Resonant oscillations of a gas in an open-ended tube

Abstract

An investigation is made of the oscillations near resonance induced in a gas contained in a tube that is open at one end and has an oscillating piston at the other. When all dissipative processes are neglected it is shown that there are discontinuities in the oscillations within a certain frequency band near resonance. The modifying effects of the following dissipative mechanisms are also investigated: (i) compressive viscosity; (ii) damping arising from radiation at the open end; (iii) the effect of the boundary layer at the wall of the tube; (iv) nonlinear dissipation such as might arise from eddy formation when separation occurs at the sharp lip of the open end. The main role of the first two mechanisms is to replace the discontinuities by sharp transitions and, more importantly, to determine their position. Under typical conditions it is possible for the boundary-layer effect to dominate the oscillations, which then differ substantially from those predicted by inviscid theory. In particular the oscillations are smooth throughout the whole range of frequencies. The oscillations are also considerably modified if a substantial proportion of the energy ejected at the open end is dissipated in eddy formation. A related problem is that of a plane wave approaching the open end from an arbitrary direction outside the tube. If the tube is closed at the other end, the boundary condition induced at the closed end by the disturbance transmitted into the tube simulates the effect of the piston. The two problems lend themselves to a similar analysis.