Numerical simulation on bubble deformation and forces at moderate Reynolds number using front-tracking algorithm coupled with proportional-integral-derivative controller

Bubbles play a crucial role in various industrial reactors and natural processes, garnering intense research interest, while the understanding of their deformation and interaction with surrounding liquid is still not enough. This study combines the front tracking (FT) method with a proportional-integral-derivative (PID) controller to achieve precise bubble fixation, enabling an in-depth analysis of bubble dynamics and steady-state variations at moderate Reynolds number (Re), including shape, surface characteristics, drag forces, and surrounding flow fields under various liquid conditions. For the first time, the form drag is separated from the skin one by means of direct numerical simulation. The results demonstrate good agreements with empirical equations from the literature for aspect ratios and drag coefficients, validating the effectiveness of the method in replicating bubble forces and deformations. As Re increases, high-velocity regions on the sides of the bubble exhibit accelerated velocities, and the low-velocity region at the rear gradually narrows. Bubble aspect ratio and sphericity vary significantly under different liquid densities, while viscosity notably affects the time required for bubble shape stabilization. The front surface area of a bubble remains consistent across various liquid conditions, while the projected area is more relevant for drag force calculations. Bubbles with larger aspect ratios show smaller projected areas and vice versa. Tangential and normal forces on the bubble surface vary with the circumferential angle, with the maximum near the front center and minimum near 90 degrees. The decreasing and increasing trend around the surface depend on fluid conditions. At moderate Re, form drag is predominant, with a ratio of approximately 7:3 between form and friction drag. Increasing the Re leads to an increase in the proportion of form drag, e.g., by increasing density or decreasing viscosity, but the former is found more effective, although the latter affects the total drag more significantly. It is interesting to notice that the proportion of form drag increases with the surface tension coefficient as well, even if it hardly changes the total drag. Finally, the surface force integration method is shown to be superior to the widely used force balance method for calculating drag coefficients.