Numerical Aeroacoustic Assessment of a Metacontinuum Device Impinged by a Laser-Generated Sound Source

Abstract

Significant research efforts have been conducted by many research groups in the last decades to circumvent the acoustic metamaterial concept limitations. Principally, the lack of a reliable methodology for the design of such metamaterial-based devices and their performance decay when they operate in a moving medium were the main topics of investigation that culminated in a number of possible approaches with a high potential of application in aeroacoustics. One of these approaches is based on the aeroacoustic spacetime reformulation of the problem to recast the governing equations in a generalized form, independent of the kinematic conditions of the supporting medium. In the present paper, the response of a spacetime-corrected metacontinuum is coupled with a high-fidelity aeroacoustic model of the hosting fluid to allow for the modelling of actual experimental realizations of a laser-generated sound source. The availability of a reliable model to couple the convective metacontinuum design with the heat-release source would make possible the systematic cross-validation of the numerical simulations with the experimental results obtained in the most advanced testing facilities, paving the way to an effective inclusion of metacontinuum-based devices in aeronautics and, finally, in a simulation-based, multidisciplinary design optimization framework. Although the method presented is valid for arbitrary acoustic responses, the numerical simulations presented in this work have been conducted using the classic Cummer–Schurig inertial cloaking as a reference application, which can now be considered a widely accepted benchmark for acoustic metamaterial applications.

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1. Introduction

Metamaterial devices have interested many research fields of both physics and engineering in the last decades. Their versatility is the attractive feature characterizing them: born in the field of electromagnetism [1–4], their applications have reached the most diverse application thanks to the advent of the transformation optics method developed by Pendry [5] and Leonhardt [6, 7] where the required properties of the metadevice started to be a consequence only of the specific application aimed. This enabled a new degree of freedom in manipulating the electromagnetic field and later led to the extension of the theory and applications in other disciplines, including acoustics. In fact, the similarity noted by Cummer and Schurig [8] between the two-dimensional acoustic equations in a fluid and the single-polarization Maxwell equation allows applying the coordinate transformation methodology for the definition of the metamaterial properties and also for acoustic applications. This led to achieving the so-called “acoustic mirage” and the direct transposition of other metabehaviours originally thought in optics. As a matter of fact, following the steps of the transformation optics, Norris developed an acoustic theory [9, 10] through which he defines a new acoustic model of continuum named metafluid. Therefore, the properties that characterize such unconventional medium are identified as anisotropic inertia and anisotropic stiffness, analytically defined as functions of the sole application (according to previous authors), and independent from the device shape.Appropriately defining the distributions of those two properties in the metadevice domain, it is possible to achieve extraordinary acoustic effects, manipulating the waves almost at will (obtaining, e.g., acoustic invisibility, also known as “cloaking” [11, 12]), the steering of incoming waves with anomalous reflection and refraction angles [13–15]. The focus on the results obtained for acoustics applications is due to the attention that they have captured among the aeronautical community. In the context of increasing air traffic and the introduction of the U-space for drone operations, aircraft noise pollution keeps being one of the main topics of research efforts, as it has been in the last decades [16–19]. Unfortunately, the use of acoustic metadevices, i.e., designed considering a quiescent hosting fluid, has been demonstrated not to be an adequate strategy to deal with applications in the presence of convection, i.e., aeroacoustic applications. With the aviation acoustic impact abatement goal, authors such as Huang et al. [20], Iemma [21], and Iemma and Palma [22–25] have tested the performance decay of acoustic metamaterials when operating in a flow. The complexity of extending acoustic metadevices to aeronautical applications is evident: due to the presence of a background flow, the fluid dynamics phenomenon couples with the acoustic propagation leading to mixed space and time derivatives terms in the governing equation for acoustic pressure propagation. Thus, the formal invariance under a general coordinates transformation of the equation that governs the acoustic propagation, previously observed by Cummer and Schurig, is lost as they do not maintain the same mathematical structure.

