A controllable canonical form implementation of time domain impedance boundary conditions for broadband aeroacoustic computation

Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations

“…In all of these studies, the IBC is linear and the Cauchy problem is formulated in the frequency domain. Although less popular than their time-harmonic counterparts, time-domain IBCs (TDIBCs) have been used in wave propagation problems including duct aeroacoustics [19][20][21][22], room acoustics [23], as well as outdoor sound propagation [24]. Richter et al [25,26] and Troian et al [27] identified an IBC in the time domain.…”

“…First, broadband discrete impedance models can be built from the so-called oscillatory-diffusive representation of physical models [5]. Second, a convenient way to compute a causal convolution is to recast it into a set of ordinary differential equations (ODEs) (or, in the sense of systems theory, to realize the corresponding LTI operator [41, § 17] [42, § 1.2]), a method sometimes named "auxiliary differential equations" [20,43], "canonical form implementation" [21,22], or "state-space model" [32] in the acoustics or fluid dynamics literature. In this way, the computation of a (in)finite-dimensional hereditary operator is reduced to the integration of a (in)finite set of time-local differential equations, which can be done with arbitrary accuracy.…”

Delayed-time domain impedance boundary conditions (D-TDIBC)

“…based on the modeling of a transfer function as a rational polynomial. The same modeling procedure was used by Zhong et al [13] to model the impedance. The transfer function represents the broadband reflection coefficient.…”

“…(10) does, in principle, allow to impose a complex reflection coefficient in the time domain; however, this results in rapidly unsustainable memory and CPU costs, especially in DNS/LES simulations. Building upon the methodologies of Fung and Ju [1,2], Scalo et al [3] and Lin et al [4], we approximate a generic target reflection coefficient R(ω) with the reflection coefficient of the boundary condition R BC expressed as a sum of rational functions: (13) where R BC (ω, n 0 ) is the reflection coefficient model and the poles and residues (p k , µ k ) must come as n 0 conjugate pairs:…”

“…in time domain solvers fall under the category of Time Domain Impedance Boundary Conditions (TDIBC). While existing TDIBC methods [1][2][3][4][9][10][11][12][13][14] have demonstrated their ability to achieve such goal, one case has not yet be considered in the Computational Aeroacoustics (CAA) community: a pure time delay. Developing Delayed-Time Domain Impedance Boundary Conditions (D-TDIBC), i.e.…”