Nonlinear resonant oscillations and shock waves generated between two coaxial cylinders

Abstract

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 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 , and then the amplitude of the oscillation reaches O(M 1/3 ). In this case the amplitude of gas oscillation is modulated periodically with a period of O(M −2/3 ). When is small ( ≈ 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.

Nonlinear analysis of periodic modulation in resonances of cylindrical and spherical acoustic standing waves

“…Recently, we have carried out the numerical analysis of resonance of cylindrical standing waves in an ideal gas between two coaxial cylinders, where the wave motion is excited from an initial quiescent gas by a harmonic oscillation of the radius of outer cylinder. 13 We have found that the shock free resonance is accompanied by a periodic modulation in amplitude and phase from the numerical solutions of the systems of Euler and Navier-Stokes equations. We have also shown that the period of modulation is approximately proportional to M −2/3 and it is a decreasing function of the radius ratio of the inner and outer cylinders.…”

“…[10][11][12][13] This is because the excitation of higher harmonics required for shock formation is impeded by the fact that the resonance frequency of each nth mode ͑n =2,3, ...͒ is not equal to the n multiple of that of the fundamental mode in cylindrical and spherical standing waves, as sometimes called dissonant. 14 As a result, the maximum wave amplitude in the shock free resonances of cylindrical and spherical standing waves attains O͑M 1/3 ͒, [10][11][12][13] which can be considerably large for M Ӷ 1 compared to the case of plane waves. The dissonant effect has been utilized for realization of large amplitude and shock free resonant oscillations.…”

“…We have also shown that the period of modulation is approximately proportional to M −2/3 and it is a decreasing function of the radius ratio of the inner and outer cylinders. 13 Furthermore, a critical acoustic Mach number, above which a shock wave is formed, has been determined numerically. 13 In the present paper, we perform a theoretical analysis of the modulation of shock free resonances of cylindrical standing waves in two coaxial cylinders and spherical standing waves in two concentric spheres.…”

“…13 Furthermore, a critical acoustic Mach number, above which a shock wave is formed, has been determined numerically. 13 In the present paper, we perform a theoretical analysis of the modulation of shock free resonances of cylindrical standing waves in two coaxial cylinders and spherical standing waves in two concentric spheres. The problem setup of two coaxial cylinders and two concentric spheres effectively works to address the geometrical effect on the resonances by changing the radius ratio.…”

“…In addition to a characteristic time scale −1 , where is an angular frequency of oscillation of radius of the outer cylinder or sphere, we introduce a slow time scale ⑀ −2 −1 inspired by the result of the previous numerical study. 13 Thus, the analysis is accomplished with the method of multiple scales 17,18 up to the third order, and a cubic nonlinear equation for complex wave amplitude is derived. Making use of the amplitude equation and its first integral, we explore several features of the modulation phenomenon.…”