol{{M}}_{\boldsymbol{u}}\,\boldsymbol{{R}}_{\!\textit{fs}}(\omega )\,\boldsymbol{{P}}\!\boldsymbol{{\hat {f}}} = \sigma ^2\,\boldsymbol{{M}}_{\!\boldsymbol{f}}\boldsymbol{{\hat {f}}}. \end{equation}

In the following, we solve numerically the here-presented problems using a first-order (pressure) and second-order (fluid velocity and solid displacement) continuous-Galerkin finite-element method through the DOLFINx library (Baratta et al. Reference Baratta, Dean, Dokken, Habera, Hale, Richardson, Rognes, Scroggs, Sime and Wells2023). The resulting eigenvalue problems are solved with SLEPc’s Krylov–Schur algorithm (Roman et al. Reference Roman, Alvarruiz, Campos, Dalcin, Jolivet and Lamas Daviña2023).

2.3. Results

Let us first examine the effect of the conventional compliant surface, made with a homogeneous viscoelastic material as illustrated in figure 1(a), on the amplification of perturbations in the boundary-layer flow. Figure 2(a) displays the optimal amplifications $\sigma$ , as a function of the frequency $f=\omega /2\pi$ , of forced harmonic flow perturbations developing in the boundary layer over the rigid surface (blue dotted curve) and the conventional homogeneous compliant surface (black dashed curve). The amplification of TSWs corresponds to the peak observed at low frequency $f \approx 210\,{\rm Hz}$ . Maximal for the rigid surface, the magnitude of this peak is clearly reduced when using the conventional compliant surface. This result, introduced in Pfister et al. (Reference Pfister, Fabbiane and Marquet2022), shows the potential of compliant surfaces for delaying the laminar-to-turbulent transition of boundary layers induced by TSWs.

Figure 2. Global, coupled, linear stability analyses of the conventional and phononic compliant surfaces in figure 1 interacting with a laminar boundary-layer flow. The optimal, harmonic, forced response (a) and eigenvalue analysis (b) are reported for the conventional compliant surface (black dashed line and crosses) and the phononic one (red solid line and white-filled, red circles). The blue dotted line in (a) indicates the response over a rigid wall. The spatial structure of the response of the fluid–structural system at the TWF amplification peak ( $f \approx 1350\,{\rm Hz}$ ) is reported for both the conventional (c) and phononic (d) compliant surfaces. The colour indicates the real part of the vertical velocity in the fluid (blue) and solid (orange) phases, while the black vectors the real part of the surface displacement at the interface. The red shaded area in (a,b) indicates the identified band gap of the phononic surface ( $1241\,{\rm Hz} \le f \le 1439\,{\rm Hz}$ ), encompassing the two truncation modes (see figure 3).

However, for the conventional compliant wall, that effect may be undermined by the appearance of a second peak around the higher frequency $f=1350\,{\rm Hz}$ . This amplification of flow perturbation at higher frequency is associated with a flow-induced surface instability first identified by Benjamin (Reference Benjamin1960). Later termed the TWF instability (Carpenter & Garrad Reference Carpenter and Garrad1985; Tsigklifis & Lucey Reference Tsigklifis and Lucey2017), it involves perturbations in both fluid and solid phases that manifest as travelling waves localised near the fluid–solid interface. As shown in figure 2(c), these waves, characterised by a (dimensional) wavelength $l_{TWF} \approx 4.5\,{\rm mm}$ , propagate along the boundary layer while undergoing amplification. The observed energy amplification of the flow response to external flow forcing at high frequencies can be interpreted as a modal resonance. The eigenvalues of the coupled fluid–structural problem, that describe the self-sustained dynamics (i.e. without external forcing) of the boundary-layer flow perturbations interacting with the conventional compliant surface, are shown in figure 2(b) (black crosses). The eigenvalue branch near the frequency $f=1350\,{\rm Hz}$ corresponds to structural vibration modes coupled with the fluid. As shown by Pfister et al. (Reference Pfister, Fabbiane and Marquet2022), this branch becomes unstable (with positive growth rates) when the compliant surface is made of a purely elastic material. In contrast, when a viscoelastic material is used, as in the present configuration, the same branch remains stable (with negative growth rates). This means that, without an external forcing, small perturbations of the coupled system will eventually decay in time because of the effect of the material viscosity. Although structural damping due to material viscosity stabilises these eigenmodes, a resonance still arises in the forced response of the flow. This is reflected in the pronounced peak centred around $f=1350\,{\rm Hz}$ in the dashed curve of figure 2(a). Designing compliant surfaces with more viscous materials may further stabilise this eigenvalue branch and thereby reduce the amplification of externally forced flow perturbations. However, an increase of the structural damping would lead to a destabilisation of the TSW (Yeo & Dowling Reference Yeo and Dowling1987). Therefore, we propose to leverage the dynamical properties of phononic surfaces to mitigate the amplification of the TWF instability at high frequency, while preserving the favourable influence of compliant surfaces on low-frequency TSW instability.

Figure 3. Truncation eigenmodes from the coupled fluid–structural interaction of the phononic compliant surface with the boundary-layer flow (see figure 2 b). The colour reports the real part of the vertical velocity in the fluid (blue) and solid (orange) phases; the black vectors indicate the real part of the surface displacement at the interface. The reported modes occur at $f = 1343\,{\rm Hz}$ (a) and $f = 1347\,{\rm Hz}$ (b).

