Frequency susceptibility of rivulets flowing down vertical

Transcription

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics
Lisbon, Portugal, 07-10 July, 2008
Frequency susceptibility of rivulets flowing down vertical plate
Sergey V. Alekseenko 1,2, Aleksey V. Bobylev 1, Sergey M. Kharlamov 1,
Dmitry M. Markovich1,2*
1: Institute of Thermophysics, Novosibirsk, Russia, dmark@itp.nsc.ru
2: Novosibirsk State University, Novosibirsk, Russia
Abstract: Field measurements of the local thickness of wavy rivulets flowing down a vertical plate were
conducted using LIF method. Cases with sufficiently different contact angles were studied. Detailed
information on the wave structure of rivulets was obtained for different wave regimes and Reynolds
numbers. For small Reynolds numbers, the cross-sectional form of smooth (waveless) rivulets is in good
agreement with theoretical predictions. For wavy rivulets, contact boundaries as well as contact angle of
rivulets are insensitive to the wave motion. For small contact angle, a significant increase in the area wetted
by a rivulet at certain frequencies of wave excitation was found. For large contact angle, all experimentally
registered waves were found to be two-humped.
1. Introduction
Liquid film flow over inclined or vertical planes, especially with the presence of thermal flux, is
often realized in the form of rivulets. For a number of practical applications (heat exchangers,
absorbers, distillation columns, coating processes) as well as natural phenomena (lava flows,
mudslides, glacier flows, etc) the knowledge about rivulet structure and possibility of its control is
rather important. Similar to two-dimensional liquid films (Alekseenko et al. 1996b; Park and
Nosoko 2003), the rivulet interface is unstable with wave formation that strongly affects heat and
mass transfer intensity through a number of mechanisms.
Irregularity and three-dimensional nature of liquid motion considerably complicate the theoretical
and experimental analysis of that kind of flow, thus it is described mainly qualitative. Only few
works exist on the study of rivulets either over flat or curved surfaces (Towel and Rothfeld, 1966,
Alekseenko et al., 1996a, 2007, Holland et al., 2001, Wilson and Duffy, 2002, Carlos et al., 2004,
Myers et al., 2004).
The value of contact angle and its hysteresis strongly affect rivulet flow regimes (Kim et al., 2004).
The aim of the present work is to explore the frequency susceptibility of straight rivulets flowing
down vertical plate and to obtain quantitative characteristics of wavy rivulets at moderate Reynolds
number of rivulet flow.
2. Experimental setup and measuring technique
The test section was represented by a vertical glass plate with dimensions of 200×650 mm, Fig.1.
Different polymer coatings on the plate were used to change contact angle of the rivulet flow for the
same working liquid. The rivulets were formed by a slot distributor with variable width. Waterglycerol and water-alcohol solutions were used as working liquids. Physical properties of the
solution were as follows: for water-glycerol solution - density ρ = 1.04·103 kg/m3, kinematic
viscosity ν = 2.4·10-6 m2/s, and kinematic surface tension σ/ρ = 53.9·10-6 m3/s2; for water-alcohol
solution - ρ = 0.925·103 kg/m3, ν = 2.65·10-6 m2/s, σ/ρ = 32.9·10-6 m3/s2. Liquid flow rates in the
experiments ranged from 0.12 to 4.8 ml/s and Reynolds numbers of investigated rivulets were in the
range Re = 7,2 ÷ 45,3 for water-glycerol and in the range Re = 25,5 ÷ 58,1 for water-alcohol.
-1-
The LIF (Laser Induced Fluorescence) method
was used to measure local thickness and wave
characteristics of rivulets. The Rhodamine 6G
was used as fluorescent dye. Hardware-based
technique on standard PIV system “PIVIT”
was used. The measuring technique provides
spatial resolution of 0.1 mm per pixel and the
thickness
measurement
accuracy
of
-2
approximately 10 mm.
