# Characteristics and Mechanisms of the Zigzag and Spiral Movement of Rising Bubbles in Still Water
): [EMAIL_ADDRESS] (Y.Z. ): [EMAIL_ADDRESS] (Z.M. ): [EMAIL_ADDRESS] (M.S.F.) Zhiqiang Mu  
1 School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS] (H.L. ): [EMAIL_ADDRESS] (Y.Z. ): [EMAIL_ADDRESS] (Z.M. ): [EMAIL_ADDRESS] (M.S.F.) Zhiqiang Mu
1 School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS] (H.L. ): [EMAIL_ADDRESS] (Y.Z. ): [EMAIL_ADDRESS] (Z.M. ): [EMAIL_ADDRESS] (M.S.F.) Zhiqiang Mu
1 School of Energy and Power Engineering, Xi'an Jiaotong University, Xi'an 710049, China; [EMAIL_ADDRESS] (H.L. ): zyz0509@stu. The instability of the wake will directly lead to instability in the movement [5; 8; 9; 10]. When bubbles rise in still water, there is an abnormal phenomenon, that is, they rise along a zigzag or spiral trajectory [11; 12], which is related to the periodic oscillation of the Karman vortex street. This trajectory is not directly generated by the movement of the wake under the corresponding \(Re\) but is generated by the close coupling between the movement of the wake and the oscillating trajectory. The trajectory also affects the shedding of the trailing vortices, which together leads to the bubble movement [9; 10]. At present, bubble flow plays an important role in a wide range of geophysical and industrial processes, including two-phase heat exchangers, oil transportation, mixing in chemical reactors, nuclear reactor cooling, and alloy refining [13]. In the metallurgical process, bubbles are injected into the liquid metal to promote liquid movement, homogenize the physical and chemical properties of the melt, and refine the melt [14]. In industrial reaction catalysis, buoyant particles or bubbles are usually released to enhance mixing in the fluid [15; 16]. Therefore, research on bubble flow is significant for guiding engineering practices scientifically. [PERSON] [17] simulated the free rising process of bubbles in still water by the VOF (volume of fluid) method and found that the wake of bubbles presents three movement states: symmetrical falling, a transitional state, and periodic falling. The two critical \(Re\) values increase with an increase in the bubble diameter, and the bubble frequency of zigzag movement decreases with an increase in the bubble diameter. He [18] simulated the rising process of a single bubble under buoyancy by the Level Set method to explore the influence of the initial shape on the bubble movement and shape. The simulation results show that the final shape and rising velocity of the bubble are determined by the Eotvos number \(Eo\) and the Morton number \(Mo\). The \(Eo\) can be considered as the ratio of buoyancy to surface tension. The \(Mo\) can be considered as the ratio of viscous force to tension. They can be used to describe the appearance of bubbles or water droplets as they move in a fluid or continuous phase. \[Eo =\frac{g\Delta\rho l^{2}}{\sigma} \tag{1}\] \[Mo =\frac{g\mu_{c}^{4}\Delta\rho}{\rho_{c}^{2}\sigma^{3}} \tag{2}\]  
where \(g\) represents gravitational acceleration. \(\Delta\rho\) represents the density difference between the two fluids. \(l\) represents the characteristic length. \(\mu_{c}\) and \(\rho_{c}\) represent the dynamic viscosity and density of the surrounding fluid, respectively. \(\sigma\) represents the surface tension. When \(Eo<52\), the shape of bubbles are not affected by the initial shape. At a high \(Eo\) and low \(Mo\), the final shape of the bubble is dominated by the initial shape, showing a spherical cap shape and a ring shape. The larger the initial aspect ratio is, the easier it is for bubbles to develop into rings and the stronger the fluctuation in the rising velocity is. [PERSON] [19] revealed that the accumulated streamwise vorticity on the bubble interface is the reason for the triggering of zigzag movement through a direct numerical simulation (DNS).