# Evolution of Flow and Streaming in Exponential Variable Cross-Section Resonators
## 4 Results and Discussion
### Effect of Driving Strength
#### 4.3.2 Transient Flow Fields  
Figures 8 and 9 show the transient flow fields in terms of streamline at typical moments (\(\theta~{}=~{}0\), \(\pi/2\), \(\pi\), and \(3\pi/2\).) during one acoustic cycle in the \(m=2.2\) exponential tube for Cases A and C, respectively. The color contours indicate the distribution of the temperature (K), and the reference velocity is the \(u\) velocity (m/s). Considering the symmetry, only the top half part of the flow fields is depicted here. The evolution processes of the flow and streaming for different driving strengths are very different, which is the most important and interesting point. The flow direction and phase shown in these figures change with increasing driving amplitude, because the fundamental modal frequency of the exponential resonator increases with increasing driving strength [10]. The figures in Figure 8 indicate that the flow fields for Case A at \(\theta~{}=~{}0\) and \(\theta~{}=~{}\pi\) are more complicated than those at the other two moments in one acoustic cycle; a vortex pair is formed on either side of the tube where it is enlarged and contracted. It appears that the vortices generate and vanish periodically. Vortices are accompanied by a larger temperature gradient and a smaller velocity gradient at \(\theta~{}=~{}0\) and \(\theta~{}=~{}\pi\), but a nearly uniform temperature field and greater velocity fluctuation, as well as smoother streamline, can been observed at \(\theta~{}=~{}\pi/2\) and \(\theta~{}=3\pi/2\). The streaming that was superposed on the main flow developed owing to the generation, evolution, and shedding of vortices, which is referred to as the third type of direct current (DC) flow in Reference [12]. The DC flow is driven by the asymmetry of the fluid flow and temperature and transmitted by the vortices, and it accompanies energy storing and releasing in a cycle. The flow regime for Case C shown in Figure 9 is different from that for Case A in Figure 8. For example, the flow fields in Figure 9 have a greater velocity gradient and smoother streamlines at \(\theta~{}=~{}0\) and \(\theta~{}=~{}\pi\), and more varied temperature fields at \(\theta~{}=~{}\pi/2\) and \(\theta~{}=~{}3\pi/2\), which indicates a greater minor loss and energy dissipation; the vortex only appears at the large end of the resonator at certain instants (\(\theta~{}=~{}0\)) and is accompanied by the biggest temperature variation in one cycle, because the pressure reaches its maximum at this moment in one cycle. The vortex shedding corresponds to an increase in the difference of pressure amplitude between the accelerating and decelerating phases, so that the cycle symmetry of the flow field is broken. The asymmetrical and non-smooth pressure and velocity fluctuation (see Figure 6) in the resonator is one of the causes of the scale and shape of the vortices, and the vortex propagation speed contributes to the position of the vortices, so the streaming flow and the process of energy storage and release in Figure 9 appear different from those in Figure 8.  
Figure 7: Temperature variation under different driving strengths at (\(x=0\), \(r(x)=0\)) in the \(m=2.2\) exponential resonator. Figure 8: Instantaneous flow fields in the \(m\) = 2.2 exponential resonator under Case A, during one acoustic cycle at four different times: (**a**) \(\theta~{}=~{}0\), (**b**) \(\theta~{}=~{}\pi/2\), (**c**) \(\theta~{}=~{}\pi\), and (**d**) \(\theta~{}=~{}3\pi/2\). The color contours indicate the temperature distributions (K); the reference velocity is the \(u\) velocity (m/s). Figure 9: Instantaneous flow fields in the \(m\) = 2.2 exponential resonator under Case C, during one acoustic cycle at four different times: (**a**) \(\theta~{}=~{}0\), (**b**) \(\theta~{}=~{}\pi/2\), (**c**) \(\theta~{}=~{}\pi\), and (**d**) \(\theta~{}=~{}3\pi/2\). The color contours indicate the temperature distributions (K); the reference velocity is the \(u\) velocity (m/s).