Well‐posedness of a generalized time‐harmonic transport equation for acoustics in flow

Abstract

We study a generalized time‐harmonic transport equation, which appears in the Goldstein equations and allows us to model the acoustic radiation in a flow. We investigate the well‐posedness of this transport problem. The result will be established under the assumption of a Ω‐filling flow, which, in 2D, is simply equivalent to a flow that does not vanish. The approach relies on the method of characteristics, which leads to the resolution of the transport equation along the streamlines, and on general results of functional analysis. The theoretical results are illustrated with numerical results obtained with a Streamline Upwind Petrov‐Galerkin finite element scheme.

Mathematical analysis of Goldstein’s model for time-harmonic acoustics in flows

“…As already seen in Section 3.1, a sufficient condition (and probably necessary) for the solvability of ( 14)(ii) is that the flow is Ω-filling (Definition 3.1). This was demonstrated in [9] and will be recalled (and extended) in Section 3.4.2. Thanks to ( 23), Goldstein's problem (14,15) is rewritten as the following "modified" convected Helmholtz problem governing the only unknown ϕ:…”

“…In dimension 2, d = 2, there exists a particularly simple characterization of Ω-filling flows provided that Ω is simply connected. The result, proven in [9], is linked to Brouwer and Poincaré-Bendixson theorems [19], exploits the fact that, roughly speaking, the existence of periodic orbits implies the existence of a stopping point. The precise statement is the following…”

“…In this work, we prove the existence and uniqueness of solutions to Goldstein's equations for subsonic Ω-filling flows satisfying an additional condition similar to the one in [15] but more explicit (in particular easy to check) and this condition can be moreover interpreted as a low vorticity condition. From the methodological point of view, our method can be seen more as a modification of the analysis made in [17] for the potential model (this is another advantage of Goldstein's model) and uses in an essential manner our previous work on the time-harmonic transport equation [9] where the Ω-filling condition, already introduced in [4] for the scalar stationary transport equation, plays a fundamental role.…”

Finite Elements III

“…Remark 61.16 (Literature). We refer the reader to Devinatz et al [104], Azerad [18], Ayuso and Marini [17], Deuring et al [103], Cantin [79], Cantin and Ern [80], Bensalah et al [30] for further results on divergence-free advection. Exercise 61.2 (Advection-diffusion, 1D).…”

Compressible fluid modes in rigid ellipsoids: towards modal acoustic velocimetry

“…Acousticians have attempted to extend the acoustic equation, to account for vortical perturbations (Goldstein 1978) and vortical mean flows U * 0 (Bergliaffa et al 2004). The resulting equations bear the name of (extended) Goldstein equations (Bensalah, Joly & Mercier 2018). They have for unknown the velocity perturbation, written as u * 1 = ∇Φ * 1 + ζ * 1 where ζ * 1 is a vortical hydrodynamic contribution.…”