Experiments in Fluids 36 (2004) 685–700
DOI 10.1007/s00348-003-0745-3
Influence of wavy structured surfaces and large scale polymer
structures on drag reduction
M. Vlachogiannis, T. J. Hanratty
685
molecular weights. Only a few (Lindgren and Hoot 1968;
Virk 1971; Bewersdorff and Thiel 1993) have considered a
roughened wall. This paper presents measurements of the
influence of a copolymer of polyacrylamide and sodium
acrylate on flow over a surface that consists of a train of six
hundred sinusoidal waves with an amplitude of 0.25 mm
and a wavelength of 5 mm. The structured surface constituted the bottom wall of a rectangular flow channel. The
top wall was flat, so comparisons of the behaviors with
smooth and wavy surfaces could be made in the same
experiment. The study differs from previous researches in
that turbulence measurements are obtained and a welldefined and well-documented roughness is used.
The experiments involved the injection of a polymer
solution with a high concentration, ci, through slots in the
wall. An important aspect of this study is the use of a
fluorescence imaging technique (Vlachogiannis and Bontozoglou 2001, 2002) to find out whether the injected
polymer solution contains sheets or large filaments of
entangled or gelatinous polymer molecules and how the
presence of these structures affects drag reduction. To the
knowledge of the authors, this type of study did not
accompany previous research that used wall slots.
There is a similarity between our experiments in which
large polymer structures were present and ‘heterogeneous’
drag reduction, first studied by Vleggaar and Tels (1973).
In this and later works (Hoyt and Sellin 1991; Bewersdorff
1982; Smith and Tiederman 1991) a highly concentrated
polymer solution (ci‡4000 ppm) was found to move
through the fluid as a thread when it was injected from a
small tube. Discussions of the effect of these threads on
drag reduction are presented in papers by Bewersdorff
1
et al. (1979, 1982, 1984), Usui et al. (1988) and Smith and
Introduction
Tiederman (1991). The experiments described in this paA large number of studies have been carried out which
show that the drag of a turbulent fluid on a smooth surface per focus on structures formed by injection through a wall
slot. They also differ from studies of heterogeneous drag
can be reduced by the addition of polymers with high
reduction, cited above, in that the concentration of the
injected solution was smaller and polymer structures
already existed in the injected solution. However, they
Received: 22 October 2002/Accepted: 20 September 2003
support the claims in several previous works that the
Published online: 13 February 2004
presence of large polymer structures can be beneficial.
Springer-Verlag 2004
Studies of turbulence in drag reducing solutions were
advanced by the works of Pinho and Whitelaw (1990), Wei
M. Vlachogiannis, T. J. Hanratty (&)
and Willmarth (1992), Harder and Tiederman (1991), and
Department of Chemical Engineering, University of Illinois,
Walker and Tiederman (1989, 1990). Their use of laser
Urbana, IL 61801, USA
Doppler velocimetry avoided problems that arise when
E-mail: [email protected]
Sponsored by the Defense Advanced Research Projects Agency, polymer solutions flowed past a probe. In the experiments
Advanced Technology Office, Friction Drag Reduction Program, of Wei and Willmarth (1992) and Harder and Tiederman
ARPA order No: MDA972-01-C-0029.
(1991) polymers were injected into the flow and
Abstract Drag reduction was studied for turbulent flow
over a structured wall that contained 600 sinusoidal waves
with a wavelength of 5 mm and an amplitude of 0.25 mm.
A concentrated solution of a co-polymer of polyacrylamide
and sodium acrylate was injected into the flow through
wall slots. Laser Doppler velocimetry was used to measure
turbulence. A fluorescence technique was developed that
enabled us to demonstrate the existence, under certain
circumstances, of large gelatinous structures in the injected polymer solution and in the flow channel.
At maximum drag reduction, the Reynolds shear stress
was zero and the velocity field was the same as found for a
smooth surface. Larger drag reductions could be realized
for a wavy wall because the initial drag was larger. The
influences of polymers on the turbulent fields are similar
for smooth and wavy boundaries. These results are of
interest since the interaction with the wall can be quite
different for water flow over smooth and wavy boundaries
(which are characterized as being completely rough). An
important effect of polymers is a decreasing relative
importance of high frequency fluctuations with increasing
drag reduction that is characterized by a cut-off frequency.
