Resonant oscillations in open axisymmetric tubes

Abstract

We study the behaviour of the isentropic flow of a gas in both a straight tube of constant cross section and a cone, open at one end and forced at or near resonance at the other. A continuous transition between these configurations is provided through the introduction of a geometric parameter k associated with the opening angle of the cone where the tube corresponds to $k=0$. The primary objective is to find long-time resonant and near-resonant approximate solutions for the open tube, i.e. $k\rightarrow 0$. Detailed analysis for both the tube and cone in the limit of small forcing $(O(\varepsilon ^{3}))$ is carried out, where $\varepsilon ^{3}$ is the Mach number of the forcing function and the resulting flow has Mach number $O(\varepsilon )$. The resulting approximate solutions are compared with full numerical simulations. Interesting distinctions between the cone and the tube emerge. Depending on the damping and detuning, the responses for the tube are continuous and of $O(\varepsilon )$. In the case of the cone, the resonant response involves an amplification of the fundamental resonant mode, usually called the dominant first-mode approximation. However, higher modes must be included for the tube to account for the nonlinear generation of higher-order resonances. Bridging these distinct solution behaviours is a transition layer of $O(\varepsilon ^{2})$ in k. It is found that an appropriately truncated set of modes provides the requisite modal approximation, again comparing well to numerical simulations.