Nonlinear resonant oscillations and shock waves generated between two coaxial cylinders

Nonlinear evolution of a gas oscillation of large amplitude generated between two coaxial cylinders is numerically studied by solving the Euler equations for an inviscid ideal gas. The resonant oscillation is excited uniformly along the axis by harmonic oscillation of the radius of the outer cylinder with the fundamental resonance frequency. Nonlinear and geometrical effects are investigated by varying two nondimensional parameters, i.e., the acoustic Mach number M at the surface of the outer cylinder and the ratio α of the radius of the inner cylinder to that of the outer cylinder. The result shows that no shock wave is formed if M is smaller than a critical value for a given α, and then the amplitude of the oscillation reaches O(M1/3). In this case the amplitude of gas oscillation is modulated periodically with a period of O(M-2/3). When α is small (α≈0.1) and M is larger than the critical value, the waveform including a shock wave is quite different from that in plane wave resonance because of the geometrical effect. Furthermore, the soaring of temperature takes place in the vicinity of the inner cylinder owing to the shock heating.