To overcome the problem, García-Meca et al. proposed the analogue transformation acoustics (ATA) approach that aims to recover the governing equations’ formal invariance property [26, 27] necessary to apply the transformation acoustic method for the identification of the metamaterial properties for a given required acoustic behaviour. As already observed by Visser in [28], a spacetime formalism guarantees the formal invariance of the governing equation under general spacetime coordinate transformations. Despite the approach’s generality, its applicability is subordinate to appropriate conformal mapping existence (available only for simple geometries). Moreover, it may lead to geometrical solutions not feasible in the aeronautical context due to the size constraints of the aeronautical constructive elements. Subsequently, the work by Iemma [21] and Iemma and Palma [22–25] proposes an extended version of such an approach that overcomes ATA limitations. The authors combine the concepts of the transformation optics techniques and the ATA method. A new definition for the metadevices properties suited for convective applications is obtained by rewriting the Norris metacontinuum model governing equation into the spacetime domain to consider convective effects. The method proposed by the authors relies on the analytical correction of statically designed metamaterials through the application of coordinate transformation dependent on the background flow characteristics. Such a correction relies on Taylor’s [29] or Prandtl–Glauert’s [30] coordinate transformation leading to encouraging results. However, more steps are necessary to validate the method: Colombo, Palma and Iemma [31], and Colombo et al. [32, 33] investigated the behaviour of the two mentioned corrections looking for a better understanding of the approach’s limitations. In fact, when coordinate transformations enclosing the background flow characteristics are introduced, it is necessary to consider which aspect of the acoustic convected phenomena are captured. The present work focuses on the assessment of the correction capabilities of the convective transformations when coupled with a higher fidelity simulation model, capable of taking into account the physical phenomena involved when sophisticated experimental techniques are used. In fact, all the works presented so far by the authors [23–25, 31–33] consider the metadevices immersed in medium where no vorticity- or entropy-induced acoustic sources are present, and the propagation of the acoustic disturbance is expressed in term of acoustic velocity potential. Even though this is a reasonable assumption in most applications, it prevents the model from providing accurate predictions when the acoustic perturbation impinging the metacontinuum device is generated by a source involving variations of entropy, especially in the close vicinity of the source location. This is the case of the most advanced devices used in experimental facilities to reproduce in real world an isotropic acoustic point source using high-energy laser beams. Although the use of this kind of sources is not new in the field of solids acoustics (see, e.g., Aussel, Le Brun, and Baboux [34]), they have had limited attention in the past in the field of aeroacoustics, probably because of the more complex physics underlying the mechanism of excitation in a flowing medium, involving the generation of a small bubble of plasma moving with the fluid (see, e.g., Bahr et al. [35] or Szöke and Devenport [36]). A high-energy laser beam is used to heat a small volume of fluid leading to an omnidirectional pressure wave generation that expands rapidly at the isentropic speed of sound, and the propagation pattern of the resulting pressure field has the characteristics of a monopole point source. The physical phenomenon underlying the generation of the acoustic perturbation is a consequence of the ionization of the molecules triggered by the electromagnetic power concentrated in a tiny volume of air by the laser beam. The chemical reactions induced generate a rapid transient that produces a very quick sequence of compression an dilatation, yielding to a resulting pressure signal of an overall duration of about 0.1 ms. As a consequence, the sound source generated by such a complex phenomenon must be modelled as a moving heat source, thus imposing the adoption of a model of the governing equations capable of taking it into account in the proper way. Therefore, the linearized Euler equation (LEE) model is used to implement the model of the laser-generated sound source derived in Rossignol et al. [37, 38]. Needless to say, the use of the LEE model inevitably increases the computational effort but in this context represents a necessary step toward the numerical assessment of the response of devices tested using this specific technique.

All the simulations in the present article are performed in the frequency domain for a monochromatic signal using the commercial FEM software COMSOL Multiphysics. The implementation of adapted acoustic metamaterials considers the variation of the Mach number from 0.1 up to 0.3 (nearly the characteristic value for take-off and landing phases for commercial aircraft). The paper is organized as follows: in Section 2, the analytical correction method of the metafluid device is presented; in Section 3, the numerical setup is shown followed by the numerical results in Section 4 and the concluding remarks in Section 5.

2. Analytical Model: Corrected Metafluid

Following the metafluid theory developed by Norris [9, 10], it is possible to identify the properties that lead to exotic behaviours of an acoustic medium (also called acoustic mirages).