The regimes both with smooth stationary
rivulets and with external periodical
perturbations of liquid flow rate were
investigated. The excitation frequency was
varied in the range from 0.5 Hz to 50 Hz. In
this range both the regimes with solitary
Fig.1 Scheme of experiment and image of wavy rivulet
waves of large amplitude and almost
sinusoidal high-frequency waves with smallscale amplitude was observed at the same liquid flow rates.
3. Results and discussion
The field measurement of rivulet local thickness allowed obtaining detailed patterns of rivulet wave
forms for different wave regimes and Reynolds numbers. Wave structure of rivulets was found to
be sufficiently different for small and large contact angles.
3.1. Frequency susceptibility of the rivulets with small contact angles will be demonstrated below
for the case of water-glycerol solution on a glass surface.
In Figure 2, the reconstructed form of rivulet free surface is shown for the smooth rivulet with the
contact angle α=5°. As seen in Fig. 2b, the contact angle is constant downstream and the crosssectional form of the
rivulet agrees well
with theoretical calculation (Carlos et al.
2004).
In Fig. 3, correlations
between
Reynolds
number of rivulet flow
and volumetric flow
rate as well as the
diagram of frequency
susceptibility of rivulet
Fig.2 Reconstructed free surface of smooth rivulet a) and its cross-sections in different
flow are shown. As
parts of flow b). Black line is the calculation by Carlos et al. 2004. Qliq=1.25 ml/s,
seen in Fig. 3а, with
Re=22.2
the growth of the flow rate, the rivulet’s Re grows linearly up to the flow rate Q = 3 ml/s. In this
range the relation between Q and Re is similar to that for film flow. For greater values of Q the
linear dependency does not hold. Experiments with wavy rivulets were conducted only in the region
of Q where the linear dependency exists. For such rivulets, susceptibility of the rivulet flow to the
external perturbations exists only in the frequency range between bottom (blue) and upper (red)
lines shown in Fig. 3b. In the vicinity of the low-frequency boundary, step-like waves with high
amplitudes are realized. In the vicinity of high-frequency boundary, almost sinusoidal waves with
small amplitudes are observed. In the middle part of the frequency range, there exist regular waves
-2-
Fig.3 Reynolds number versus liquid flow rate a) and region of frequency susceptibility
of rivulets versus Reynolds number b). Contact angle α=5º
Fig.4 Reconstructed form of large solitary wave on the free surface of rivulet at
excitation frequency F=15 Hz a) and cross-sections for different parts of the wave b).
Qliq=3 ml/s, Re=42.6
Fig.5 Reconstructed form of the step-like wave at excitation frequency F=1 Hz. a) and
cross-sections for different parts of the wave b). Qliq=0.2 ml/s, Re=7.2
-3-
with a well-developed
capillary
precursor (Fig. 3b).
It is interesting that
the
width
and
contact angle of
rivulets
are
not
sensitive to the phase
of passing waves. As
an example, in Fig. 4
the 3D form of the a
large-amplitude solitary wave (Fig. 4a)
and
the
crosssections
of
the
rivulets for different
wave phases (Fig.
4b) are shown. As
clearly seen in Fig.
4b, the side walls of
the
rivulet
and
contact angle are
constant
for all
phases.
Additional
peculiarity of the
rivulet flow is the
existence of stable
step-like waves (Fig.
5).
As seen in Fig. 5b,
the amplitude of the
“step” may exceed
the thickness of the
residue layer several
times. The existence
of step-like waves is
a distinctive feature
of rivulets. As know,
in film flow excited
step-like waves are
unstable and quickly
disintegrate with the
formation
of
a
number of solitary
waves (Alekseenko
et al., 1994).
Fig. 6 illustrates the
case
that
is
characteristic for all
high-frequency regular wave regimes. For
such regimes, the
wave amplitude is
small
and
its
longitudinal
crosssections are close to
sinusoidal.
