We present a numerical framework to predict the impedance of acoustic liners subject to a grazing mean flow using the inverse Helmholtz solver, which is a novel computational technique that can evaluate the broadband acoustic impedance at the open surface of any given geometry (e.g. a cavity) for a given spatial distribution of pressure at the same surface. The latter is a problem-dependent closure condition to the iHS that needs to be supplied externally. The iHS relies on a unstructured spectral-element-based discretization of the geometry of the cavity and it assumes (in the current version) linearity of the acoustics. The iHS does not solve the governing equations for the overlying flow but only focuses on the cavity itself. To verify the technique, we have performed companion two-dimensional laminar fully compressible Navier-Stokes simulations of flow over (and inside) the acoustic liner analyzed by Tarn et al. in JSV 2014. These fully resolved calculations serve as a reference since they provide the 'true' impedance of the liner directly extracted via Fourier analysis of the pressure and velocity data at the mouth of the cavities. We investigate three different centerline Mach numbers (0.0, 0.3, and 0.85) at two different frequencies of excitation (1750 Hz and 3000 Hz) with three different amplitudes of perturbations (120 dB, 140 dB, and 160 dB). Results therefore range from linear acoustic regime with low mean shear to nonlinear acoustic amplitudes and high shear. We observe that, in the linear regime, the proposed model is able to obtain liner impedances that are sufficiently close to those educed directly from the fully resolved and coupled Navier-Stokes calculations, even with approximate pressure closure conditions derived from either analytical or simple numerical duct acoustic models. It is noted that the spatial distribution of the impedance at the mouth of the cavities is, in fact, very dependent on the velocity induced by the pressure fluctuations. The velocity field at the mouth of the cavities, in fact, closely follows the local geometrical details of the cavity itself, which are captured by the iHS due to its unstructured nature. The iHS results, in fact, capture the overshoot in admittance magnitude near solid, viscous edges (as in the Stokes 2nd problem) as well as the streamwise gradient in the real part of admittance, which is connected to grazing flow effects. Future steps involve accounting for nonlinear acoustic wave amplitudes and flows with higher mean shear, which are not captured by the current version of the iHS framework.
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