In order to establish design guidelines for an acoustic compressor`s resonator, it is necessary to investigate nonlinear oscillations of gas in an axisymmeric closed tube at its resonance. The relationship between the tube shapes and the pressure waveforms has to be also studied.
When the acoustic resonance occurs in the tube, acoustic variables such as density, velocity, and pressure undergo very large perturbation, often described as nonlinear oscillation. In general, this nonlinear motions cause a portion of input energy at resonant frequency to leak into its higher harmonics, so that the waveforms become distorted from a pure sinusoidal form. Specially, in a cylindrical tube, shock waves are often generated and propagate periodically back and forth if interior gas is excited with its resonant frequency. This nonlinear phenomenon has been extensively investigated theoretically and experimentally. Recent studies, done by MacroSonix Corp., revealed that the waveform is strongly related to the tube shape, and it is possible to reduce the nonlinearity of the waveform by modifying the tube shape. This RMS(Resonant Macrosonic Syntesis) technique makes it possible to get extremely high acoustic pressure without shock formation in the resonant tube. In addition, the feasibility of an acoustic compressor as a commercial pump or compressor was also demonstrated.
In this paper, the nonlinear acoustic phenomena in the resonant tube such as harmonics generation, shock formation, resonant frequency shift, etc., are extensively studied for various tube shapes. First, in order to observe distortion patterns of nonlinear pressure waves in different shape tubes, experiments were accomplished for a cylindrical tube and a conical tube with varying the amplitude of driving acceleration. They were filled with air or R-134a gas of different static pressure. In the cylindrical tube, shocks were generated in spite of small amplitude of driving acceleration, and compression ratio was very low because the energy leakage into higher harmonics caused the acoustical saturation. On the other hand, in the conical tube filled with 5atm mixture gas (air and R-134a), very high pressure difference about 11 atm(peak-to-peak) was obtained when the tube was driven with acceleration of 50g.
In order to analyze these nonlinear acoustic phenomena effectively, it needs to derive a nonlinear analytical model for the interior acoustic field and develop numerical codes for solving nonlinear governing equations. If a wavelength is much larger than the maximum diameter of a tube, the interior acoustic field can be modeled as one-dimensional form since plane waves dominate in the tube. At both ends of the tube, `characteristic boundary conditions` are implemented in order to set boundary conditions properly without any kind of approximate extrapolation. The governing equations show that the interior acoustic field is strongly affected by the term of $\frac{1}{S}(\frac{dS}{dx})$, where S is cross sectional area. In addition, inherent nonlinearity in fluids can be estimated by the specific heat ratio γ as well as the parameter of nonlinearity $\frac{B}{A}$. If these values increase, the inherent nonlinearity of acoustics increases, too. The nonlinear governing equations are solved numerically by using a higher-order finite difference scheme, which consists of a fourth-order central difference method for spatial differentiation and a fourth-order Runge-Kutta method for time evolution. Numerical simulations were accomplished for various tube shapes such as cylindrical, conical, $\frac{1}{2}$ cosine, $\frac{3}{4}$ cosine, and exponential horn tubes with the same volume and length each other. Under the identical conditions, the $\frac{1}{2}$ cosine shape tube shows much higher compression ratio and much smaller distortion of pressure waveforms than the other tubes, so that it seems to be more suitable to the acoustic compressor.
In addition, the axisymmetric two-dimensional model is also developed in order to describe much more realistic wave motion in the axisymmetic resonant tube. Similar to one-dimensional analysis, several numerical simulations were performed for various tube shapes such as cylindrical, conical, $\frac{1}{2}$ cosine, and $\frac{3}{4}$ cosine tubes with the same volume and length each other. The results well agree to those of one-dimensional analysis, and show that the one-dimensional simple model analysis is quite useful and reasonable in practice.
A perturbation theory is also used to simply estimate the patterns of nonlinear resonant phenomena according to the tube shape and to do physical interpretation of them. The results are well consistent with those of fully nonlinear analysis. The design guidelines of the resonant tube to get high acoustic pressures are proposed by using the design parameters such as `input effort,` `second harmonics ratio,` and `tube surface area.