The open red circles in figure 2(b) display the fluid–solid eigenvalue spectrum obtained with the phononic compliant surface. The bump in the growth rate previously observed along the branch of fluid–solid modes for the conventional compliant surface (black crosses in figure 2 b) is no longer present. Instead, a gap appears in the eigenvalue branch between $1241$ and $1439\,{\rm Hz}$ , as highlighted by the red shaded region (see § 3 for definition and further details). Within this frequency range, only two isolated eigenmodes are found at $f = 1343\,{\rm Hz}$ and $f = 1347\,{\rm Hz}$ , both having near-identical eigenvalues. Their spatial structures, shown in figure 3, reveal high amplitudes localised at the upstream and downstream edges of the compliant surface. The modes result from truncation resonance (Rosa et al. Reference Rosa, Davis, Liu, Ruzzene and Hussein2023), a phenomenon inherent to finite-sized phononic materials, which is further discussed in a later section (see § 3.3). The band gap in the eigenvalue spectrum is centred (by design, see § 3) precisely around the frequency at which TWF instabilities are strongly amplified in the case of the conventional compliant surface. This effect arises from a key property of phononic materials: their ability to inhibit the propagation of waves with specific wavelengths, thereby suppressing the emergence of natural vibration modes at targeted frequencies.

We now examine the impact of this band gap on the amplification of flow perturbations over the phononic compliant surface when excited by an external harmonic forcing. The red solid curve in figure 2(a) presents the optimal amplification $\sigma$ in the presence of the phononic compliant surface. The prominent peak observed near $f=1350\,{\rm Hz}$ in figure 2(a) has entirely vanished, while the attenuation of Tollmien–Schlichting instabilities is preserved at lower frequency ( $f \approx 210\,{\rm Hz}$ ). Figure 2(d) shows the spatial structure of the fluid and solid perturbations in response to optimal external flow forcing at that frequency. A spatially growing surface wave characterised by a distinct wavelength of approximately $l_{TWF} \approx 4.5\,{\rm mm}$ is still observed but with a magnitude strongly reduced compared with the case of the conventional compliant surface (figure 2 c). One should note that the observed wavelength is approximately twice the spacing $l_I$ between the periodic inserts that constitute the phononic surface (grey lines in figure 2 d). This correspondence can be attributed to the relationship in phononic media between the frequency intervals of band-gap appearances and the wavelength of the blocked solid wave, which in turn is linked to its intrinsic periodicity (Hussein et al. Reference Hussein, Leamy and Ruzzene2014). These results demonstrate that the proposed phononic compliant surface, specifically designed to introduce a band gap in the eigenvalue spectrum at the frequency of TWF instability, effectively suppresses its amplification. Crucially, this is achieved without compromising the compliant nature of the surface, and therefore retain its beneficial influence on the control of low-frequency TSWs.

Figure 4. Spatial attenuation of TSWs. (a) The perturbation fields over the phononic surface for the optimal, harmonic, forced response at the amplification peak of TSWs, $f = 210\,{\rm Hz}$ in figure 2. The colour shows the streamwise component of the perturbation velocity in the fluid ( $u^\prime _{f,x}$ , in blue) and the vertical component of the solid velocity ( $\dot {\xi }^\prime _{s,y}$ , in orange). The magnitude of the streamwise velocity perturbation field at (b) $x = 35\,{\rm mm}$ and (c) $x=66\,{\rm mm}$ . The dotted, dashed and solid curves indicate, respectively, the rigid, conventional compliant and phononic compliant surfaces. (d) The integrated amplitude of the velocity perturbation in the fluid phase; the grey area indicates the streamwise extent of both the conventional and the phononic compliant walls.

It is then of interest to evaluate the performance of the phononic compliant surface not only on the attenuation of TWF, but also of TSWs. Figure 4(a) reports the response of the coupled system over the phononic surface for the optimal response at the TSW amplification peak ( $f = 210\,{\rm Hz}$ , as identified in figure 2 a); the blue colours indicate the streamwise component of the velocity perturbation in the fluid, while the orange the vertical component in the solid. It is observed that the TSW interacts with the compliant wall over more than three wavelengths and the solid responds to the perturbation in a similar manner. According to the analyses by Pfister et al. (Reference Pfister, Marquet and Carini2019), the response of the compliant wall induces a counter-phase TSW, which results in an attenuation of the perturbation in the fluid. This is reflected in the attenuation of the amplification factor $\sigma$ (reported in figure 2 a), which indicates that the integrated magnitude of the velocity perturbation field in the fluid domain is reduced and, therefore, a global attenuation of TSWs is observed. Nevertheless, this quantity does not provide any information on spatial amplification. To better characterise this aspect, the streamwise component of the velocity perturbation in the fluid is extracted, for the responses at $f = 210\,{\rm Hz}$ , at two different streamwise locations: in the middle of the compliant walls at $x = 35\,{\rm mm}$ (figure 4 b) and downstream of them at $x = 66\,{\rm mm}$ (figure 4 c). In accordance with Pfister et al. (Reference Pfister, Fabbiane and Marquet2022), at both locations the TSW presents with a reduced amplitude with respect to the rigid-wall case (blue dotted line) when a compliant wall is considered (black dashed line, conventional; dashed red line, phononic). Notably, the TSW over the compliant walls presents with its typical modal shape, as over the rigid wall. To complete the analysis, figure 4(d) illustrates the streamwise development of the velocity perturbation energy, integrated along the wall-normal direction. For context, the blue dotted line represents the spatial growth of TSW over a rigid wall, which, in accordance with conventional stability theory, exhibits exponential-like behaviour along the streamwise direction (Schmid & Henningson Reference Schmid and Henningson2001). In contrast, the dashed and solid lines denote the responses for the conventional compliant wall and the phononic surface, respectively. Notably, both compliant surfaces demonstrate comparable attenuation of TSW growth within the compliant wall’s extent (grey region), relative to the rigid-wall scenario. This reduced amplitude persists downstream of the patch location, indicating that TSWs interacting with either the conventional or the phononic compliant surface will exhibit delayed breakdown to turbulence, compared with those interacting with a rigid wall. We note that a localised increase in amplitude is present at the trailing edge of the compliant surfaces, specifically at $x \approx 59\,{\rm mm}$ . This phenomenon bears resemblance to observations in studies focused on controlling TSWs through localised flow interaction with a phononic subsurface (Michelis et al. Reference Michelis, Putranto and Kotsonis2023). In those contexts, such localised amplification limits the effective implementation of the proposed flow control principle. Recent developments, exploring combined streamwise and spanwise patterns of localised phononic subsurfaces, suggest potential strategies for mitigating this issue (Hussein et al. Reference Hussein, Roca, Harris and Kianfar2025). Nonetheless, in the current configuration featuring a streamwise-extended phononic surface, despite a slight reduction in overall performance, this aspect does not pose a significant limitation to utilising compliant walls for TSW attenuation. While TSW attenuation is common to the two investigated compliant surfaces, conventional and phononic, the phononic one avoids the TWF amplification characteristic of its conventional counterpart. Given that TWF can trigger premature laminar-to-turbulent transition (Lucey & Carpenter Reference Lucey and Carpenter1995), a genuine transition delay is only achievable with the employment of the phononic compliant surface.

