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Investigation of capillary waves on the surface of Taylor bubble propagating in
vertical tubes
Department of Fluid Mechanics and Heat Transfer, Tel Aviv University
By
Dan Liberzon
Under the supervision of:
Prof. Dvora Barnea & Prof. Lev Shemer
Scope of the presentation
• Taylor bubbles
• Wind/Wave, Wave/Current interaction
• Theoretical model
• Experimental research
• Future plans
Sir Geoffrey Ingram Taylor1886-1975
Taylor bubbles
• Oil / Gas industry:
Flow rates, monitoring equipment, pumps health/efficiency.
• Chemical industry:
Design / Exploitation of heaters, boilers, etc.
•Power plants
Boiling processes, heaters / coolants transport.
Understanding Taylor bubbles
• Generation / Creation
• Flow parameters
• Propagation
• Breaking
• Coalescence
• Theoretical models
• Numerical models
• Experimental research
Experimental facilities
• Nd: YAG laser producing short, sec pulse
• Ability to capture relatively fast motion
• Sharp air-water interface
810
Short bubble in 44mm diameter pipe
Rising in stagnant water
Waves characteristics
• Waves on a vertical water surface
• Capillary waves, few millimetres in length
• Absent on longer bubbles and at higher water Re numbers
• Wave length decreases on longer bubbles and at higher water Re numbers
• Shorter at the rear part, longer near the nose
Previous Studies
• Nigmatulin and Bonetto (1997): study observing the nature of capillary waves on
standing short Taylor bubbles. Suggested the presence of standing capillary waves
on the bubble interface. The waves amplitude seemed to increase on shorter
waves.
•Kockx et al. (2005): experiments conducted on elongated air Taylor bubble
standing inside a relatively wide pipe against opposite flowing water. Suggested
the presence of downward traveling capillary waves of equal length.
Waves on the Taylor bubble interface
Taylor bubble
driftmtr UCUU
mtrfilmtrf UU)UU(R
smeantr
filmfilmw
filmtr
L
G
film
RUU
|)x(U|)x(UfD
2
)x(UU
1g
dx
))x(U(d
Translational velocity
gD.Udrift 350
Liquid holdup in the film
Drift velocity
g)x(R)(A
)x(S)x(
dx
))x(UU(d))x(UU)(x(R fGL
wwfilmtrfilmtrfL
Momentum equation on the liquid film, Barnea (1990)
,UU,0x mfilm B.C.
)x(U filmCross-sectional average velocity in the liquid film
Bottom oscillations as wave maker
HzfHz 152
Polonsky (1998)
Wave – current interaction should be taken in to consideration
Pure capillary waves
Dispersion relation of capillary waves traveling on water.
23
2 )f2(=ρ
Tk+gk=
Waves are traveling on vertical surface:
f2,2
k,ea )tkx(i0
λ – Wave length
T – Water-air surface tension
f – Wave frequency
• X : frame of reference moving with the bubble
•X’: frame of reference moving with liquid film
Wave traveling in frame of reference X :
The same wave traveling in frame of reference X’:
Doppler shift
ωt)η(kx U rel =U tr +U film
σt)η(kx=ωt)kUη(kx rel ''
tU+x=x rel`
(x)U+U=(x)U filmtrrel
Dispersion relation
In X’ coordinate system:
relk(x)U(x)+ω=ρ
Tk(x)3
relkU+ω=σ
, Barnea (1990)
Results in calculation of :k(x)
π=λ(x)
2
For pure capillary wave on current in water
σt)η(kx'=ωt)kUη(kx' rel
Frequency sensitivity
relk(x)U(x)+ω=ρ
Tk(x)3
Experiments
• Large (140~200) series of bubbles
• Image processing to detect wave lengths
• Ensemble average
Image processing
• Enhancement
• Edge detection
• Wave lengths and position detection
Error Factors
• Unequal illumination (pipe diameter, distance from the laser)
• Optical distortions
• Image resolution
• Algorithm accuracy
Ø 44 mm Ø 26 mm
Ensemble average
Averaging bin
Ensemble average
• Random phase causes spectra widening
The results, stagnant water
Results for non-zero Reynolds numbers
Waves Dissipation
c2
3
k2
3
dk
Tkd
dk
dC
3
)3
kL4exp(a)L(a
3
0
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
)texp(a)t(a 0
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.