This cut-off is the same for smooth and wavy walls at
maximum drag reduction. The sensitivity of drag reduction to the method of preparing and delivering the polymer solution suggests that aggregation of polymers could
be playing an important role for the system that was
studied. For example, drag reduction was enhanced when
large polymer structures are present.
686
measurements were stopped when the polymer concentration in the holding tank built to an unacceptable level.
The flow was recycled through a large pump in order to
degrade the polymers to a point where the pressure drop
was the same as would be measured for water flow. The
experiments were then resumed. Wei and Willmarth
introduced solutions of Polyox with concentrations of 500
to 2000 ppm. The mixed polymer concentration in the test
section was 10 ppm and the amount of drag reduction was
about 30 percent. They found that the energy of the
normal velocity fluctuations is greatly reduced and that
there is a redistribution of energy from high frequencies to
low frequencies.
Warholic et al. (1999b) used this approach and a
solution of a co-polymer of polyacrylamide and sodium
acrylate to explore a wider range of conditions for flow
over a flat surface. Of particular interest is their finding
that the Reynolds shear stress was zero, at all locations in
the channel when maximum drag reduction was realized.
The work described in this paper used the same technique
and the same flow loop that was employed by Warholic
et al. (1999a).
When a fluid flows turbulently over a smooth wall, it is
sustained by elongated flow-oriented vortices located near
the wall. (See, for example, the collection of papers edited
by Panton [1997]). Laboratory studies of flow over a wavy
boundary, with large enough amplitude, a, that separation
occurs, reveal that turbulence is generated in shear layers
that form behind the crest. (Buckles et al. 1984; Hudson
et al. 1996). The study of Hudson was carried out with a
train of waves with a wave height, 2a, to wavelength ratio
0.1 that was the bottom wall of a rectangular channel. The
Reynolds number based on the half-height of the channel,
h, and the average velocity was Re=3380. The dimensionless height of the waves based on the friction velocity and
the kinematic viscosity of the liquid was 2a+=73.5. Hudson
et al., concluded that flow near the boundary is ‘‘fundamentally different from what is found for flow over a flat
wall in that production is not associated with the floworiented vortices described by Brooke and Hanratty (1993)
and by Kline and Robinson (1989).’’ Detailed descriptions
of these separated flows have been obtained in direct
numerical simulations by Cherukat et al. (1998) and by Na
(Nakagawa et al. 2003a) for a wavy wall with a wave height
to wavelength ratio of 0.1, Re=3400, and a+=69.6. Not
surprisingly, vortical structures close to the wall were
different from what is observed for a flat wall. Na showed
contours of the streamwise fluctuating velocity in an x-z
plane located just above the wave crest. No evidence of a
streaky structure was observed. Cherukat et al. (1981) also
observed that ‘‘coherent streak-like structures..are not
found anywhere above the wave.’’ Na showed that floworiented vortical structures, identified by the scheme of
Yong et al. (1990), form on the upstream side of the wave
and disappear as they progress over the crest. They,
therefore, appear to be the result of a centrifugal instability.
Studies of water flow over the wavy surface used in this
research (2a/k=0.1, k=5 mm) were carried out by Nakagawa and Hanratty (2003a, b) with LDV techniques for
Re=46,000, Re=11,000 and Re=6000. Comparison PIV
measurements were also made by Nakagawa and Hanratty
(2001) at Re=46,000. The rationale in these experiments
was that the dimensionless wave amplitude, a+, at
Re=46,000 was approximately the same as for the DNS
studies at Re=3400. Therefore, the flow pattern observed
very close to wall with the DNS might be similar to what
exists for laboratory measurements at Re=46,000, since
2a/k and a+ are the same. Nakagawa and Hanratty (2003b)
carried out visual studies for which dye was injected at the
crest. The dye streamer separated from the crest both for
Re=46,000 and Re=11,000 but the mixing was much more
vigorous for Re=46,000. The streamer followed the wave
contour for Re=6000.
The surfaces studied by Nakagawa were characterized
by equivalent sand roughnesses of kþ
s ¼ 104; 22:4; 6:9.
They were described as being completely rough, as possessing an intermediate roughness and as being hydraulically smooth. The experiments described in this paper
were for Re=48,000, Re=20,700, Re=11,000 and Re=6000
for which kþ
s ¼ 104; 46:8; 22:4; 3:4 lower flow. The results
are presented in separate sections for high and low Reynolds numbers. The experiments at the two largest Reynolds numbers are in or close to the ‘‘fully rough’’ regime.