In the following section, we recall and discuss in further detail the principles that led to the design of the phononic surface, namely the relation between the periodic structure of the phononic surface and its dynamical properties by means of Bloch analysis.

3.1. Bloch analyses of the phononic compliant surface

We here consider two different scenarios for the Bloch analysis of the phononic compliant surface. The first involves only the periodic elastic medium, isolated from the fluid domain and subject to a stress-free boundary condition (§ 3.1.1). The second incorporates the interaction with the fluid, but it considers as a base state a fluid with no flow, i.e. at rest (§ 3.1.2). The Bloch analysis of the coupled interaction problem between the fluid flow and the phononic surface presents substantial theoretical and numerical challenges, and is therefore left for future investigation.

3.1.1. Bloch analysis without fluid interaction

We first investigate the dynamical properties of the ideal viscoelastic structure displayed in figure 5. As illustrated in figure 5(a), this structure is infinitely long in the $x$ direction and clamped at the bottom boundary. It is obtained by periodically repeating in the $x$ direction the unit cell displayed in figure 5(b). The height of this unit cell is fixed to $h=2.35\,{\rm mm}$ while its length $l_I$ will be varied. This unit cell is made from two different materials. The matrix (orange) is made with an elastomer characterised by Young’s modulus $E_s$ , structural viscosity $\nu _s$ and density $\rho _s$ defined before. The circular inclusion (red) of diameter $d_I= 0.94\,{\rm mm}$ that is inserted into the middle of this matrix is made from an elastomer of the same density ( $\rho _I=\rho _s$ ) and damping ( $D_I=D_s$ ) but with different Young’s modulus $E_I$ , here larger than the matrix’s Young’s modulus, i.e. $E_I \gt E_s$ .

Figure 5. Infinite, periodic, phononic surface. The periodic cell (b) – clamped at the bottom and free at the top – with span $l_I$ is periodically and infinitely replicated along the $x$ direction (a).

The solid state is defined by the variable $\boldsymbol{{q}}_s^\prime = [\boldsymbol{\xi }_s^\prime , \boldsymbol{u}_s^\prime , p_s^\prime ]^T$ , where $\boldsymbol{\xi }_s^\prime$ is the displacement field, $\boldsymbol{u}_s^\prime = \partial _t \, \boldsymbol{\xi }^\prime _s$ is the velocity field and $p_s^\prime$ is the pressure field introduced to describe the incompressible structure. They satisfy the viscoelastic equations

(3.1) \begin{align} \rho _s \,\partial _t \, \boldsymbol{u}^\prime _s &= \boldsymbol{\nabla }\boldsymbol{\cdot }\left ( p_s^\prime \,\boldsymbol{I} + \frac {E(\boldsymbol{x})}{3}\left (\boldsymbol{\nabla }\,\boldsymbol{\xi }_s^\prime + (\boldsymbol{\nabla }\,\boldsymbol{\xi }_s^\prime )^T\right ) + \frac {D_s}{2}\left (\boldsymbol{\nabla }\,\boldsymbol{u}_s^\prime + (\boldsymbol{\nabla }\,\boldsymbol{u}_s^\prime )^T\right ) \right ), \end{align}

(3.2) \begin{align} 0 &= \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{\xi }_s^\prime , \end{align}

where the spatially varying Young’s modulus $E(\boldsymbol{x})$ is a $l_I$ -periodic function in the $x$ direction defined as $E(\boldsymbol{x}) = E_s$ for $\boldsymbol{x}$ in the matrix’s material and $E(\boldsymbol{x}) = E_I$ for $\boldsymbol{x}$ in the inclusion. The structure is clamped at the bottom boundary at $y=-h=-2.35\,{\rm mm}$ , i.e. $\boldsymbol{\xi }^\prime _{s}=\boldsymbol{u}^\prime _{s}=\boldsymbol{0}$ . Here, we do not consider any interaction with the fluid and therefore impose a stress-free boundary condition at the top $y=0$ , i.e. $\partial _x \xi ^\prime _{s,y} + \partial _y \xi ^\prime _{s,x} = 0$ and $-p^\prime _s + 2 \partial _{y} \xi ^\prime _{s,y} = 0$ .