With changes
in
Fig.6 Reconstructed form of sinusoidal regular waves at excitation frequency F=15 Hz
excitation frequency
a) and cross-sections for different parts of the wave b).. Qliq=0.2 ml/s, Re=7.2
in the region of
existence of regular
high-amplitude waves a change in the width of the rivulets downstream was observed. The 3D form
and cross-sections in the upper and lower parts of the flow for one of such regimes are shown in
Fig. 7a-c. It is seen that the boundary location and rivulet contact angle are insensitive to the phase
of the passing wave. At the same time, the rivulet width downstream is sufficiently higher than that
Fig.7 Reconstructed form of developed regular waves at excitation frequency F = 23 Гц a) and its cross sections for
crest and trough of the waves in the upper b) and lover c) region of the flow. Qliq=1.25 ml/s, Re = 22.2
upstream (22 and 18 mm, respectively, for the above case). For the same regime, longitudinal crosssections are shown in Fig. 8 and the values of the wave’s amplitude downstream of the given
regimes – in Fig. 8b. As seen in Fig. 8b, in this case the amplitude of waves grows nonmonotonically downstream,
which
is
apparently due to the
rivulet
widening
down-stream.
In
Fig.
9
the
dependence of the
rivulet width in the
bottom region of
rivulet flow on the
is shown. The width
Fig.8 Transversal cross sections of developed regular waves at excitation frequency
of smooth rivulet was
F = 23 Гц a) and amplitudes of the waves down stream b). Qliq=1.25 ml/s, Re = 22.2
taken for 100%. As
-4-
Fig.9 Related width of rivulet in dependence of excitation frequency. The width of
smooth rivulet accepted as 100%. Qliq=1.25 ml/s, Re = 22.2
follows from the Fig.
9, the rivulet width and
consequently the area
wetted by rivulet grow
with the growth of
frequency and reaches
a maximum at F = 43
Hz. After that with
further
growth
of
the
rivulet
width
sharply decreases and
practically drops to the
initial value of smooth
rivulet width at F=50
Hz, which corresponds
to the upper boundary of the rivulet frequency susceptibility range.
3.2 The case of large contact angle
For large contact angle, the rivulet wave structure is fundamentally different from that for small
angles. In the region of rivulet frequency susceptibility, no step-like or sinusoidal waves were
observed. For all frequencies of excitations within the region of susceptibility the developed waves
have a distinct two-humped form. Characteristic properties of wave rivulets with large contact angle
will be described below by the example of the rivulet with the contact angle 23º (in this case
ethanol-water solution was the working liquid and fluorocarbon polymer coating on the glass plate
was used). The form of smooth rivulet is still in good agreement with theoretical calculations.
Similarly to the case of small contact angles, the side walls of the rivulet are also insensible to the
phase of passing waves. The 3D form of the waves at different excitation frequencies is shown in
Fig. 10 and longitudinal cross-sections for this flow regime at different excitation frequencies are
shown in Fig. 11. As can be seen, at all frequencies of excitation the waves have a very similar twohumped form with a high-frequency front and low-frequency back humps. At low excitation
frequencies, the amplitude of the front hump is sufficiently higher than that of the back one. With
the growth of excitation frequency, the amplitude of the humps decreases at different rates and the
humps become almost equal in amplitude near the upper frequency boundary of the existence of
excited waves (Fig. 12). It is interesting that two-humped waves of similar shape were observed
earlier for the rivulets flowing down the outer part of an inclined cylinder (Alekseenko et al.,
1996a). The results of these authors obtained with the use of the shadow method are presented in
Fig. 13. With the growth of Re, the form of the waves at all excitation frequencies is similar to that
Fig.10 Reconstructed form of excited waves in the case of big contact angle at different excitation frequencies.
Qliq=0.12 ml/s, Re = 25.5: a) F=7 Hz; b) F=17; c) F=28 Hz
-5-
Fig.11 Profiles of waves at different excitation frequencies. Qliq=0.12 ml/s, Re = 25.5
described above (Figs. 14-16) with the only difference that the back hump at higher Re becomes
less pronounced.
