Nonlinear standing waves and acoustic streaming in an axial-symmetrical resonator with exponentially varying cross-sectional area were studied. A two-dimensional gas-kinetic Bhatnagar-Gross-Krook scheme based on the non-structure triangular grid was established to simulate nonlinear acoustic oscillations in the resonator. Details of the transient and steady flow fields and streaming were developed. The effects of winding index of the exponential-shape resonator, the displacement amplitude of the acoustic piston on the streaming, and the vortex pattern were analyzed. The results demonstrate that the acoustic streaming pattern in such resonators is different from the typical Rayleigh flow in a constant cross-sectional area resonator. No obvious shock wave appeared inside the exponential-shape resonator. The comparison reveals that with increasing the displacement amplitude of the acoustic piston energy dissipation is accompanied by vortex break-up from a first-level to a second-level transition, and even into turbulent flow. This research demonstrates that the exponential-shape resonator, especially that with a winding index of 2.2 exhibits better acoustic features and suppression effects on shock-wave, acoustic streaming, and the vortex.

The temporal-spatial evolution of gas oscillation was simulated by gas kinetic scheme (GKS) in a cylindrical resonator, driven by a piston at one end and rigidly closed at the other end. Periodic shock waves propagating back and forth were observed in the resonator under finite amplitude of gas oscillation. The studied results demonstrated that the acoustic pressure is a saw-tooth waveform and the oscillatory velocity is a square waveform at the central position of the resonant tube. Moreover, it was found by harmonic analysis that there was no presence of obvious feature for pressure node in such a typical standing wave resonator, and the distribution of acoustic fields displayed a one-dimensional feature for the acoustic pressure while a quasi-one-dimensional form for oscillatory velocity, which demonstrated the nonlinear effects. The simulation results for axial distribution of acoustic intensity showed a good consistency with the published experimental data in the open literature domain, which provides a verification for the effectiveness of the GKS model proposed. The influence of displacement amplitude of the driving piston on the formation of shock wave was numerically investigated, and the simulated results revealed the cascade process of harmonic wave energy from the fundamental wave to higher harmonics. In addition, this study found that the acoustic intensity at the driving end of the resonant tube would increase linearly with the displacement amplitude of the piston due to nonlinear effects, rather than the exponential variation by linear theory. This research demonstrates that the GKS model is strongly capable of simulating nonlinear acoustic problems.

The temporal-spatial evolution of the sound field in a cylindrical resonator driven by a piston was simulated by gas-kinetic scheme (GKS), and periodic shock waves propagating back and forth were observed in the resonator with finite displacement amplitude of the driving piston. It can been seen from the instantaneous spatial distribution of the oscillatory pressure and velocity in the resonator that the oscillatory pressure is a one-dimensional,but the oscillatory velocity exhibits a quasi-one-dimensional distribution due to the effect of the boundary layer.The evolution process of shock wave from the sinusoidal wave to the zigzag wave was demonstrated with the increasing of the driving displacement amplitude of the piston. Especially, simulation results revealed that the acoustic intensity was linearly proportional to the driving displacement amplitude of the piston, rather than the square type. The studied results demonstrated well the process of harmonic energy cascade by the FFT analysis, at the same time, the physical explanations has been also given. The simulation results of acoustic intensity were well consistent with the existing experimental results, which verifies the validity of GKS modeling method.