The use of anisotropic inertia and stiffness is the key enabler of such unconventional acoustic responses. While the peculiar design of these properties relies on the specific application aimed at, the governing equation that describes the propagation within such a medium can be written in the general form:$$\begin{array}{}\text{(1)}& c_{ref}^2\widehat{K}\mathbf{S}:{\widehat{\mathbf{\rho }}}^{-1}\mathbf{S}\nabla p-{\partial }_{\text{tt}}p=0,\end{array}$$where $c_{\text{ref}}$ is the reference speed of sound defined as $c_{\text{ref}}^2=K_{\text{ref}}/{\rho }_{\text{ref}}$, $\widehat{K}$ is the adimensionalized space-varying bulk modulus, and with the second-order tensors $\widehat{\mathbf{\rho }}$ and $\mathbf{S}$ embedding the inertial and the elastic anisotropic features of the metacontinuum, respectively. The interested reader can find the actual expressions for the tensors introduced in (1) referring to some specific applications including acoustic invisibility in [9]. It is worth noting that the analytic convective corrections described and tested in the present paper are, in principle, independent of the specific “mirage” produced by the static design of the metamaterial. In fact, the method first proposed by Iemma [21] and Iemma and Palma [22–25] extends the transformation acoustics approach by correcting the existing acoustic metamaterial designs with convective coordinate transformations widely used in aeroacoustics. In order to attain the desired effect, i.e., avoid the performance decay of the metamaterial-based device in the presence of a flow, it must be ensured that their application would preserve the mathematical structure of the governing equation. Hence, an invariant form of (1) is needed. This can be obtained by recasting the governing equation (1) in the four-dimensional spacetime. It is worth noting that (1) is obtained by following the procedure in [10] that necessarily leads to the description of the propagation of acoustic perturbations in a generic metacontinuum only in terms of pseudo-pressure. Hence, reformulating the phenomenon in a pseudo-Riemannian manifold, a point is now defined as a point event through the four-vector $\mathbf{\xi }={\xi }_0,{\xi }_1,{\xi }_2,{\xi }_3=c_{\text{ref}}t,x_1,x_2,x_3$ and the four-gradient operator is defined as ∂ = $1/c_{\text{ref}}{\partial }_t,\nabla$. Subsequently, equation (1) is rewritten in a new form that lives in a pseudo-Riemannian manifold leading to the following general formalism:$$\begin{array}{}\text{(2)}& {\partial }_{\mu }{W}^{\mu \nu }{\partial }_{\nu }p=0,\end{array}$$with$$\begin{array}{}\text{(3)}& \mathbf{W}=\begin{array}{ccc}-1/c_{\text{ref}}^2& ⋮& 0\\ \cdots & \cdot & \cdots \\ 0& ⋮& \widehat{K}\mathbf{S}{\widehat{\mathbf{\rho }}}^{-1}\mathbf{S}\end{array},\end{array}$$where the property of $\mathbf{S}$ to be divergence-free is exploited. The second-order tensor $\mathbf{W}$ is linked to the contravariant metric tensor ${\mathbf{g}}^{-1}$, being $1/g$ its determinant, as $\mathbf{W}=\sqrt{-g}{\mathbf{g}}^{-1}$. The description of (1) in a (3+1)-dimensional manifold allows deriving the governing equation (2) which has a conservative form meaning that is invariant under general spacetime coordinate transformations [22–25, 31–33]. This property leads to an easy introduction of a spacetime transformation that modifies the components of the $\mathbf{W}$ tensor and, consequently, changes the metric of the propagating phenomenon. Specifically, the purpose is to adjust the $W^{\mu \nu }$ components introducing the background flow information in the acoustic metacontinuum design. The coordinate mappings considered follow from the ones proposed by Iemma and Palma [24] based on Prandtl–Glauert’s [30] and Taylor’s [29] coordinate change. In a spacetime formalism, the former assumes the following form:$$\begin{array}{c}\text{(4)}& {\xi }_0^{\prime }=\beta {\xi }_0+M_{\infty }\frac{{\xi }_1}{\beta },{\xi }_1^{\prime }=\frac{{\xi }_1}{\beta },\\ {\xi }_2^{\prime }={\xi }_2,\\ {\xi }_3^{\prime }={\xi }_3,\end{array}$$being $\beta =\sqrt{1-M_{\infty }^2}$ the characteristic Prandtl–Glauert’s parameter indicating the presence of a subsonic uniform-free stream aligned with the $x_1$-axis whereas the latter has the form of$$\begin{array}{}\text{(5)}& {\xi }_0^{\prime }={\xi }_0+M_{\infty }\widehat{\Phi }\xi ={\xi }_0+M_{\infty }{\xi }_1+\widehat{\varphi },{\xi }_i^{\prime }={\xi }_i,\end{array}$$where $\widehat{\Phi }$ is the adimensionalized velocity potential $\widehat{\Phi }=\Phi /{\mathbf{v}}_{\infty }$ that takes into account the nonuniformity of the flow in the limitation of the $OM$-terms. Recasting (4) and (5) in a matrix form, it is possible to obtain the associated Jacobian matrices ${\mathbf{\Lambda }}_{PG}$ and ${\mathbf{\Lambda }}_T$$$\begin{array}{c}\text{(6)}& {\mathbf{\Lambda }}_{PG}=\begin{array}{cccc}\beta & \frac{M_{\infty }}{\beta }& 0& 0\\ 0& \frac{1}{\beta }& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array},{\mathbf{\Lambda }}_T=\begin{array}{ccc}1& ⋮& M_{\infty }\nabla \widehat{\Phi }\mathrm{x}\\ \cdots & \cdot & \cdots \\ 0& ⋮& \mathrm{I}\end{array}.\end{array}$$

