Convective correction of metafluid devices based on Taylor transformation

The paper presents a method to modify the mechanical properties of a statically--designed metamaterial device in order to preserve its performance when operating in a non--uniform aerodynamic flow. The objective of the research is to contribute to the disclosure of the acoustic metamaterial potential in aeroacoustic applications, where acoustic propagation and scattering are deeply affected by aerodynamic convection. The emphasis is on aeronautical applications aiming at the development of methods for the design of breakthrough solutions for aircraft noise mitigation. The present approach is based on the application of the inverse Taylor transformation to the static design space in order to obtain the properties of an equivalent metamaterial in the convected space. The metamaterial so--obtained is capable to guarantee the target acoustic response in presence of a non--uniform background flow at low Mach number and with negligible vorticity. Numerical results obtained through finite--element simulations are presented for a free stream Mach number $\leq 0.35$, which is compatible with the take--off and landing operation of a commercial airplane. The benchmark used is the widely assessed problem of the scattering cancellation (cloaking) of a circular obstacle. The acoustic disturbance is here assumed to be generated by an isotropic point source located in the vicinity of the scatterer and co--moving with it. The numerical results reveal that the effect of the proposed correction strongly depends on the relative position of the source and the metamaterial device, with the worst performance obtained when they are aligned with the free stream. The reason of this dependence is analytically explained and verified numerically. When the alignment of the source and the treated object is orthogonal to the flow, the Taylor--corrected metamaterial recovers almost completely its expected behaviour. This result makes the approach appealing for all those engineering applications where the relative position of sources and moving boundaries is fixed and compatible with the most favourable conditions.