将下面进行学术翻译：图9给出了玻璃珠和FCC颗粒在扩径段不同高度上的功率谱密度分布。从图中可以看出所有径向位置上两种颗粒的功率谱振幅均随着床高的增加而降低在扩径段底部振幅最大当高度增加到H2和H3时功率谱振幅大幅降低。这是由于气固相互作用沿轴向减弱从而造成扩径段上部的能量波动小。而且功率谱主峰随着高度增加逐渐变得模糊在H3高度时已观察不到明显的主峰存在。上述现象说明气固两相流动在扩径段内沿轴向发展

Figure 9 presents the power spectral density distributions of glass beads and FCC particles at different heights in the expansion section. It can be observed that the power spectral amplitudes of both particles decrease with the increase of bed height at all radial positions. The amplitude is largest at the bottom of the expansion section, and decreases significantly at heights H2 and H3. This is due to the weakening of the gas-solid interaction along the axial direction, resulting in smaller energy fluctuations in the upper part of the expansion section. Moreover, the main peak of the power spectrum gradually becomes blurred with increasing height, and no obvious main peak is observed at height H3. These phenomena indicate the weakened periodic motion of gas-solid two-phase flow along the axial direction in the expansion section, which may be attributed to the gradual smoothing of the flow as the turbulent energy dissipation increases with height. The variation of power spectral curves radially at different heights is also consistent, with the periodic characteristics of power spectra decreasing in strength from the wall towards the center of the expansion section. It is also noteworthy that the power spectral fluctuation range of glass beads is larger than that of FCC particles at both the wall and center regions. The power spectral amplitude of glass beads is significantly higher than that of FCC particles at the bottom of the expansion section, and the main peak is more pronounced, indicating that the increase in particle density enhances the periodic flow in this region of the expansion section. In the center of the pipe, the power spectral amplitude of glass beads is lower than that of FCC particles at the bottom of the expansion section, indicating different effects of increasing particle density on the periodic flow process radially. As the height increases, the differences in power spectra between the two types of particles decrease.