Each coordinate transformation acts upon the space within which an acoustic perturbance propagates, associating the aeroacoustic solution in the presence of aerodynamic convection with its equivalent in a quiescent medium. Particularly, Prandtl–Glauert’s transformation compresses the acoustic field along the propagation axis of the free stream transforming a total derivative of the form $D_t^{\infty }={\partial }_t+{\mathbf{U}}_{\infty }\cdot \nabla$ into the standard time-derivative ${\partial }_t$. On the other hand, Taylor’s transformation is able to map back an $O{M}_{\infty }$ equation (that includes the effect of a low-speed, nonuniform flow on the acoustic perturbation propagation) to the static d’Alembertian. Looking at the wavefronts of the acoustic propagating waves, the transformations convert the convective pattern into the static isotropic one. Therefore, the inverse transform is needed for the metacontinuum to mimic the propagation in the presence of a flow inside itself, and the $\mathbf{W}$ tensor is hence corrected as follows:$$\begin{array}{}\text{(7)}& {\mathbf{W}}_C={\mathbf{\Lambda }}^{-1}\mathbf{W}{\mathbf{\Lambda }}^{-T}.\end{array}$$

The new properties of the adapted metadevice are obtained directly from the components of ${\mathbf{W}}_C$ originated from (7). As previously stated, in this process, there are no assumptions on the acoustic metamaterial design, and thus, the method based on analytical spacetime corrections is general and limited only by the range of validity of the coordinate transformation proposed.