Those for the two smallest Reynolds numbers are in the
intermediate and hydraulically smooth regimes. Another
reason for this method of presentation is that drag
reduction was not realized (for the range of polymer
concentrations used in this research) at the two lowest
Reynolds numbers unless large scale polymer structures
were present in the solution.
The contributions of this paper are (1) a study of the
influence of a drag reducing polymer on flow over a welldefined and well-documented roughened wall, (2) the
demonstration that the delivery of polymers through wall
slots could result in the formation of large polymer
structures, and (3) the further documentation that the
presence of these structures can enhance drag reduction.
However, some of the results could impact theoretical
understanding of drag reduction. The interaction of a
turbulent fluid with a wavy wall can be quite different from
the interaction with a smooth wall; yet, there is a similarity
in the turbulence measurements in drag reducing flows.
The effect of polymers on spectral functions is of particular interest since attention has been given to the notion
that the changes in turbulence with the addition of polymers is due to the elastic behavior of stretched or partiallystretched polymers (Tabor and de Gennes 1986; de Gennes
1990). This is manifested by a decrease in the relative
importance of high frequency (or low wavenumber) fluctuations and by the existence of a cut-off frequency. The
documentation of these effects in this paper is, therefore,
useful.
2
Description of the experiments
2.1
The flow facility
The flow facility, which is depicted in Fig. 1, has been
described in several previous papers (Warholic et al. 1999;
Niederschulte et al. 1990; Nakagawa et al. 2001). The
polymers after this treatment was determined to be completely destroyed, by comparing the pressure drop of the
degraded polymer solution and the pressure drop obtained
with a water flow. Also, agreement was realized between
measurements made at the beginning and the end of an
injection period.
Fig. 1. Diagram of the flow facility
closed loop water channel has an aspect ratio of 12:1 and a
height, 2 h, of 50 mm. The temperature was maintained at
25.0±0.8C, by using the reservoir’s cooling coils.
The rectangular channel had a length of 11 m. The test
section consisted of the final 3 m. As depicted in Fig. 1,
three sets of optical grade windows were located on both
sides of the channel in order to allow the insertion of laser
light. Measurements were carried out at the third set of
windows. Figure 2 is a sketch of the test section. The top
wall (referenced as 1) was flat and the removable bottom
wall (referenced as 2), contained six hundred sinusoidal
waves that extended over the whole width of the channel.
2.2
Polymer solution
A master polymer solution of Percol 727, a copolymer of
polyacrylamide and sodium acrylate, was prepared over a
period of 12–16 hours before each experiment. Details of
the mixing procedure and the design of the system are
given in the paper by Warholic et al. (1999). The concentrated polymer solution could be injected without
operating a pump, by using a mixing tank that was located
10 m above the flow loop. The polymer was admitted to
the system 8 m upstream of the test section through two
inclined slots in the bottom of the channel. (Tests which
used two slots in the top wall or all four slots gave the same
results.) The concentration of the injected solution, ci,
varied from 100 to 2500 ppm. The concentration of the
polymer in the test section, cm, was adjusted by changing
the flow rate of the injected solution.
The operational procedure was similar to that developed in the laboratory of Tiederman. The liquid was circulated with a 5 hp centrifugal pump, which did not
completely degrade the polymers in a single pass. The
injected polymers were diluted to such an extent in the
piping and holding tank that the concentration of undegraded polymers at the inlet did not change appreciably
during the injection period. In order to avoid the buildup
of undegraded polymers, the injection was stopped after
7–10 min and the liquid was circulated through a 60 hp
pump for more than 15 min. The effectiveness of the
2.3
Velocity measurements
The velocity field was measured with the three-beam, two
color LDV system used by Günther et al. (1998) and
Warholic et al. (1999). The accuracy of the measurements
is discussed in these papers. A beam-expanding module
(TSI model 9832) was used with a fiber optic probe (TSI
model 9253) to produce a measuring volume, which had a
diameter of 45 lm and a length in the spanwise direction
of 0.44 mm. The water and the injected polymer solution
were seeded with polystyrene spheres with an average
diameter of 500 nm and a specific gravity of 1.005. These
were manufactured by a process described by Goodwin
et al. (1973). The receiving optics (TSI model 9176), which
were located at the opposite side of the channel, gathered
the scattered light from the particles. Both the transmitting
and receiving optics were located on an I-beam, which was
mounted on a transversing mechanism. The latter was
used in order to change the position of the measuring
volume and to assure that all of the optics were moved
together.
