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Assigning boundary conditions, such as acoustic impedance, to the frequencydomain thermoviscous wave equations (TWE), derived from the linearizedNavier-Stokes equations (LNSE) poses a Helmholtz problem, solution to whichyields a discrete set of complex eigenfunctions and eigenvalue pairs. Theproposed method -- the inverse Helmholtz solver (iHS) -- reverses suchprocedure by returning the value of acoustic impedance at one or more unknownimpedance boundaries (IBs) of a given domain, via spatial integration of theTWE for a given real-valued frequency with assigned conditions on otherboundaries. The iHS procedure is applied to a second-order spatialdiscretization of the TWEs on an unstructured staggered grid arrangement. Onlythe momentum equation is extended to the center of each IB face where pressureand velocity components are co-located and treated as unknowns. The iHS isfinally closed via assignment of the surface gradient of pressure phase overthe IBs, corresponding to assigning the shape of the acoustic waveform at theIB. The iHS procedure can be carried out independently for differentfrequencies, making it embarrassingly parallel, and able to return the completebroadband complex impedance distribution at the IBs in any desired frequencyrange to arbitrary numerical precision. The iHS approach is first validatedagainst Rott's theory for viscous rectangular and circular ducts. The impedanceof a toy porous cavity with a complex geometry is then reconstructed andvalidated with companion fully compressible unstructured Navier-Stokessimulations resolving the cavity geometry. Verification against one-dimensionalimpedance test tube calculations based on time-domain impedance boundaryconditions (TDIBC) is also carried out. Finally, results from a preliminaryanalysis of a thermoacoustically unstable cavity are presented.

Surface Impedance Determination via Numerical Resolution of the Inverse Helmholtz Problem