Aeroacoustic catastrophes: upstream cusp beaming in Lilley's equation

Abstract

The downstream propagation of high-frequency acoustic waves from a point source in a subsonic jet obeying Lilley's equation is well known to be organized around the so-called 'cone of silence', a fold catastrophe across which the amplitude may be modelled uniformly using Airy functions. Here we show that acoustic waves not only unexpectedly propagate upstream, but also are organized at constant distance from the point source around a cusp catastrophe with amplitude modelled locally by the Pearcey function. Furthermore, the cone of silence is revealed to be a cross-section of a swallowtail catastrophe. One consequence of these discoveries is that the peak acoustic field upstream is not only structurally stable but also at a similar level to the known downstream field. The fine structure of the upstream cusp is blurred out by distributions of symmetric acoustic sources, but peak upstream acoustic beaming persists when asymmetries are introduced, from either arrays of discrete point sources or perturbed continuum ring source distributions. These results may pose interesting questions for future novel jet-aircraft engine designs where asymmetric source distributions arise.

Cones of silence, complex rays and catastrophes: high-frequency flow–acoustic interaction effects

“…Complex rays are extensions to the familiar real ray solutions, with the paths described by the ray equations lying on a surface in complexified space (see Chapman et al 1999). The complexification of rays facilitates the propagation of exponentially growing and decaying waves, the latter of which are critical for generating shadow zones, such as the cone of silence (see Goldstein 1982;Stone et al 2017;Hubbard 1991, chapter 5), and as we show, capturing the finer details of oscillatory behaviour in the point source field. However, one cannot realistically perform global searches by seeding a Newton method with a complex ray, as is possible with real rays.…”

“…There is clearly a discontinuity in the number of real rays crossing θ S that is not accountable to numerics but rather a physical effect of ray refraction (see § 7.3). Although we show in § 7.4 that the true local behaviour is governed by a catastrophe-like integral first described in Stone et al (2017) and that closer inspection reveals a complex ray generated caustic nearby, the uniform asymptotics of (6.6) does not incorporate this integral and we must proceed with a piecewise description according to figure 5. Now, using the caustic structures of figure 4a, we make a total ray calculation whose amplitude is shown in figure 4b.…”

“…The theoretical position of the line θ S in an isothermal parallel flow may be deduced from the local form (7.3) and the mappings fromξ to (ϕ, θ S ) shown in Stone et al (2017). As shown thereinξ…”

“…Although the framework of § 2 to § 6 may be applied more generally, as we show in § 8 for a spreading jet, such flows are a useful testbed for our programme, and have been used extensively as mean flow models of slowly spreading unheated jets (see Lilley 1958). To exemplify the efficiency of our ray approach in this section we carry out a parametric study over a range of off-axis source locations and Mach numbers, establishing the persistence and stability of the novel caustic structures first seen in Stone et al (2017).…”

“…figure 4). Here we have used the mappings defined in Stone et al (2017) to express the coefficientsξ as functions of (ϕ, θ). The mapping describes the caustic structure in the region ϕ = [0, π], with no loss of generality for describing the behaviour about the evaporation point in the region ϕ = [π, 2π].…”