According to Bloch’s theorem (Ziman Reference Ziman1972), the waves propagating in this infinitely long $x$ -periodic structure can be represented as a superposition of Bloch waves in the $x$ direction in the form

(3.3) \begin{align} \boldsymbol{{q}}_s^\prime (\boldsymbol{x}, t) = \int _{0}^{\infty } \boldsymbol{{q}}_{s}(\boldsymbol{x}, t; \beta _x)\,{\rm e}^{\mathrm{i}\beta _{x}\,x}\,{\rm d}\beta _x + \mbox{c.c.} \quad {\rm with}\quad \boldsymbol{{q}}_s(\boldsymbol{x}, t; \beta _x) = \boldsymbol{{q}}_s(\boldsymbol{x}+l_I\boldsymbol{e}_x, t; \beta _x). \end{align}

The complex field $\boldsymbol{{q}}_{s}(\boldsymbol{x}, t; \beta _x)$ , associated with the wavenumber $\beta _x= 2\pi / \lambda _x$ , is $l_{I}$ -periodic in the $x$ direction, but the wavenumber $\beta _x$ is a real parameter that allows for the solid perturbation $\boldsymbol{{q}}_s^\prime (\boldsymbol{x}, t)$ to break this spatial periodicity of the lattice. Without any simplification of the problem, the range of values for $\beta _x$ can be restricted to the irreducible first Brillouin region, i.e. $-\pi /l_I \le \beta _x \le \pi /l_I$ (Brillouin Reference Brillouin1946). Moreover, as $\boldsymbol{{\tilde {q}}}_s(-\beta _x) = \boldsymbol{{\tilde {q}}}_s^*(\beta _x)$ from the above definition, the investigated range of values can be further reduced to $0 \le \beta _x \le \pi /l_I$ . The structure being made with viscoelastic material, we further decompose the periodic complex fields as

(3.4) \begin{equation} \boldsymbol{{q}}_s(\boldsymbol{x},t;\beta _x) = \boldsymbol{{\tilde {q}}}_s(\boldsymbol{x};\beta _x)\,{\rm e}^{(\gamma - \mathrm{i} \omega )t}, \end{equation}

where $\gamma$ and $\omega$ represent the temporal growth rate (decay) and frequency of the solid mode $\boldsymbol{{\tilde {q}}}_s$ . Note that the phase and group velocities of the Bloch waves are then classically given by $v_{\phi } = \omega / \beta _x$ and $v_{g} = \partial \omega / \partial \beta _x$ , respectively. By injecting the decompositions (3.3) and (3.4) into the viscoelastic model (3.1)–(3.2), we obtain the following eigenvalue problem:

(3.5) \begin{align} (\gamma - \mathrm{i} \omega )\,\boldsymbol{\tilde {\xi }}_s &= \boldsymbol{\tilde {u}}_s ,\end{align}

(3.6) \begin{align} (\gamma - \mathrm{i} \omega )\,\boldsymbol{\tilde {u}}_s &= \frac {1}{\rho _s} \boldsymbol{\nabla} _{\beta } \boldsymbol{\cdot }\left ( \tilde {p}_s\,\boldsymbol{I} + \frac {E(\boldsymbol{x})}{3}\big (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {\xi }_s} + (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {\xi }}_s)^T\big) + \frac {D_s}{2}\big(\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {u}}_s + (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {u}}_s)^T\big)\! \right)\! , \end{align}

(3.7) \begin{align} 0 &= \boldsymbol{\nabla} _{\beta } \boldsymbol{\cdot }\boldsymbol{\tilde {\xi }}_s , \end{align}

defined in the unit cell shown in figure 5(b). Note that the spatial operator has changed and is now defined as $\boldsymbol{\nabla} _{\beta } = ( \partial _x + \mathrm{i} \beta _x , \partial _y )$ . As previously defined, zero shear stress and clamped boundary conditions are, respectively, imposed at the top and bottom boundaries of the unit cell. On the left and right boundaries of the unit cell, periodic boundary conditions are finally imposed for all variables, i.e.

(3.8) \begin{equation} \boldsymbol{\tilde {q}}_{s}(-l_I/2,y)=\boldsymbol{\tilde {q}}_{s}(l_I/2,y). \end{equation}

The right-hand-side operator in the eigenvalue problem introduced above is invariant by a combination of the spatial transformation $x \rightarrow -x$ and the conjugate operation $()^*$ . Consequently, the eigenvalue spectrum obtained for a given value of $\beta _x$ is symmetric with respect to frequency axis $\omega$ . For a solid mode $\tilde {\boldsymbol{q}}_{s}$ associated with the eigenvalue $(\gamma - \mathrm{i} \omega )$ that propagates in the $x$ direction with positive phase velocity $v_{\phi } = \omega / \beta _x$ , we obtain that the complex conjugate solid mode $\tilde {\boldsymbol{q}}_{s}^*$ associated with the eigenvalue $(\gamma + \mathrm{i} \omega )$ propagates with negative phase velocity $v_{\phi } = - \omega / \beta _x$ . The full eigenvalue spectrum of the infinitely long $x$ -periodic structure illustrated in figure 5(a) is finally constructed by repeatedly solving the eigenvalue problem in the unit cell with real wavenumber varying in the range $0 \le \beta _x \le \pi / l_I$ .

3.1.2. Bloch analysis with fluid interaction

We now investigate the dynamical properties of the infinitely long $x$ -periodic structure interacting at its top boundary with the fluid, as shown in figure 6(a), that is characterised by density $\rho _{\!f}$ and dynamic viscosity $\mu _{\!f}$ . The fluid–structural unit cell shown in figure 6(b) is now composed of the solid unit described in the previous subsection with a fluid column at the top. Unlike the analysis performed in § 2, we simplify the analysis by considering that the fluid is at rest. This analysis allows us to take into account the fluid added mass in the study of the natural dynamics of a solid, as recently addressed by Zhang et al. (Reference Zhang, Cong, Sellam and Chpoun2024) for a two-dimensional phononic plate.

