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

Abstract

In this paper we develop a novel ray solver for the time-harmonic linearized Euler equations used to predict high-frequency flow–acoustic interaction effects from point sources in subsonic mean jet flows. The solver incorporates solutions to three generic ray problems found in free-space flows: the multiplicity of rays at a receiver point, propagation of complex rays and unphysical divergences at caustics. We show that these respective problems can be overcome by an appropriate boundary value reformulation of the nonlinear ray equations, a bifurcation-theory-inspired complex continuation, and an appeal to the uniform functions of catastrophe theory. The effectiveness of the solver is demonstrated for sources embedded in isothermal parallel and spreading jets, with the fields generated containing a wide variety of caustic structures. Solutions are presented across a large range of receiver angles in the far field, both downstream, where evanescent complex rays generate the cone of silence, and upstream, where multiple real rays are organized about a newly observed cusp caustic. The stability of the caustics is verified for both jets by their persistence under parametric changes of the flow and source. We show the continuation of these caustics as surfaces into the near field is complicated due to a dense caustic network, featuring a chain of locally hyperbolic umbilic caustics, generated by the tangency of rays as they are channelled upstream within the jet.

Aeroacoustic catastrophes: upstream cusp beaming in Lilley's equation

“…We can locate the caustic structures that organize these diffraction patterns by using the ray tracing apparatus of [11,12]. The caustics, where the ray amplitude is singular, can be located by calculating the loci of points where the ray Jacobians vanish.…”

“…This provides closure to the large θ ray contributions missing from [6], and describes the rays' behaviour from an analytical perspective. The ray trajectories are numerically integrated solutions of the initial value problem in Cartesian coordinates (x, y, z) as shown in [11,12] and dp where I is the unit matrix, ⊗ denotes an outer product and ∇ x is the gradient operator in (x, y, z).…”

“…Alternatively, the Green function may be described using ray theory or geometrical acoustics [9,10]. Recent developments have shown that one can now seek full-scale ray solutions to the Green function field in a generic flow [11,12], including the important effects of complex rays [13]. The trade-off between a computationally efficient symmetric WKB expression and a full-asymmetric solution is not automatically required [11,12].…”

“…Recent developments have shown that one can now seek full-scale ray solutions to the Green function field in a generic flow [11,12], including the important effects of complex rays [13]. The trade-off between a computationally efficient symmetric WKB expression and a full-asymmetric solution is not automatically required [11,12]. In those papers, we provide a practical numerical algorithm to calculate acoustic fields in the presence of generic subsonic flows (symmetric or asymmetric), using both real and complex rays.…”

“…Near to the loci of the peak acoustic field, the ray approximations generate a singular, divergent amplitude. This is due to the tangency of pairs of rays emitted from the point source [15] causing the vanishing of a Jacobian [11,12] in the denominator of the local approximation. The effective boundary of the cone of silence is thus a fold caustic of the rays, being the simplest member of the catastrophe hierarchy of bifurcation sets ( [15], [16], ch.…”