3. Numerical Setup

This section explains the numerical setup of the aeroacoustic simulations that are performed in the frequency domain, through the FEM commercial solver COMSOL Multiphysics. The simulation domain recalls the one used in [31, 33] and includes three concentric circumferential regions centre in the axis origin: the innermost one is occupied by an impermeable sound-hard cylindrical obstacle (SHO) of radius $R_{\text{body}}$; a toroid occupied by the metacontinuum surrounds the scatterer with an external radius of $R_{\text{cloak}}=2R_{\text{body}}$; finally, the outer domain is occupied by the hosting fluid with an outer radius of $R_{\text{ext}}=60R_{\text{body}}$ with a resulting geometry depicted in Figure 1(a). As said, to improve the numerical simulations’ fidelity with respect to previous works where the hosting fluid is considered a potential fluid, here it is modelled through the linearized Euler equations and a laser-generated sound source is introduced as the forcing term $S_e$ of the energy conservation equation as follows:$$\begin{array}{c}\text{(8)}& i\omega \rho +\nabla \cdot \rho {\mathbf{v}}_0+{\rho }_0\mathbf{v}=0,{\rho }_{0}i\omega \mathbf{v}+{\mathbf{v}}_0\cdot \nabla \mathbf{v}+{\rho }_0\mathbf{v}\cdot \nabla {\mathbf{v}}_0+{\rho }_0\mathbf{v}\cdot \nabla {\mathbf{v}}_0+\nabla p=0,\\ i\omega p+{\mathbf{v}}_0\cdot \nabla p+\mathbf{v}\cdot \nabla p_0+\gamma p_0\nabla \cdot \mathbf{v}+\gamma p\nabla \cdot {\mathbf{v}}_0=S_e,\end{array}$$where the subscript “0” indicates the equilibrium quantities for the acoustic perturbation problem, evaluated from the aerodynamic solution. The acoustic perturbations emitted by the heat release travel onto an aerodynamic background flow characterized by a Mach value associated with a uniform free stream of values $M_{\infty }\in 0.1;0.3$. The aerodynamic field is generated under the assumption of a compressible potential flow by the presence of the SHO. The source term $S_e$ in (8) is modelled in the frequency domain as in [38]$$\begin{array}{}\text{(9)}& S_e=\gamma -1{\dot{\vartheta }}^{\prime }=\gamma -1f\mathbf{x}{\vartheta }_p\omega ,\end{array}$$with$$\begin{array}{c}\text{(10)}& f\mathbf{x}=\frac{1}{{\sqrt{\pi b}}^3}\exp -\frac{{\mathbf{x}-{\mathbf{x}}_0}^2}{b^2},{\vartheta }_p\omega =c_0^{2}P\frac{{\tau }_h}{\sqrt{\ln 4}}\exp i\omega {\tau }_L+3{\tau }_h\exp -\frac{{\tau }_h^2}{4\ln 2}{\omega }^2,\end{array}$$where $P={10}^{4}W$ is the laser intensity, ${\tau }_h={10}^{-6}s$ is the half-length of the Gaussian distribution, ${\tau }_L$ is the activation time of the laser set to $0s$, $\omega$ is the frequency of the monochromatic acoustic wave, $b$ is the spatial amplitude of the Gaussian distribution set to $0.025m$, and ${\mathbf{x}}_0=0;7.5R_{\text{cloak}}$ is the source location. This analytical model provides a full description of the pressure pulses generated by a laser energy deposition in the medium experimentally used in aeroacoustic wind tunnel testing as reported in [38]. A first set of simulations is dedicated to assessing the Gaussian pulse source, comparing it to a distributed compact monopolar energy source in the Helmholtz number range $2.93<{\text{kR}}_{\text{cloak}}<5.86$. After that, the metacontinuum formulation is studied using the benchmark of the aeroacoustic scattering cancellation, for which the shape of the tensor ${\mathbf{W}}_C$ in (7) is given both from the background flow characteristics (through $\mathbf{\Lambda }$) and from the acoustic mirage’s coordinate transformation. For numerical implementation simplicity, the assumption of anisotropic inertia is used in defining the static components of $W^{\mu \nu }$ in (2), i.e., defining the specific $\widehat{\mathbf{\rho }},\mathbf{S},$ and $\widehat{K}$. Specifically, in (2), the components $W^{\mu \nu }$ are set as in [8]. In this way, the specific $\widehat{\mathbf{\rho }},\mathbf{S},$ and $\widehat{K}$ assume the form$$\begin{array}{c}\text{(11)}& \widehat{\mathbf{\rho }}=\begin{array}{cc}\frac{r}{r-R_{\text{body}}}& 0\\ 0& \frac{r-R_{\text{body}}}{r}\end{array},\mathbf{S}=\mathbf{I},\widehat{K}={\frac{R_{\text{cloak}}-R_{\text{body}}}{R_{\text{cloak}}}}^2\frac{r}{r-R_{\text{body}}},\end{array}$$being $r=\mathbf{x}$. This choice does not limit the potentiality of the aforementioned spacetime method for the adaptation of metadevices to convective applications. Following Norris’ metacontinuum model, in fact, it is possible to achieve acoustic invisibility also by moving the anisotropic feature to both the inertia and the bulk modulus or even only to the latter. The configuration with anisotropic inertia is known as inertial cloaking (IC) and it is largely investigated by the research community with a vast reference bibliography associated.

