Surface Impedance Determination via Numerical Resolution of the Inverse Helmholtz Problem

Abstract

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.

Trapped waves in supersonic and hypersonic turbulent channel flow over porous walls

“…Although few in number, there have been efforts to apply TDIBC to flow simulations to model the effect of porous coatings over hypersonic vehicles. For instance, Sousa et al (2019) applied the currently adopted TDIBC technique to simulate the effect of UAC in a Mach 7.5 cone flow, with a broadband impedance (Patel, Gupta & Scalo 2017) fitted to match benchtop acoustic absorption measurements of UAC samples. The simulations have demonstrated the ability of TDIBC to model UAC's acoustic absorption in hypersonic environments.…”

Numerical Investigation of Second Mode Attenuation over Carbon/Carbon Surfaces on a Sharp Slender Cone

“…The inverse Helmholtz solver (iHS) 41,42 From a high resolution image of the C/C block (figure 3), the edges of large surface are autoand their volumes meshed to be provided as an input geometry to the iHS, which evaluates the broadband impedance of each individual surface cavity. These impedances are then combined, noting that the admittance of the surrounding hard surface is zero, to yield the effective surface averaged impedance for the C/C block in consideration.…”