Waves Propagation
Conclusions
Pure capillary waves
• Development of a theoretical computational model and comparison with experimental results
• Effect of wave-current interaction
• Determination of wave inception condition
New facilities
5 m
Acknowledgments
My supervisors:
Prof. Lev Shemer & Prof. Dvora Barnea
Faculty technical and administrative stuff
Waves Breaking
Capillary wave steepness 2
akaS
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)
Short bubble in 26mm diameter pipe rising in stagnant water
Liquid film velocity
smtr
filmfilmw
filmtr
L
G
film
RUU
|)x(U|)x(UfD2
)x(UU
1g
dx
))x(U(d
,UU,0x mfilm
D.E.
B.C.
)x(U filmCross-sectional average velocity in the liquid film
Points for discussion
• Critical Re number and/or bubble length sustaining presence of pure-capillary waves
• To develop more accurate approach to account for film velocity profile and interface shape
• Exact calculations and/or measurements of bottom oscillation frequency
Numerical Calculations
The method: Fluent software CFD model:
• Axisymmetric at average
• Taylor bubble profile calculated by Barnea (1990) model
• Stationary bubble and moving walls of the pipe
• Separate model for each hydrodynamic conditions
The goal: Exact liquid film velocity distribution
1000hU
4ReL
filmL
f
Transition criterion, Wallis (1969):
000,11Re f 000,10Re f
Ø 44 mm, stagnant water Ø 14 mm, stagnant water
CFD Results Comparison
Stagnant water
Laser
Predicted waves lengths on bubbles rising in stagnant water
relk(x)U(x)+ω=ρ
Tk(x)3
Kelvin-Helmholtz instability
.Tk+)ρk(ρ)sin(g=ω)k(Uρ+ω)k(Uρ 321
222
211
2gw
22gwgw
gw
3gw
gw
ggww
)ρ+(ρ
k)U(Uρρ
ρ+ρ
Tk+k)sin()gρ(ρ±
ρ+ρ
)Uρ+Uk(ρ=ω
0)(
k)UU(Tkk)sin(g)()k(F
2gw
22gwgw
gw
3gw
Convection of the
wave in the x direction
0)(
)UU(Tk)k(F
2gw
2gwgw
gw
,0)sin(g,0
Previous Researches
Wave-current / Wave-wind interaction
• Peregrine (1976): Theoretical basis
• Plate and Trawle (1970) and Long and Huang (1976): experiments of wind
generated large scale waves in presence of water currents, no quantitative
kinematic results.
• Thomas (1981): numerical and experimental studies of wave-current
interaction, depth averaged mean current velocity used as interacting velocity.
• Lai et al. (1989): experimental work of wave-current interaction on large
scale waves.
• Yao and Wu (2004): experiments on large scale wave-group dynamics and
breakings on following and opposite currents.
Goals
• Waves generation mechanism
• Hydrodynamic conditions allowing waves existence
• Calculation model predicting waves characteristics
ε)+πfti(+kz
r
rss e
t)πf+os(sθ
t)πf+(sθ(kr)JA=t)z,θ,Φ(r, 2
2c
2sin
Bottom oscillations
f
Open sheet circular basin:
Potential:
f r
-- Sloshing oscillations frequency
-- Rotating frequency
0=Φ
Sir Horace Lamb, 1879
Rotating bottom oscillations
f r 2 f f r12
f
Kelvin-Helmholtz instability
Long waves mode
K=0, λ∞
Most unstable mode
[m]=k
=λ]m
[=km
mm 4.422πrad
1.42 No KH
instability
KH instability range
KH instability range upper boundK=2.13 [rad/m], λ=2.95 [m]
F(k)
Sloshing frequency calculations
0.11
1
0
kR(kR)J
(kR)J=
zΦrΦ
=U
U
axial
rad k1 , k 2 , k 3
From the disperse relation:
π
ρTk
+gk
=f
3
1,2,3
21,2,3
B.C. :
)tf2(ikz
r
rss
ve)tf2ssin(
)tf2scos()kr(JB
t)t,,r(