[figure(s) omitted; refer to PDF]

To guarantee the continuity of the acoustic information propagation across the boundary separating the conventional (in the hosting fluid domain) and unconventional (in the metacontinuum domain) media, the boundary conditions are imposed at the domains interface (see Figure 1(b)) similarly to [31].$$\begin{array}{c}\text{(12)}& p_h=p_c,\widehat{\mathbf{n}}\cdot \frac{1}{{\rho }_0}{\widehat{\mathbf{\rho }}}_c^{-1}\nabla p_c-\mathbf{\alpha }{p}_c=\frac{\nabla p_h}{{\rho }_0}\cdot \widehat{\mathbf{n}}-\mathbf{\alpha }\cdot \widehat{\mathbf{n}}\frac{p_c}{{\rho }_0},\hspace{1em}\mathbf{x}\in \Gamma \end{array}$$with $\mathbf{\alpha }=-{\widehat{\mathbf{\rho }}}^{-1}{\mathbf{v}}_0$. Considering the difficulties in defining a proper Neumann condition for the communication of different continuum models in the presence of convection, the main criteria for the selection of the normal acceleration continuity in (12) are to restore a flow condition analogous to its equivalent static case. Moreover, to avoid any reflection on the outer boundary $R_{\text{ext}}$, a nonreflecting boundary condition (asymptotic far-field radiation condition) is imposed to reproduce the free space propagation in the FEM implementation. This analysis is limited to a single frequency, ${\text{kR}}_{\text{cloak}}=4$.

4. Numerical Results

This section is dedicated to the numerical results obtained with the numerical setup shown in Section 3. First, a check of the consistency of the noise emitted in the free field by the laser-generated source model compared with a homogeneous distributed monopole source is needed. The sound pressure levels emitted by the two sources are evaluated in correspondence with three virtual probes arbitrarily chosen in the computational domain for a range of frequencies associated with a Helmholtz number in the range $2.93<{\text{kR}}_{\text{cloack}}<5.86$. The probes are located at points $x_1=0;-3,x_2=-1.5;0$ and $x_3=0,3$. The results are compared in Figure 2 for each of the free-stream Mach numbers tested $M_{\infty }\in 0.0;0.3$, normalizing the levels with the highest value for each virtual probe. Some deviations are present in the acoustic fields generated by the two sources. However, it should not be surprising since the laser-beam analytical model introduces a noncompact acoustic source that is spatially truncated by its definition in a limited circular domain of radius $R_{\text{source}}/R_{\text{cloak}}=0.25$ while the distributed monopolar source requires a homogeneous distribution of the intensity in the entire source domain.

[figure(s) omitted; refer to PDF]

Once the consistency between the sound emissions of a monopole source and the heat release sound wave is confirmed, the value of the emission frequency for the aeroacoustic problem is set to correspond to the intermediate Helmholtz number of ${\text{kR}}_{\text{cloak}}=4$ to simulate the acoustic cloaking benchmark. A cylindrical obstacle is introduced in the domain, surrounded by a metacontinuum to implement acoustic invisibility. The parameters of the classic metacontinuum for quiescent fluids are set as in (11) and corrected through the spacetime framework presented. The acoustic pressure field obtained is shown in Figure 3 for $M_{\infty }=0.3$ to highlight the ability of the corrections to recover the performance of the IC when it operates in a moving medium. The residual scattering is abated from the application of ${\mathbf{\Lambda }}_{PG}$ and ${\mathbf{\Lambda }}_T$ as can be seen in Figures 3(c) and 3(d), respectively. The results are consistent with what was obtained in previous works [24], and therefore, the analytical corrections effectively adapt the metamaterial device for the convective application considered. A very small residual scattering can be seen when Taylor’s transformation is applied: the $O{M}_{\infty }$ approximated equation is assumed to be valid for Mach values less than 0.3. Therefore, this is a confirmation of the coherent behaviour of the analytical corrections.

