Nonlinear resonant oscillations in closed tubes of variable cross-section

Abstract

An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations in an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation – the amplitude–frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).

Resonant oscillations of an inhomogeneous gas between concentric spheres

“…It also yields the result for the experimentally important case of the cone, see Lawrenson et al (1998) and Mortell & Seymour (2004).…”

“…This phenomenon was termed resonant macrosonic synthesis (RMS). Following a number of numerical investigations of the governing equations (Ilinskii et al 1998Bednarik & Cervenka 2000;Chun & Kim 2000), Mortell & Seymour (2004) provided the first analytical (as distinct from numerical) explanation for these experiments, reproducing in broad terms the experimental findings. The time scale for the evolution of the amplitude is t = 3 2 t, where 3 3 is the small dimensionless Mach number.…”

“…To solve the nonlinear problem defined by equations (2.6) and (2.9)-(2.11) for a continuous output, the following perturbation expansion is proposed (see Mortell & Seymour 2004) …”

“…The eigenvalue equation for a cone, as in the experiments in Lawrenson et al (1998), also emerges. Then the assumption that the flow in the cone is quasi-one-dimensional (see Ilinskii et al 1998;Mortell & Seymour 2004) is not necessary as it is a radial flow in a segment of a spherical shell. Thus, the new results from the flow of an inhomogeneous gas in a spherical shell also provide a synthesis to a range of previous disparate results.…”

Resonant hydraulic sloshing in a tank of variable depth

“…shockless) pressure output. Mortell and Seymour [6] gave a theoretical explanation of that phenomenon. Here we focus on the role of geometry in the response of a similar resonant system and show that when the bottom topography of a closed tank varies a hydraulic jump can be obviated and the fluid motion remains continuous.…”