# Evolution of Flow and Streaming in Exponential Variable Cross-Section Resonators
## 1 Introduction  
Owing to its high reliability and environmental friendliness, thermoacoustic engines without moving parts have attracted increasing attention as a major technology to develop more efficient energy conversing or generating systems [1; 2]. The working gas in thermoacoustic engines is characteristic of the alternating current (AC) gas flow, which alternately moves backward and forward with pulsating pressure at certain frequencies. Until recently, the thermoacoustic conversion efficiency had been continually improving with both novel configurations and insight into the physics of flow and heat transfer in resonators [3; 4]. The geometry of the resonator is a crucial factor in determining the non-linear standing pressure waveform and amplitude generated within it. If the shape of a resonator is a straight cylinder, a shock wave accompanied by acoustic saturation will appear when the inner gas oscillates at its resonance frequency [5]. In recent years, researchers have been focusing on developing measures and approaches to suppress these non-linearities [3; 4]. It is well-known that one can obtain high-amplitude and shock-free acoustic oscillations by varying the shape of resonators. This method was first presented by [PERSON] et al. [6] and is now called resonant macrosonic synthesis (RMS). [PERSON] et al. [7]and [PERSON] et al. [8] applied variable cross-sectional area resonators to thermoacoustic devices, i.e., energy-focused heat engines or acoustic pressure amplifiers, and got a higher pressure ratio and better performances. Optimization is often involved in engineering applications. The optimal shape of variable cross-sectional area resonators is investigated by mathematical modeling and numerical simulations of RMS. To optimize the axisymmetric resonator shape by maximizing the pressure compression ratio, [PERSON] et al. [9] solved the one-dimensional linear wave equation, considering the boundary-layer dissipation numerically; the optimal resonator shapes were obtained in case of a piston, shaker, or loudspeaker driving. [PERSON] et al. [10, 11] presented an approximate formula to calculate the natural frequency, pressure, pressure ratio, and amplification ratio of the pressure amplitude for some axisymmetric resonators and suggested the optimal shape parameters for these resonators. Additionally, the streaming in the resonator is an important non-linear flow and heat-transfer phenomenon, which greatly affects the performance of thermoacoustic systems. Suppressing the acoustic streaming or DC gas flow in thermoacoustic engines is also an important measure that could improve the thermoacoustic conversion efficiency [12]. Many studies have been conducted on the utilization of tubes with varying cross-sections to suppress the acoustic streaming [13, 14, 15]. [PERSON] and [PERSON] [13] showed that a tapered tube can suppress the secondary streaming; compared with cylindrical and double-tapered tubes, pulse tubes with optimal taper angles can improve the cooling power more significantly. [PERSON] et al. [14] performed numerical simulations and experiments with tapered tubes and achieved results consistent with those of [PERSON] and [PERSON]. [PERSON] et al. [15] showed that increasing the taper angle can decrease the enthalpy flow loss owing to the mass streaming. Numerical simulation is an alternative method to investigate the fluid flows and heat transfer in different resonator configurations. However, theoretical and numerical analyses on the flow and streaming features inside variable cross-sectional resonators have been rarely reported. So far, considerable investigations on the shaped resonator have been based on the one-dimensional model. [PERSON] [16], [PERSON] [17], and [PERSON] [18] solved their one-dimensional models and obtained satisfying results by using the finite difference method, the Galerkin method, and the finite element method, respectively. By applying the Galerkin method, [PERSON] et al.