[figure(s) omitted; refer to PDF]

With respect to other works where the hosting fluid is modelled as a potential fluid flow [24, 31–33] and the boundary conditions use the linearized, nonstationary Bernoulli relation between pressure and velocity potential, here the condition at the interface between the two media are set in terms of primitive variable, as shown in (12). This allows the reduction of the analytical and numerical approximation degree introduced in the simulations, aiming at more reliable results. However, despite this, the insertion loss (IL) parameter evaluation in Figures 4 and 5 still shows deviations from the ideal case, having defined the IL as$$\begin{array}{}\text{(13)}& IL=20{\log }_{10}\frac{p_i}{p_t},\end{array}$$with $p_i$ the incident pressure field and $p_t$ the total one. For perfect acoustic cloaking, the incident and total pressure at the probes should be identical, leading to a null IL. It is worth noting that for this kind of numerical simulation, careful attention is needed in the setup choices. In fact, the tested application not only involves the implementation of an unconventional model but also has to deal with the computational burden of coupling a linearized Euler model with such metacontinuum through “unconventional” boundary conditions. Therefore, this nonstandard analysis required careful attention to the numerical setup choices with the verification of the effective minimization of the numerical error in the simulations. Specifically, the attention must be higher, mainly on the mesh handling, if one desires to also implement the Gaussian laser-beam source analytical model: not only the computational effort required notably increase but it also brings higher risks of spurious reflection at the boundaries. Furthermore, it is necessary to guarantee the symmetry of the mesh in the domain where the source is defined. In general, all the domains required a characteristic dimension far beyond the canonical $\lambda /10$. Specifically, the meshing is divided into three different regions: for $r<R_{\mathrm{cloak}}$, the source domain and the remaining hosting fluid where the average element dimension is $\lambda /45,\lambda /30$ and $\lambda /20$, respectively. Keeping this in mind, Figures 4 and 5 show the IL polar diagrams for the scattering cancellation benchmark evaluated in the near-field (NF) and far-field (FF), respectively. These are obtained with the calculation of the IL along two circumferential lines of virtual microphones of radius $R_{\text{mic}.\text{NF}}=1.5R_{\text{cloak}}$ for the NF and $R_{\text{mic}.\text{FF}}=20R_{\text{cloak}}$ for the FF. Since acoustic cloaking aims at scattering cancellation, the reference case for the IL evaluation is the free-field propagation of acoustic perturbations $p_i$.

[figure(s) omitted; refer to PDF]

Figures 4 and 5 show that both ${\mathbf{\Lambda }}_{PG}$ (red curve) and ${\mathbf{\Lambda }}_T$ (blue curve) are recovering the performance decay of the statically designed metamaterial (magenta curve), as expected.

In general, the maximum deviation from the reference case appears at $M_{\infty }=0.3$: for the near-field evaluations in Figure 4(c), a 1 dB deviation appears for the IC corrected with ${\mathbf{\Lambda }}_{PG}$ and approximately 2 dB for the IC-adapted design with ${\mathbf{\Lambda }}_T$. This discrepancy between the behaviour of the two corrections in disfavour of Taylor’s transformation is not surprising since the condition $M_{\infty }=0.3$ approaches and exceeds the validity limit $M^2\ll 1$ of the latter in the presence of a circular obstacle, where the maximum local $M$ exceeds 0.4. Meanwhile, the statically designed IC still has a maximum deviation that exceeds 3 dB. The residual presence of scattering in all configurations where the metadevice is applied could be due to the limitations of the coordinate transformation to cope with convective effects. In fact, the introduction of a nonperfect spacetime coordinate transformation implies the introduction of analytical error, which is sensitive to the variation of characteristic parameters for aeronautical problems (such as the relative position between source, body, background flow, and the intensity of the latter) [33]. Therefore, even when smooth IL curves are achieved, the deviations seen could not be entirely ascribed to the presence of numerical noise but also to the analytical approximation introduced. Finally, regarding the far-field polar diagrams in Figure 5, it is possible to see that the approximation introduced by both the $\mathbf{\Lambda }$ seems to affect less the performances in far-field evaluations showing the effective improvement of the analytically corrected metadevices with respect to the statically designed ones. In any case, the choice of a high-fidelity model and its computational effort is rewarded by an improved consistency of the solutions (compared to previous works) and by the extension of the aeroacoustic spacetime framework moving toward more realistic and actual applications.

