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We have carried out axisymmetric numerical simulations of a spatially developing hypersonic boundary layer over a sharp 7• -half-angle cone at M∞ = 7.5 inspired by the experimental investigations by Wagner (2015). Simulations are first performed with impermeable (or solid) walls with a one-time broadband pulse excitation applied upstream to determine the most convectively-amplified frequencies resulting in the range 260kHz -400kHz, consistent with experimental observations of second-mode instability waves. Subsequently, we introduce harmonic disturbances via continuous periodic suction and blowing at 270kHz and 350kHz. For each of these forcing frequencies complex impedance boundary conditions (IBC), modeling the acoustic response of two different carbon/carbon (C/C) ultrasonically absorptive porous surfaces, are applied at the wall. The IBCs are derived as an output of a pore-scale aeroacoustic analysis -the inverse Helmholtz Solver (iHS) -which is able to return the broadband real and imaginary components of the surface-averaged impedance. The introduction of the IBCs in all cases leads to a significant attenuation of the harmonically-forced second-mode wave. In particular, we observe a higher attenuation rate of the introduced waves with frequency of 350kHz in comparison with 270kHz, and, along with the iHS impedance results, we establish that the C/C surfaces absorb acoustic energy more effectively at higher frequencies.

We present a numerical method to determine the complex acoustic impedance at the open surface of an arbitrarily shaped cavity, associated to an impinging planar acoustic wave with a given wavenumber vector and frequency. We have achieved this by developing the first inverse Helmholtz Solver (iHS), which implicitly reconstructs the complex acoustic waveform-at a given frequency-up to the unknown impedance boundary, hereby providing the spatial distribution of impedance as a result of the calculation for that given frequency. We show that the algebraic closure conditions required by the inverse Helmholtz problem are physically related to the assignment of the spatial distribution of the pressure phase over the unknown impedance boundary. The iHS is embarassingly parallelizable over the frequency domain allowing for the straightforward determination of the full broadband impedance at every point of the target boundary. In this paper, we restrict our analysis to two-dimensions only. We first validate our results against Rott's quasi one-dimensional thermoacoustic theory for viscid and inviscid constant-area rectangular ducts, test our iHS in a fully unstructured fashion with a geometrically complex cavity, and finally, present a simplified, two-dimensional analysis of a sample of carbon-carbon ultrasonically absorptive coatings (C/C UACs) manufactured in DLR-Stuttgart, and used in the hypersonic transition delay experiments by Wagner et al. in AIAA 2012-5865. The final goal is to model C/C UACs with multi-oscillator Time Domain Impedance Boundary Conditions (TDIBC) developed by Lin et al. in JFM (2016) to be applied in direct numerical simulations (DNS) focused on the overlying boundary layer, eliminating the need to simultaneously resolve the microscopic porous structure of the C/C UACs.

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Assigning boundary conditions, such as acoustic impedance, to the frequency domain thermoviscous wave equations (TWE), derived from the linearized Navier-Stokes equations (LNSE) poses a Helmholtz problem, solution to which yields a discrete set of complex eigenfunctions and eigenvalue pairs. The proposed method -the inverse Helmholtz solver (iHS) -reverses such procedure by returning the value of acoustic impedance at one or more unknown impedance boundaries (IBs) of a given domain, via spatial integration of the TWE for a given real-valued frequency with assigned conditions on other boundaries. The iHS procedure is applied to a second-order spatial discretization of the TWEs on an unstructured staggered grid arrangement. Only the momentum equation is extended to the center of each IB face where pressure and velocity components are co-located and treated as unknowns. The iHS is finally closed via assignment of the surface gradient of pressure phase over the IBs, corresponding to assigning the shape of the acoustic waveform at the IB. The iHS procedure can be carried out independently for different frequencies, making it embarrassingly parallel, and able to return the complete broadband complex impedance distribution at the IBs in any desired frequency range to arbitrary numerical precision. The iHS approach is first validated against Rott's theory for viscous rectangular and circular ducts. The impedance of a toy porous cavity with a complex geometry is then reconstructed and validated with companion fully compressible unstructured Navier-Stokes simulations resolving the cavity geometry. Verification against one-dimensional impedance test tube calculations based on time-domain impedance boundary conditions (TDIBC) is also carried out. Finally, results from a preliminary analysis of a thermoacoustically unstable cavity are presented.