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(