For a more complete characterization, the residual scattering is quantified by evaluating the parameter $\sigma$ defined as$$\begin{array}{}\text{(14)}& \sigma =\frac{1}{2\pi R_{\text{mic}}}{\int }_C\frac{p_i-p_t}{p_i}\mathrm{d}l,\end{array}$$obtained as the integral of the difference between the incident pressure field $p_i$ and the total one $p_t$, evaluated along the circumferences $C$ of virtual microphones of radii $R_{\text{mic}.\text{NF}}$ and $R_{\text{mic}.\text{FF}}$. It represents the acoustic pressure field fraction of the residual scattering to the incident field, being $p_i-p_t=p_s$ the resulting scattered pressure field: in the case of a perfect cloaking $\sigma =0$; whereas, if there is a performance lack of the device, it generates a $\sigma \ne 0$. Figure 6 shows the trends of $\sigma$ to the variation of the background flow intensity for each configuration considered both for the near field (Figure 6(a)) and the far field (Figure 6(b)). For low $M_{\infty }$, the cloaking device has a very low residual scattering for all the configurations considered since it is a case near to the static applications (correspondent to a $M_{\infty }=0$ condition) whereas, with the Mach increasing, the performance decay becomes evident. This behaviour is likely due to the higher influence of the convective terms in the background flow the acoustic perturbance is propagating onto. As expected, the static design performance decay is partially recovered by both the spacetime-corrected designs in the near-field and far-field evaluations. As said, the incomplete recovery of the performance of the metacontinuum device is due to the analytical approximation introduced, and the performance gain obtained through the spacetime analytic corrections is evident and satisfactory also when coupled with the LEE model in the hosting medium. It is worth emphasizing one more time that this result is the main achievement of the present work and confirms the validity of the approach suggested by the authors in a simulation context much more accurate than that used so far, and paves the way to an extensive campaign of validation of different metacontinuum devices in wind tunnel experiments using the laser excitation.

[figure(s) omitted; refer to PDF]

5. Conclusions

In the present work, the analysis of the coupling of the linearized Euler equations and Norris’ metacontinuum model is presented. The aeroacoustic invisibility benchmark is used involving the application of aeroacoustic spacetime analytical corrections for the adaptation of statically designed acoustic metamaterial to convective applications. Specifically, the acoustic mirage is implemented through an inertial cloaking design corrected with Taylor’s or Prandtl–Glauert’s coordinate transformations. The choice of this specific test case does not limit the validity of the spacetime approach since the analytic corrections depend solely on the background flow characteristics. Therefore, all the considerations made are generally valid and independent of the specific acoustic mirage implemented. The LEE allows the imposition of the boundary conditions at the interface in terms of primitive variables leading, in principle, to a reduction of the analytical approximation degree with respect to previous works. Furthermore, this high-fidelity framework also permits the use of a model of a realistic laser-generated sound source. This is analytically modelled as a heat release spatially modulated with a Gaussian-beam shape and it is fully representative of a real sound source experimentally used in aeroacoustic wind tunnel testing reported in the literature. Its usage in the present work is considered a step towards more realistic simulations of possible experiments of applied aeroacoustic metacontinua. The simulations presented in this paper bring additional complexity to the previous state of the art due to the increase of their computational cost and the coupling of the two different physics (LEE and metacontinuum), requiring special attention in the definition of the numerical setup to avoid undesired and nonphysical numerical errors. This residual scattering is quantified by evaluating the insertion loss and the $\sigma$ parameter as merit factors: numerical results show that the spacetime correction of the statically designed acoustic cloaking produces the recovery of its performances in the presence of a background flow. However, discrepancies from the ideal behaviour in the benchmark are expected, and both the IL and $\sigma$ are nonzero values that get higher with the Mach increasing: these trends are due to the limited capability of the aeroacoustic corrections to account for the effects of convection. In conclusion, the results presented in the paper confirm the effectiveness of the approach proposed by the authors when a more sophisticated model of the acoustics in the host is used. The assessment of the coupling with a LEE model makes possible the planning of an extensive campaign of validation of the simulation results with experiments conducted with laser acoustic sources. It also opens the possibility to analyse operating conditions characterized by higher Mach number, when the background flow may exhibit high levels of diffused vorticity or even shock waves. Furthermore, it can be also foreseen a further improvement of the external aeroacoustics modelling by adopting the linearized Navier–Stokes equations to model the effects of the flow viscosity and the presence of a boundary layer, not considered so far, that can have a not negligible impact on the device performances and design.

Acknowledgments

This work was partially supported by internal resources and the European Commission through Project AERIALIST (AdvancEd aicRaft-noIse-AlLeviation devIceS using meTamaterials), Grant Agreement no. 723367 until May 31, 2020.

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