The output signals from the photo-multipliers (TSI
model 9162) were analyzed with a two-channel correlation
processor (TSI model IFA655). We acquired four samples
(files) of 5000 points, with a maximum data rate of 1000 to
3000 Hz, for each y-location. Very small differences were
noted between a 5000 point file and the sum of the four
files so this assured that 20000 points provided converged
statistics. The position in the spanwise direction was kept
constant at 210 mm away from the sidewall, where the
transmitting optics were located. Previous measurements
of the mean streamwise velocity and the root-mean-square
of the fluctuating velocity at different locations indicate
that the results were not affected by the proximity of the
sidewall.
The signals were corrected for the presence of white
noise, which was estimated from measurements of the
frequency spectra. The procedure used is described by
Günther et al. (1998) and by Warholic et al. (1999). The
noise plateau was clearly seen in the spectra measured with
the polymer solutions, so that the method of correction
was easily implemented.
2.4
Measurements of the pressure drop
Measurements of the pressure gradient were obtained with
a Validyne reluctance pressure transducer (model DP103).
The pressure range can be varied by changing the diaphragm, so that measurements of high-accuracy were
obtained for all the Reynolds numbers. The pressure taps
were located in the top flat wall; they had a separation
distance of 1.01 m.
As depicted in Fig. 2, the location of zero Reynolds
shear stress may not be at the center of the channel
687
688
very long length of channel. This was not possible in the
system used in this study. However, wall shear stresses
determined from pressure drop measurements and by
extrapolating Reynolds shear stress measured in the central regions of the channel were the same for most of the
runs. This suggests that the flow in the test section is
approximated by a fully-developed condition at the locations where measurements were made. The possibility has
been mentioned that the turbulence observed at maximum
drag reduction are remnants of upstream turbulence
events. This would mean that a fully developed flow at
Fig. 2. Sketch of the test section and schematic representation of maximum drag reduction is laminar. But this has never
the measurement of sw,1 and sw,2
been mentioned in the literature.
because of the difference in the roughnesses of the top and
bottom walls. The distortion of the profile, for the same
wavy wall and for a range of Reynolds numbers, is described in detail in a paper by Nakagawa et al. (2003a). A
force balance across the test section was used in order to
determine the average wall shear stresses:
DP
TopwallðsmoothÞ: and sw1 ¼ h1
ð1Þ
Dx measured
DP
ð2Þ
BottomwallðwavyÞ: and sw2 ¼ h2
Dx measured
where hi is the distance between the position of zero stress
and the wall (i=1 for the top wall and i=2 for the bottom
wall).
The wall shear stress was also obtained from the
extrapolation of measurements of the Reynolds shear
stress in the central regions of the channel to the wall. A
very good agreement with measurements obtained from
pressure drops was found in studies with water and with
polymer solutions for all runs except those close to maximum drag reduction. The amount of drag reduction (DR)
is defined in terms of the wall shear stresses for the
polymer solution and for the Newtonian fluid,
sw2;water sw2;polymer
%DR2 ¼
100
ð3Þ
sw2;water
The use of a channel with different walls has the
advantage that drag reductions with flat and structured
surfaces could be determined simultaneously. The results
for the flat surface were obtained from Eq. 3 with sw2
replaced by sw1.
The total shear stress consists of the sum of the Reynolds shear stress, the viscous stress and the polymer
stress (Warholic et al. 1999). Except for maximum drag
reduction or conditions close to it, the viscous stress and
the polymer stress are negligible in a region around the
center of the channel. The determination of the location of
zero shear stress was also made by using the observation
of Nakagawa and Hanratty (2003) that it is the same as the
location of the maximum in the velocity profile. For
maximum drag reductions the location of zero shear stress
is at the center of the channel.
The usual method for determining whether a flow is
fully-developed is to measure the pressure profile over a
3
Visual studies of aggregated polymers
In order to visualize the flow of the injected polymer
solutions, a fluorescence imaging method, described by
Vlachogiannis et al. (2001), was used. The injected polymer solution was doped with 100–300 ppm, of a sodium
salt of fluorescein, C20H10O5Na2. The resulting solution
fluoresces under ultraviolet illumination and emits light in
a specific wavelength range of 520–575 nm. The ultraviolet
source consisted of four high-intensity lamps (Philips,
TL20/05), which were located above the top wall of the
channel.
