Toroidal bubbles

Yasuda, A., H. Takahira, Numerical analysis of the dynamics of toroidal bubbles considering the heat transfer of internal gas, JSME Int. J. Ser. B 46, 600-609 (2003). 

Walters, J.K. and J.F. Davidson, 1%3, The initial motion of a gas bubble formed in an inviscid liquid. Part 2. The three-dimensional bubble and the toroidal bubble. J. Fluid Mech. 17,321-336. 

Geometrical symmetry requires that the boundary conditions as well as the equations are symmetrical. Problems including gravity can only be symmetrical in a cylinder with gravity in the axial direction. Reduction to a ID or 2D problem may also change the physical appearance, e g. bubbles do not exist in axisymmetric 2D except on the symmetry axis they become toroid in 3D elsewhere. A toroid bubble moving in a radial direction must alter the diameter to maintain the volume, and the forces around the bubble will be unphysical. 

Owing to the high computational load, it is tempting to assume rotational symmetry to reduce to 2D simulations. However, the symmetrical axis is a wall in the simulations that allows slip but no transport across it. The flow in bubble columns or bubbling fluidized beds is never steady, but instead oscillates everywhere, including across the center of the reactor. Consequently, a 2D rotational symmetry representation is never accurate for these reactors. A second problem with axis symmetry is that the bubbles formed in a bubbling fluidized bed are simulated as toroids and the mass balance for the bubble will be problematic when the bubble moves in a radial direction. It is also problematic to calculate the void fraction with these models. 

Bhaga (B3) determined the fluid motion in wakes using hydrogen bubble tracers. Closed wakes were shown to contain a toroidal vortex with its core in the horizontal plane where the wake has its widest cross section. The core diameter is about 70% of the maximum wake diameter, similar to a Hill s spherical vortex. When the base of the fluid particle is indented, the toroidal motion extends into the indentation. Liquid within the closed wake moves considerably more slowly relative to the drop or bubble than the terminal velocity Uj, If a skirt forms, the basic toroidal motion in the wake is still present (see Fig. 8.5), but the strength of the vortex is reduced. Momentum considerations require that there be a velocity defect behind closed wakes and this accounts for the tail observed by some workers (S5). Crabtree and Bridgwater (C8) and Bhaga (B3) measured the velocity decay and drift in the far wake region. 

MASS TRANSFER TO DROPS AND BUBBLES. When small drops of liquid are falling through a gas, surface tension tends to make the drops nearly spherical, and the coefiBcients for mass transfer to the drop surface are often quite close to those for solid spheres. The shear caused by the fluid moving past the drop surface, however, sets up toroidal circulation currents in the drop that decrease the resistance to mass transfer both inside and outside the drop. The extent of the change depends on the ratio of the viscosities of the internal and external fluids and on the presence or absence of substances such as surfactants that concentrate at the interface. ... 

Benjamin and Ellis predicted that a ring vortex would emerge from the jet flow (a phenomenon observed by Lauterborn), which leads to the formation of a bubble ring from the toroidally deformed bubble. ... 

On a solid surface, the effect of the liquid jet depends on the ratio y = d/R, where d is the distance between the bubble and the surface, and R the maximum bubble diameter. When y is lower than 0.3, the bubble becomes toroidal in shape, and the jet propagates at a velocity estimated to 200m s"i and the erosion is maximum, even on hard materials such as steel. Activated pits are formed, and the particles ejected should react more efficiently than the bulk solid due to the larger active surface area. The collapse is associated with a shock wave, and pressures as high as several tens of kbars are produced locally. 

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