Figure 6. Infinite, periodic, phononic surface interacting with a fluid at rest. The periodic cell (b) – clamped at the bottom and interacting with the fluid at the top – with span $l_I$ is periodically and infinitely replicated along the $x$ direction (a).

The flow perturbation is defined by the variable $\boldsymbol{{q}}_{\!f}^\prime = [\boldsymbol{u}_{\!f}^\prime , p_{\!f}^\prime ]^T$ , where $\boldsymbol{u}_{\!f}^\prime$ is the fluid velocity perturbation and $p_{\!f}^\prime$ is the fluid pressure perturbation. Following the two decompositions introduced in the previous subsection for the solid perturbation, we decompose the fluid perturbation as

(3.9) \begin{align} \boldsymbol{{q}}_{\!f}^\prime (\boldsymbol{x}, t) =& \int _{0}^{\infty } \!\boldsymbol{{\tilde {q}}}_{\!f}(\boldsymbol{x}; \beta _x)\,{\rm e}^{\mathrm{i}(\beta _{x}\, x - \omega \, t)} {\rm e}^{\gamma t} \, {\rm d} \beta _x + \mbox{c.c.} \!\quad{\rm with}\!\quad \boldsymbol{{\tilde {q}}}_{\!f}(\boldsymbol{x}; \beta _x) = \boldsymbol{{\tilde {q}}}_{\!f}(\boldsymbol{x}+l_I\boldsymbol{e}_x; \beta _x). \end{align}

By introducing this decomposition into the Stokes equation governing the fluid perturbation, we obtain

(3.10) \begin{align} (\gamma - \mathrm{i} \omega )\,\boldsymbol{\tilde {u}}_{\!f}^\prime &= \frac {1}{\rho _{\!f}} \boldsymbol{\nabla} _{\beta }\boldsymbol{\cdot }\left (-\tilde {p}_{\!f}^\prime \boldsymbol{I} + \mu _{\!f} \left (\boldsymbol{\nabla} _{\beta }\boldsymbol{\tilde {u}}_{\!f}^\prime + (\boldsymbol{\nabla} _{\beta }\boldsymbol{\tilde {u}}_{\!f}^\prime )^T\right )\right ) & {\rm in}\ \varOmega _{\!f}, \end{align}

(3.11) \begin{align} 0 &= \boldsymbol{\nabla} _{\beta }\boldsymbol{\cdot }\boldsymbol{\tilde {u}}_{\!f}^\prime & {\rm in}\ \varOmega _{\!f}, \end{align}

where $\varOmega _{\!f}$ denotes here the fluid domain in the unit cell. These fluid equations are coupled at the fluid–solid interface to the solid equations (3.5), (3.6) and (3.7) defined in the solid domain $\varOmega _s$ . The continuity of the velocity and stress perturbation at the interface $y=0$ is simply written

(3.12) \begin{align} \boldsymbol{\tilde {u}}_{\!f}^\prime &= \boldsymbol{\tilde {u}}_{s}^\prime & {\rm on}\ \varGamma _{\!\textit{fs}} \end{align}

(3.13) \begin{align} \left (-\tilde {p}_{\!f}^\prime \boldsymbol{I} + \mu _{\!f} \left (\boldsymbol{\nabla} _{\beta }\boldsymbol{\tilde {u}}_{\!f}^\prime + (\boldsymbol{\nabla} _{\beta }\boldsymbol{\tilde {u}}_{\!f}^\prime )^T\right )\right ) (-\boldsymbol{e}_{y}) &\nonumber\\=\left ( \tilde {p}_s\,\boldsymbol{I} + \frac {E(\boldsymbol{x})}{3}\left (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {\xi }_s} + (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {\xi }}_s)^T\right ) \right .&+\left . \frac {D_s}{2}\left (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {u}}_s + (\boldsymbol{\nabla} _{\beta }\,\boldsymbol{\tilde {u}}_s)^T\right ) \right ) \boldsymbol{e}_{y}\quad &{\rm on}\ \varGamma _{\!\textit{fs}}. \end{align}

Equations (3.12) and (3.13) ensure that the fluid and solid perturbations respect the kinematic and dynamic conditions at the interface. The coupled eigenvalue problem is defined on the fluid–solid unit cell by (3.5), (3.6) and (3.7) coupled to (3.10) and (3.11) through the interface conditions (3.12) and (3.13). Periodic boundary conditions are applied to all variables on the left and right boundaries. At the top boundary of the fluid domain, the stress-free condition is imposed. Finally, the dispersion relation of fluid–solid Bloch waves in this infinitely long $x$ -periodic structure interacting with the fluid at rest is obtained by solving this eigenvalue problem for values of the real wavenumber in the range $0 \le \beta _x \le \pi /l_I$ . Note that the same symmetry argument holds for the Stokes equations, i.e. the fluid–structural mode $(\tilde {\boldsymbol{q}}_{\!f}, \tilde {\boldsymbol{q}}_{s})$ associated with the eigenvalue $(\gamma - \mathrm{i} \omega )$ is related to the complex conjugate mode $(\tilde {\boldsymbol{q}}_{\!f}^*, \tilde {\boldsymbol{q}}_{s}^*)$ associated with the eigenvalue $(\gamma + \mathrm{i} \omega )$ , with the two modes propagating with the same velocity but in opposite directions.

