Statistics Evaluation of Terminal Oscillations of Rising Bubble Shapes, Paths and Velocities

The bubble, immediately after its formation, behaves like an axisymmetric body moving straight upward. Respective simplifications of CFD calculations may be used to compute its shape and motion. An asymptote, i.e. the steady state terminal shape and the steady state velocity of the bubble represents unfortunately a singular solution for the axisymmetric case. According to the experimental experience, this may be adequate solely for the low-Reynolds number systems, i.e. either for small, nearly spherical bubbles, or for several larger bubbles in high-viscosity liquids.

More frequently, medium size bubbles are met in chemical reactors and in industrial mixing and separation processes.

Actually, the trajectory of larger bubble (with the Eötvös number Eo>1, where Eo≡d²ρ g/σ) rising in low-viscosity liquids is not a straight vertical line. It has apparently helical or zigzag pattern. Prosperetti noted that such behavior of common bubbles in water had been mentioned yet in the notebooks of Leonardo da Vinci and introduced the term „Leonardo’s paradox“.

Such movement cannot be predicted by standard CFD software. However few top-level two-phase flow research groups mastered sophisticated modified methods how to approach to the unsteady three-dimensional movement of deformable bubbles. Nevertheless, the programming is too complex, respective computation is quite time demanding and adequacy of the calculated results may be still questionable. Therefore, experimental investigation seems to be the most efficient way to the understanding the problem.

Our experimental project was targeted to systematic study of single bubble or small group of bubbles rising in liquid column. To be able to observe bubbles for considerably long period, experimental setup with levitating bubbles was suggested. Here, liquid is pumped downward through the entrance region of the conical diverging channel. It was proved that with exception of a narrow boundary layer, the plug liquid flow occurs in the channel. When the volume flow rate is properly adjusted, bubbles remain in definite space of the channel. Vertical position of the bubble can be used for determination of the bubble rising velocity and its changes are connected with its acceleration during the oscillations. The channel is placed in the square vessel to eliminate an optical distortion, and two perpendicular bubble projections are observed simultaneously. The bubble shape and position is recorded by high speed camera Olympus allowed a period 60 seconds to be recorded at the frequency 150 frames per second.

Image analysis by tailor-made program provides time series of quantities related to the position, size, and shape of the bubbles.

Systematic experiments were carried out with various liquids with aim to elucidate the effect of viscosity and surface tension. Particular results were obtained also with non-Newtonian liquids. Possible effects of electrolytes were investigated as well.

Small bubbles (Eo<1) are essentially spherical, their rising is quite straightforward, and is subjected to classical theories of Stokes and Hadamard – Rybczinski. Shape of the large bubbles (Eo>20) oscillates irregularly and the bubbles are subjected to the breakup very fast, while a coalescence occurs quite rarely. The effect of liquid properties on the bubble half-life was a subject of our previous papers.

Therefore, the medium size bubbles with 1<Eo<20, classified usually as the ellipsoidal ones, are the most frequent in industrial gas-liquid processes, and our present research has been oriented to this class of bubbles in low- and moderate-viscosity liquids (Re>30, where Re≡u d ρ/μ).

Bubble size is usually characterized by the diameter d of equivalent volume sphere. The best one parameter approximation of the shape uses aspect ratio a/b of semi-axes of equivalent oblate ellipsoid. From the statistics of two perpendicular bubble projections, we concluded that (2 a/d)² = 1 + 0.095 Eo^0.75. Nevertheless, actually bottom of part of the bubbles is a little bulkiest, which is valid for inclined wobbling bubbles as well. The difference is small, but statistically meaningful, and it depends on the bubble inclination. On this bases a theoretical “dumbbell model” has been developed, explaining probable mechanism of the bubble wobbling and of bubble path zigzagging.

The autocorrelation and cross-correlation analysis of the bubble path and bubble shape prove, that the rising bubbles are all the time oriented toward the direction of motion by their lower semi-axis. The hydraulic resistance is therefore controlled by the maximum front area that is πa². Then, we have developed a successful prediction of the average bubble rising velocity, u, on the basis of the drag coefficient defined as C_D ≡ (Δρ g π d³/6)/[½ ρ u² π(2 a)²]. Such a drag coefficient is simply an universal function C_D(Re) of the Reynolds number, and the only effect of Eötvös number is implicitly included in the determination of a.

Other goal of autocorrelation analysis has been determination of the frequency of bubble shape and path oscillation. This frequency is usually 5 10 Hz and it is practically independent on viscosity (or Reynolds number). A modified Strouhal number was suggested which is only slightly dependent on Eo. The suggested correlation fits well also all the literature data available.

Statistical evaluation of horizontal velocity and acceleration enables us to determine also the amplitude and pattern of the bubble wobbling. The amplitude is usually within the range 0.4 d‑0.8 d, and the wobbling pattern is significantly zig-zag at Eo<5, while the helical motion prevails at Eo>10.

Recently we have continued in investigation of interactions in a group of rising bubbles and their ordering in a liquid column.

Generous support of the present paper by EC project ICT no. ED2.1.00/03.0082 is gratefully acknowledged.