Asymptotic solutions for shocked resonant acoustic oscillations between concentric spheres and coaxial cylinders

Abstract

For resonant oscillations of a gas in a straight tube with a closed end, shocks form and all harmonics are generated, see Chester [“Resonant oscillations in a closed tube,” J. Fluid Mech. 18, 44 (1964)]10.1017/S0022112064000040. When the gas is confined between two concentric spheres or coaxial cylinders, the radially symmetric resonant oscillations may be continuous or shocked. For a fixed small Mach number of the input, the flow is continuous for sufficiently small L, defined as the ratio of the inner radius to the difference of the radii, see Seymour et al. [“Resonant oscillations of an inhomogeneous gas between concentric spheres,” Proc. R. Soc. London, Ser. A 467, 2149 (2011)]10.1098/rspa.2010.0576. However, shocks appear in the resonant flow for either larger values of L or larger input Mach number. A nonlinear geometric acoustics approximation is used to analyse the shocked motion of the gas when L ≫ 1. This approximation and the exact numerical solution are compared for the shocked wave profiles and shock strengths, and the approximation is valid for surprisingly small values of L. The flow in the plane wave case for a straight tube is recovered in the limit L → ∞ for both the spherical and cylindrical cases, providing a check on the results. The shocked solutions given here complement those continuous solutions previously derived from a dominant first mode approximation.

On the transition between continuous and shocked solutions for resonant gas oscillations in the frustum of a closed cone

“…Seymour et al. (2012) provided an estimate for the parametric range of the transition in the case of a spherical annulus and for a specific forcing amplitude. More recently Amundsen, Mortell & Seymour (2017) considered the analogous transition in the context of an open axisymmetric tube.…”

“…The same authors (Seymour et al. 2012) considered the case of nonlinear resonant oscillations of a gas contained between two concentric spheres or coaxial cylinders when shocks are present and the dominant first harmonic approximation is not valid. A nonlinear geometric acoustics approximation predicted shocked flows that are confirmed by numerical solutions for a range of – the slope of a cone.…”

Resonant Response in Bounded Nonlinear Wave Systems

“…Subsequently, Seymour et al [57] conducted research to describe shocked resonant oscillations between concentric spheres and coaxial cylinders when the dominant firstharmonic approximation is not valid. They explained how a nonlinear geometric acoustics approximation could predict shocked flows.…”

“…One of the most important questions is what happens if the dominant first harmonic approximation is not valid. Recently, Seymour et al [57] have shown that a nonlinear geometric acoustics approximation predicts shocked flows between concentric spheres and coaxial cylinders. They compared the results with full numerical solutions and found good agreement.…”

“…Figure 1from Amundsen[57] illustrating the comparison of numerical solution (solid) and geometric acoustic approximation (dashed) for gas in spherical shell L = 30, M = 0.005 for u(0.5, θ).…”