3.2. Results of Bloch analyses

Results of the Bloch analyses are first presented for the infinitely long phononic compliant surface with geometrical and material properties similar to those of the finite-length phononic compliant surface described in the previous section (§ 2). The dispersion relation and modes are described for the case without fluid interaction in § 3.2.1 and with interaction with a fluid at rest in § 3.2.2. The effect of the inclusion’s spacing $l_I$ and Young’s modulus $E_I$ on the frequency band gap is then investigated in § 3.2.3.

3.2.1. Infinitely long phononic compliant surface without fluid interaction

Figure 7 presents the eigenvalues computed for the infinitely long periodic structure, with $l_{I}=2.35$ mm and $E_I = 10 E_s$ and without fluid interaction. These parameters correspond to the phononic compliant surface studied in § 2. In the top-left panel, the frequencies $f=\omega /2\pi$ associated with the identified eigenvalue branches are plotted as a function of the wavenumber $\beta _x$ . These same branches are also visible in the eigenvalue spectrum shown in the top-right panel, where the frequency is plotted against the normalised growth ratio $\gamma /\omega$ . All eigenvalues exhibit negative damping due to the intrinsic viscous properties of the materials, leading to natural solid waves whose amplitudes decay over time. For $f \ge 600\rm \, Hz$ , the frequency axis is not continuously populated, revealing the presence of band gaps between the identified branches (highlighted in red areas). Within these band gaps, only evanescent waves can propagate, effectively preventing long-range energy transport.

Figure 7. (Top) Dispersion relation from Bloch analysis for an isolated, $x$ -periodic phononic medium. The shaded red areas highlight the bands of angular frequencies for which no eigenvalues are found (band gaps). (a–c) The time–space evolution of the modes labelled in the top figures. On the left, the colour map reports the vertical displacement field in the solid domain at $t=0$ , while the vectors the displacement at the interface. On the right, the colour map reports time–space evolution of the vertical displacement at the interface over one oscillation.

The eigenmode of the lowest-frequency branch corresponding to $\beta _x = \pi /(2l_I)$ is displayed in figure 7(a). In the left-hand panel, the vertical component of the displacement perturbation field is displayed with colours at the (arbitrary) instant $t=0$ . This mode is shown on a domain composed of $10$ unit cells (inclusions) for visualisation purposes, but was computed on a single unit cell. The displacement at the top boundary is also displayed with arrows. In the right-hand panel, the vertical displacement of the interface is shown as a function of space $x$ and time $t$ . This Bloch mode is characterised by a spatially periodic pattern (colour) modulated by the wavelength $2\pi /\beta _x = 4\,l_I$ of the envelope that is clearly visible at the interface. As can be seen from the right-hand panel, this is a travelling wave propagating along the $x$ direction with a positive phase velocity $v_\phi = \omega /\beta _x$ and a positive group velocity $v_{g} = \partial \omega / \partial \beta _x$ . Note that the complex conjugate eigenmode (obtained for the same value of $\beta _x$ ) provides a Bloch wave travelling in the opposite direction with negative phase and group velocities. When increasing the wavenumber on this low-frequency branch, the group velocity corresponding to the speed of the envelope decreases. At $\beta _x = \pi /l_I$ , this group velocity is zero and the envelope of the Bloch mode is a standing wave. The corresponding mode is displayed in figure 7(b). Inside the standing-wave envelope of wavelength equal to $2 l_I$ , a spatially periodic wave propagates with positive phase velocity equal to $v_{\phi }= 2 f_{min} l_I$ , where $f_{min}$ is the lowest frequency of the first band gap. As seen in the left-hand panel, the maximal and minimal displacements of the interface are reached in between the inclusions (circles). The Bloch mode displayed in figure 7(c) corresponds to the highest frequency $f_{max}$ of the first band gap. This mode is also characterised by a standing-wave envelope of wavelength $2 l_I$ but for a larger phase velocity now equal to $v_{\phi }= 2 f_{max} l_I$ . The maximal and minimal displacements of the interface are now reached above the inclusions. For frequencies $f_{min} \le f\le f_{max}$ in between these two standing Bloch waves, no wave may propagate, thus confirming the wave-blocking properties of the phononic surface along the $x$ direction.

The wavelength $2 l_I$ of these two standing Bloch modes occurring at the edges of the band gap is similar to that of the TWF instability, as observed in figure 2. This establishes the first principle for designing our phononic surface: its spatial periodicity should be half of the wavelength of the TWF wave to control. The second principle is that the bang-gap frequency also matches the range of frequency where TWF instabilities are amplified. The frequency band gap predicted by the present Bloch analysis extends in the range $1540\,{\rm Hz} \le f \le 1780\,{\rm Hz}$ , while that observed for the finite-length phononic surface in § 2.3 sits between $1241$ and $1439\,{\rm Hz}$ . We anticipate that this overestimation of the frequency band gap is mainly due to the interaction of the compliant surface with a heavy fluid (water). Added-mass effects can then significantly alter the frequencies of the natural vibration modes of structures (Pfister & Marquet Reference Pfister and Marquet2020) and should be taken into account to design the phononic compliant surface.

