**5. Conclusions**

To sum up, we use numerical simulation method to study the offset characteristics of a bubble plume in a bubble column, in which momentum exchange induced by gas–liquid interaction leads to bubble offset. The detailed flow field shows that the characteristics of bubble offset need time to fully develop. It is similar to the period of bubble oscillation, and also presents periodic range changes. In addition, from the height surface (Y-coordinates unchanged) of the maximum offset position, the offset characteristic exhibits periodic fluctuations in each section. The vortices are mainly concentrated around the bubble flow and on the wall in the domain. Furthermore, we apply the maximum offset position to construct the quantitative analysis of the offset characteristics of the bubble plume, and successfully draw the correlations between the dimensionless offset distance and angle in the oscillation characteristics of different flux and aspect ratio. The fitting degree is good as well. With the increase of volume flux, due to the limited area space, the dimensionless offset distance has an upper limit, so that the value will not change after 1, and the offset angle gradually increases; with the increase of aspect ratio, the dimensionless offset distance shows the same trend, but the angle presents a decreasing trend when the volume flow is large, and other situations fluctuate greatly without obvious regularity. Considering the effect of a limited closed region on the offset of bubble plume, the conditions without oscillation period are found as well.

A series of closure models are applied to the case of bubbly flow, especially to examine bubble plume oscillation based on previous works, with a focus on the dynamic bubble plume process. The simulation results agree quite well with the experimental data, including the velocity profile and POP. Some of the observed deviations could occur because a fixed bubble diameter was used in the simulations, and the change in bubble size was neglected. The models can be further optimized, because the resulting effects are not ye<sup>t</sup> truly reliable. The dynamic details of the bubble plume, including the velocity, volume fraction, vortex intensity and dynamic bubbly flow variations, are observed in detail. It is found that the column is not a real three-dimensional model because its depth is very small. Therefore, the intermediate plane (Z = 0) is chosen to describe the dynamic behavior of bubble oscillation. Furthermore, the influence of the aspect ratio and gas volume flux on the oscillation characteristics is examined based on our simulation results, and qualitative fitting correlations are provided. It is noteworthy that when the dimensionless offset distance (η) is 1, no POP is observed. In other words, POP does not occur in bubble columns at a high gas volume flux and aspect ratio. A more detailed investigation of the specific conditions resulting in the absence of POP has a grea<sup>t</sup> potential. Additional research on bubble plume oscillation is required for the development of accurate bubble interaction and turbulence models. The gas–liquid mixing and mass transfer can be enhanced around the maximum offset position. Gas–liquid mixing is mainly concentrated in the middle region. Furthermore, the maximum offset position is related to the aspect ratio and volume flux. These should have a wide application range in the chemical industry.

**Author Contributions:** Y.C.: Software, Investigation, Writing—original draft preparation. Q.Z.: Resources, Data curation, Visualization. P.J.: Supervision, Project administration. K.Z.: Validation, Formal analysis. W.W.: Conceptualization, Methodology, Funding acquisition, Writing—review and editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Natural Science Foundation of China, gran<sup>t</sup> number 51709210 and 51679178.

**Conflicts of Interest:** The authors declare no conflict of interest.