4. Conclusion
Field measurements of the local thickness of wavy rivulets flowing down a vertical plate were
conducted using LIF method. Cases with sufficiently different contact angles were studied. Detailed
information on the wave structure of rivulets was obtained for different wave regimes and Reynolds
numbers.
It was found that general property of rivulet flow is insensitivity of contact angle of rivulets to the
wavy motion at all contact angles. At the same time wave structure of rivulets was found to be
sufficiently different for small and large contact angles.
-6-
Fig.14 Profiles of waves on free surface of rivulet.
Qliq=0.25 ml/s, Re=34.5
Fig.12 Amplitude versus frequency of excitation
Qliq=0.12 ml/s, Re = 25.5.
Qliq=0.5 ml/s, Re=42.4
Qliq=1 ml/s, Re=58.1
For small contact angles a wide variety of wave
patterns in dependence on excitation frequency is
observed. In this case the range of frequencies was
observed when the amplitude of excited waves was
maximal and comparable with the rivulet maximal
thickness to all studied flow rates. For high
frequencies of excitation, the small amplitude
waves with sinus-like shape were observed. For
frequencies in the vicinity of lover limit of the
frequency susceptibility of rivulets step-like waves
with high amplitude are formed. Also for the case
of small contact angles existence of narrow range of
excitation frequencies was found inside which
considerable increase of rivulet width (without
Fig.13 Profiles of stationary exited waves on the
changing of contact angle), and consequently
inclined cylinder at α=15º, Q=0.82 ml/s, ethanol.
increase of wetting area occurs.
Alekseenko et al., 1996a
For large contact angles, the rivulet wave structure
is fundamentally different from that for small
angles. In this case, in the region of frequency susceptibility of rivulets, no step-like or sinusoidal
waves were observed. Instead, for all frequencies of excitations within the region of susceptibility
the developed waves are similar and have a distinct two-humped form.
-7-
Acknowledgements
08-01501-a.
This work is supported by RFBR Foundation, grants N 06-01-00762-a, 06-
References
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measurements in a liquid film flowing down an inclined cylinder. Experiments in Fluids 43:197207
Alekseenko SV, Markovich DM, Shtork SI (1996a) Wave flow of rivulets on the outer surface of an
inclined cylinder. Phys Fluids 8:3288–3299
Alekseenko SV, Nakoryakov VE, Pokusaev BG (1994) Wave Flow of Liquid Films. Begell House,
New York
Alekseenko SV, Nakoryakov VE, Pokusaev BG (1996b) Wave effect on the transfer processes in
liquid films. Chem. Eng. Comm 141(142): 359-385
Carlos A, Perazzo A, Gratton J (2004) Navier-Stokes solutions for parallel flow in rivulets on an
inclined plane. J Fluid Mech 507:367–379
Holland D, Duffy BR, Wilson SK (2001) Thermocapillary effects on a thin viscous rivulet draining
steadily down a uniformly heated or cooled slowly varying substrate. J Fluid Mech 441:195–221
Kim H, Kim J, Kang BH (2004) Meandering instability of a rivulet. J Fluid Mech 498:245-256
Myers TG, Liang HX, Wetton B (2004) The stability and flow of a rivulet driven by interfacial
shear and gravity. Int J Nonlinear Mech 39:1239–1249
Park CD, Nosoko T (2003) Three-dimensional wave dynamics on a falling film and associated mass
transfer. AIChE J 49(11):2715-2727
Towell GD, Rothfeld LB (1966) Hydrodynamics of rivulet flow. AICHE J 12:972–980
Wilson SK, Duffy BR (2002) On the gravity-driven draining of a rivulet of fluid with temperaturedependent viscosity down a uniformly heated or cooled substrate. J Engineering Mathematics
42:359-372
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Frequency susceptibility of rivulets flowing down vertical

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