3.2.2. Infinitely long phononic compliant surface with fluid interaction

To capture the added-mass effect of the heavy fluid (water) on the frequencies of the Bloch modes, we now present results of the Bloch analysis for an infinitely long phononic compliant structure that interacts with a still fluid as discussed in § 3.1.2. Figure 8 displays with solid curves the branch of eigenvalues computed with the coupled fluid–solid eigenvalue problem, while dashed curves correspond to the original one. The branches are modified and the expected shift towards lower frequency is observed. The first band gap now sits in the range $1270\,{\rm Hz}\le f \le 1440\,{\rm Hz}$ , to be compared with $1241\,{\rm Hz}\le f \le 1439\,{\rm Hz}$ from the global stability analysis of the finite-length phononic surface in § 2.3. Additionally, the frequency range of the band gap in the phononic compliant surface is now in much better agreement with the targeted value $f=1350\,{\rm Hz}$ , at which the maximal amplification of TWF occurs. The agreement also validates the general principle of the present phononic compliant surface that is to forbid the propagation of the solid waves in the range of frequencies where TWF is most amplified. From the right-hand panel, a slight increase in the growth ratio is also noted. Looking at the shapes of the modes limiting the band gap (figure 8 a,b), they present the same structure as those identified for the isolated solid (figure 7 a,b). When the interface vibrates, it drives the fluid in its upward/downward motion, which means the solid has to displace the mass of the fluid, which qualitatively explains the lower natural frequency. In the case considered, this effect is important since the mass ratio between the solid (elastomer) and the fluid (water) is unitary; in air, it would be less pronounced because of the lower density of the fluid.

Figure 8. Dispersion relation from Bloch analysis for a viscoelastic, periodic solid interacting with a fluid at rest (solid lines). The shaded areas highlight the bands of angular frequencies for which no eigenvalues are found (band gaps). The dashed lines report the dispersion relation for the isolated periodic solid (see figure 7). (a–c) The modal shapes of the modes labelled in the top figures. On the left, the colour maps report the vertical displacement field in the solid (orange) and fluid (blue) domains at $t=0$ , while the vectors the displacement at the interface. On the right, the time–space evolution of the vertical displacement at the interface is reported over one oscillation.

3.2.3. Effect of inclusion spacing, Young’s modulus and diameter on frequency band gaps

The results previously reported have been obtained for unit cells of size $l_I = 2.35\,{\rm mm}\approx l_{TWF}/2$ and Young’s modulus $E_I = 1\, {\rm MPa} = 10E_s$ that lead to the attenuation of the TWF reported in the first section. We now investigate the impact of these two parameters on the positioning and extent in frequency of the band gaps in relation to the TWF amplification peak, providing a comprehensive explanation for the choice of the selected design point.

Figure 9. Parametric study on the spacing $l_I$ (a), Young’s modulus $E_I$ (b) and diameter $d_I$ (c) of the inserts. The error bars indicate the location of the band gaps at each computed point. The data in (a) are for $E_I = 10E_s$ , $d_I = 0.94\, {\rm mm}$ and varying $l_I$ ; those in (b) for $l_I = 2.35\,{\rm mm}$ , $d_I = 0.94\, {\rm mm}$ and varying $E_I$ ; those in (c) for $E_I = 10E_s$ , $l_I = 2.35\,{\rm mm}$ and varying $d_I$ . The vertical grey lines indicate the design point, reported in figure 8. The horizontal dashed grey lines indicate the frequency at which the TWF peak occurs.

Figure 9(a) presents a parametric investigation into the effect of insert spacing $l_I$ on the dynamic behaviour of the phononic compliant surface interacting with the still fluid, for insert rigidity $E_I = 1\, {\rm MPa}$ . The error bars in this figure represent the frequency range delimiting the band gaps that is displayed as a function of the dimensional value of $l_I$ (bottom axis) and the non-dimensional value $l_I/l_{TWF}$ (top axis) where $l_{TWF}$ is the wavelength of TWF instabilities for $f=1350\,{\rm Hz}$ (horizontal dashed line). The design point explored so far is highlighted by the vertical solid line at $l_I = 2.35\,{\rm mm}\approx l_{TWF}/2$ . For this spacing, the first band gap is centred at the target frequency and wavelength of the TWF waves, thus confirming again the successful design based on Bloch analysis. Increasing the spacing $l_I$ is observed to shift the band gaps towards lower frequencies. For the first band gap (lowest frequency), this phenomenon can be explained by recalling that the limiting frequencies correspond to Bloch modes of two different branches occurring at wavenumber $\beta _x = \pi /l_I$ (see figure 8). As $l_I$ decreases, the wavenumber of the limiting Bloch modes thus increases and consequently the frequencies delimiting the bang gap also increase. Conversely, increasing the spacing $l_I$ leads to lower wavenumbers, resulting in lower frequencies. This general trend is also observed for the second band. Interestingly, its frequency range is centred around the targeted frequency of TWF for $l_{I}=4.28\,{\rm mm}$ , suggesting that the TWF may also be attenuated with this new compliant phononic surface. As the bang gap is narrower than for $l_I=2.35\,{\rm mm}$ , we may also anticipate that its effect is more limited.

Once the inserts’ spacing $l_I$ is fixed, their rigidity can be exploited to pilot the extension of the band gap. This effect is illustrated in figure 9(b) where band gaps are reported for a varying Young’s modulus $E_I$ of the insert material, while the spacing is fixed at $l_I = 2.35\,{\rm mm} \approx l_{TWF}/2$ . As before, the horizontal dashed line indicates the target frequency and the vertical solid line the chosen design point, i.e. $E_I = 1\, {\rm MPa} = 10E_s$ , where $E_s$ is the Young’s modulus of the material surrounding the insert. The width and location of the band gap stabilise for $E_I \gt 0.8\,\, {\rm MPa}$ and no significantly different behaviour is observed at higher values than the design one. For lower values, instead, the band gap shrinks and shifts to lower frequency. At $E_I = 0.1\,\, {\rm MPa} = E_s$ , no band gaps are observed, the case being equivalent to a homogeneous surface.

