1. Investigation of capillary waves on the surface of Taylor bubble propagating in vertical tubes By Dan Liberzon Under the supervision of: Prof. Dvora Barnea & Prof. Lev Shemer Department of Fluid Mechanics and Heat Transfer, Tel Aviv University

6. Short bubble in 44mm diameter pipe Rising in stagnant water

9. Taylor bubble Translational velocity Liquid holdup in the film Drift velocity Momentum equation on the liquid film, B.C. Cross-sectional average velocity in the liquid film Barnea (1990)

10. Bottom oscillations as wave maker Polonsky (1998)

11. Pure capillary waves Dispersion relation of capillary waves traveling on water. λ – Wave length T – Water-air surface tension f – Wave frequency Wave – current interaction should be taken in to consideration Waves are traveling on vertical surface:

14. Frequency sensitivity

18. Ensemble average Averaging bin

20. The results, stagnant water

21. Results for non-zero Reynolds numbers

22. Waves Dissipation The group velocity C relates the spatial and the temporal wave amplitude decay rate. Amplitude variation of the pure-capillary wave subjected to the viscous dissipation

23. Waves Dissipation No waves shorter than 0.5 mm were present, causing the shift in the average values The ensemble is the bubbles rising in 26 mm diameter pipe inside stagnant water . The red curve is the Gaussian distribution with mean at 0.9 mm .

24. Waves Propagation

26. New facilities 5 m

28. Waves Breaking Capillary wave steepness In our case the steepness did not exceed S =0.5 The critical steepness for capillary waves on clean water is S =0.730 , Crapper (1957)

29. Short bubble in 26mm diameter pipe rising in stagnant water

30. Liquid film velocity D.E. B.C. Cross-sectional average velocity in the liquid film

33. Transition criterion, Wallis (1969) : Ø 44 mm, stagnant water Ø 14 mm, stagnant water

34. CFD Results Comparison Stagnant water

35. Laser

36. Predicted waves lengths on bubbles rising in stagnant water

37. Kelvin-Helmholtz instability Convection of the wave in the x direction

40. Bottom oscillations Open sheet circular basin: Potential: -- Sloshing oscillations frequency -- Rotating frequency Sir Horace Lamb, 1879

41. Rotating bottom oscillations

42. Kelvin-Helmholtz instability KH instability range upper bound K=2.13 [rad/m], λ=2.95 [m] F(k) Long waves mode K=0, λ  ∞ Most unstable mode No KH instability KH instability range

43. Sloshing frequency calculations From the disperse relation: B.C. :