Impedance Models in Time Domain, Including the Extended Helmholtz Resonator Model

Abstract

The problem of translating a frequency domain impedance boundary condition to time domain involves the Fourier transform of the impedance function. This requires extending the definition of the impedance not only to all real frequencies but to the whole complex frequency plane. Not any extension, however, is physically possible. The problem should remain causal, the variables real, and the wall passive. This leads to necessary conditions for an impedance function.Various methods of extending the impedance that are available in the literature are discussed. A most promising one is the so-called z-transform by Ozyoruk & Long, which is nothing but an impedance that is functionally dependent on a suitable complex exponent e −iωκ . By choosing κ a multiple of the time step of the numerical algorithm, this approach fits very well with the underlying numerics, because the impedance becomes in time domain a delta-comb function and gives thus an exact relation on the grid points.An impedance function is proposed which is based on the Helmholtz resonator model, called Extended Helmholtz Resonator Model. This has the advantage that relatively easily the mentioned necessary conditions can be satisfied in advance. At a given frequency, the impedance is made exactly equal to a given design value. Rules of thumb are derived to produce an impedance which varies only moderately in frequency near design conditions. An explicit solution is given of a pulse reflecting in time domain at a Helmholtz resonator impedance wall that provides some insight into the reflection problem in time domain and at the same time may act as an analytical test case for numerical implementations, like is presented at this conference by the companion paper AIAA-2006-2569 by N. Chevaugeon, J.-F. Remacle and X. Gallez.The problem of the instability, inherent with the Ingard-Myers limit with mean flow, is discussed. It is argued that this instability is not consistent with the assumptions of the Ingard-Myers limit and may well be suppressed. wherev = (v · n),v is the acoustic velocity vector and n denotes the normal vector of the wall that points into the wall. Pressure is scaled on ρ 0 c 2 0 , velocity on c 0 and impedance on ρ 0 c 0 , while we use throughout the e +iωt * Associate professor,

Numerical study of acoustic modes in ducted shear flow

“…These appear to be much more difficult than one might think on the face of it and require special investigation, Rienstra & Tester [14], Rienstra [15,16], Brambley & Peake [17]. Therefore, we decided to leave these issues outside the scope of the present work.…”

A full discrete dispersion analysis of time-domain simulations of acoustic liners with flow

“…Similar analyses could be performed to assess and compare new methods or to revisit existing methods and better understand their properties, including impedance models [36,28,33], implementations of the boundary conditions [21], and numerical schemes (such as finite element, finite volume or spectral methods).…”

“…This implementation of the impedance condition satisfies all the conditions given in [33]. The second auxiliary variable ξ corresponds to the displacement of the mass-spring-damper (this is also the displacement of the fluid).…”

“…Other popular impedance models include the Extended Helmholtz Resonator proposed by Rienstra [33], the three-parameter model by Tam and Auriault [36], and broadband impedance models [28]. They differ by the way they are formulated and implemented in the time domain, as well as their ability to represent realistic impedance functions.…”

“…This must be done in such a way that a number of basic physical principles such as causality are satisfied [33]. The implementation of the impedance depends strongly on the type of impedance functionẐ(ω) that is assumed.…”