This study employs a simple circular geometry for the scatterer, chosen arbitrarily to illustrate the fundamental principle behind using phononic surfaces to suppress TWF. Up to this point, the scatterer’s diameter ( $d_I$ ) has been set to $0.94\,{\rm mm}$ , balancing between the overall wall compliance and the band-gap size. Figure 9(c) presents the location of band gaps as a function of scatterer diameter ( $d_I$ ), with fixed parameters $l_I = 2.35\,{\rm mm}$ and $E_I = 1\, {\rm MPa}$ . At the design point ( $d_I = 0.94\,{\rm mm} = 0.4\,h$ ), the band gap is centred at the target frequency corresponding to TWF. As the diameter increases, the band gaps widen but also shift towards higher frequencies due to the stiffening effect caused by the increasing proportion within the unit cell of the space occupied by the stiffer material making the scatterer. Notably, a stiffer structure would lead to a reduced attenuation of the TSW, as documented in Pfister et al. (Reference Pfister, Fabbiane and Marquet2022); therefore, if a geometrical optimisation of the shape of the scatterers is performed, a constraint should be considered to take into account this effect.

3.3. Comparison with eigenvalue analyses of finite-length phononic compliant surfaces

Thus far, this section has outlined the design process for the unitary cell of the phononic surface using Bloch analysis, with the primary objective of suppressing the propagation of waves responsible for TWF. This analysis is based on the assumption of an infinitely extended surface; in the following, this hypothesis is dropped and the dynamics of a finite phononic surface is compared with the results of the Bloch analysis.

Figure 10. Comparison between the fluid–structural eigenvalue branches computed by means of the Bloch analysis (lines, see figure 8) and the eigenvalues of the finite-length surface (squares). A fluid at rest is considered for the two analyses. The coloured shaded areas indicate the band gaps predicted by the Bloch analysis. (a–d) The respective modal shapes of the modes labelled in the top figure (for a finite-length structure).

The coupled interaction of a finite patch with a flow at rest is considered. The eigenvalue problem (2.3) is considered with a vanishing base solution $\boldsymbol{{Q}} = \boldsymbol{0}$ : the velocity of the base flow is zero ( $\boldsymbol{U}_{\!f}(\boldsymbol{x}) = \boldsymbol{0}$ ) as well as the displacement of the structure, i.e. $\boldsymbol{\varXi }_s(\boldsymbol{x}) = \boldsymbol{0}$ , in order to be consistent with the analysis in § 3.2.2 for the infinite plate interacting with a fluid at rest. Note that, in the absence of fluid flow, the concept of a Reynolds number derived from boundary-layer characteristics becomes inapplicable. However, for consistency, the fluid’s viscosity remains identical to that of water used in previous analyses. Figure 10 presents a comparative analysis of eigenvalues obtained for the finite-length phononic surface (red squares) and from the Bloch analysis of the infinite-length phononic surface (red lines), revealing both similarities and key differences. The primary distinction stems from the discrete nature of the eigenvalues in the finite-length case, as opposed to the continuous branches predicted by the Bloch analysis – an expected outcome due to the finite spatial extent of the surface. Despite this, the discrete eigenvalues lie precisely on the continuous branches, demonstrating that the Bloch analysis reliably predicts both frequency and damping characteristics of the finite-length surface.

Regarding modal shapes, figures 10(a) and 10(b) show the eigenmodes near the first band gap, as indicated in the top panel of the figure. These modes exhibit a structure consistent with those predicted by the Bloch analysis: in particular, they feature surface oscillations occurring between the inserts (figure 10 a versus figure 8b ) and above the inserts (figure 10 b versus figure 8c ). An amplitude modulation along the $x$ direction is observed, which is attributed to the lateral clamped boundary conditions imposed on the finite-length phononic surface. A second notable difference with respect to the Bloch analysis is the appearance of a double eigenvalue within the band gap, referred to as truncation resonance (Rosa et al. Reference Rosa, Davis, Liu, Ruzzene and Hussein2023). This phenomenon arises from the breaking of periodicity induced by the finite length of the surface and the clamped boundary conditions at its lateral ends. These eigenmodes shown in figures 10(c) and 10(d) are spatially localised near the left and right boundaries of the surface. One exhibits a symmetric structure, while the other is anti-symmetric. Importantly, both mode share the same eigenvalue, which implies that they can be linearly combined to form spatial structures fully localised at either boundary, while still satisfying the coupled eigenvalue problem for the same eigenvalue. These truncation modes also appear when the compliant phononic surface interacts with the boundary-layer flow (§ 2.3). In this configuration, the fluid is no longer at rest; instead a boundary layer develops over the surface. Two isolated eigenmodes are observed at close frequencies ( $f = 1343\,{\rm Hz}$ and $f = 1347\,{\rm Hz}$ ; see figure 2 b) each localised near one of the lateral boundaries (see figure 3). Unlike in the static fluid case, where symmetric and anti-symmetric pairs of eigenmodes are found, the base flow velocity oriented in the streamwise direction breaks the symmetry and yields two distinct, localised modes. The slight frequency difference between the two modes ( $f = 1343\,{\rm Hz}$ versus $f = 1347\,{\rm Hz}$ ) is likely due to the streamwise development of the boundary layer, which interacts differently with the compliant surface at the upstream and downstream edges.

Beyond the isolated truncation modes, the proposed fluid–structure Bloch analysis accurately captures the frequency range of the band gaps, a key feature for the effective suppression of TWF. This is clearly demonstrated by the strong agreement between the predicted band gap from the Bloch analysis ( $1270\,{\rm Hz} \le f \le 1440\,{\rm Hz}$ ) and the observed band gap for the finite-length phononic surface ( $1241\,{\rm Hz} \le f \le 1439\,{\rm Hz}$ ). This close correspondence underscores the reliability of the Bloch framework as a predictive and practical design tool for compliant phononic surfaces aimed at TWF control, even when applied to finite-length configurations.