The polymer particles have an average diameter of
550 nm, which is at least ten times greater than the dye
particles. The powder of the polymer was mechanically
mixed with the dye before adding it to water. The color of
the mixed powder changed from white to a yellow–green
color.
Since the dye adhered to the polymer particles, by using
sieves of different sizes we assured that only dyed polymer
particles, and not pure dye grains, were mixed with the
water. The mixing of the dyed powder with the water
produced a green polymer solution. The concentration of
the dye in the final polymer solution varied between 100 to
2500 ppm.
A spectrometer (Perkin-Elmer Lambda 40) was used to
examine the resulting polymer solution. The intensity of
the light was higher for a dyed polymer solution than for
dyed water because of the adherence of dye particles to the
polymer. This behavior provided a method that used
measurements of spatial variations of light intensity to
determine how the polymers distribute in the fluid.
A CCD video camera (shutter speed, 1/2000 s) and a
digital video camera were employed to acquire flow images
in the test section. The light intensity in the images varies
linearly with both the flow of the injected polymer solution
and the concentration of dispersed dye in the flowing
water. The use of tap water, doped with the same amount
of dye as was contained in the injected solution, showed
that the dye was uniformly mixed with the circulated liquid. This measurement was the baseline for the image
processing, which used commercial (CorelDraw) as well as
in-house software (built in MatLab). The digitization noise
was eliminated by applying appropriate convolution filters
to the incoming images.
A representative example of the fluorescence imaging
method is depicted in the image series of Fig. 3. The
distance from the side or top wall were kept constant in
the different experiments.
4
Homogeneous and heterogeneous drag reduction
Two methods were used to introduce the polymers into the
mixing tank. One involved pre-wetting of the polymer
particles with a venturi injector (Warholic et al. 1999).
Injected concentrations of ci £ 500 ppm could be obtained
in this way. Solutions for which the fluorescence imaging
technique did not reveal any large structures are called
‘homogeneous’. This was the case for all experiments that
used the venturi mixing device. The ability to observe
structures with the fluorescence imaging method depends
on the light that is emitted from the dyed polymer solution. Small-scale structures of the order of 200 lm or less,
are not clearly seen since the emitted light intensity is not
enough to distinguish these structures from the surrounding fluid. It is possible (or, perhaps, likely) that
smaller aggregates exist in solutions that are called
‘homogeneous’. For situations in which maximum drag
reduction existed, thin threadlike structures were visible to
the eye under intense illumination. These were not picked
up by imaging techniques.
The other method of mixing involved the sprinkling of
polymer powder that was sieved on the top of the liquid in
the mixing tank. The particles had a size range of 400 lm
to 800 lm. Large structures were observed if ci‡500 ppm.
These solutions are called ‘heterogeneous’. It is to be noted
that either homogeneous or heterogeneous solutions could
be obtained at ci=500 ppm depending on whether the
venturi device was used. For ci>500 ppm mixing was done
only by sprinkling the polymer particles.
Sections 3 and 5.3 present observations of polymer
structures in the channel. An experiment was performed in
which the concentrated solution was allowed to flow down
an inclined plane rather than to flow through the injection
slots in the flow channel. This produced a thin flowing film
with a free surface. The polymer damped the waves that
would be present for a water flow. Therefore, filaments
were easily observed in the film flow under conditions that
a heterogeneous solution would exist in the flow channel.
This indicates that filaments were present in the solutions
Fig. 3. The implementation of the fluorescence imaging method that were injected, when large polymer structures were
in polymer drag reduction studies at Re=11,000. a Original image observed in the flow channel.
acquired by the color digital camera. b The light reflection from
the wall in the image plane. c The baseline image obtained from 5
dyed water experiments. d The resulting image after the
Results
implementation of the image processing analysis
original image is shown in Fig. 3a, with the wavy wall
clearly seen at the bottom. Some of the dye in the injected
polymer escaped from the filaments to produce a greenish
tinge to all of the fluid. The light that is reflected because
of the wavy wall, Fig. 3b, was subtracted from the final
images. The baseline images, Fig. 3c, were used in order to
subtract the light that is transmitted from the scattered dye
in the circulated liquid. Thus, only the light that is associated with the dyed polymers remained in the final
grayscale image, as illustrated in Fig. 3d. The concentration of dye, the intensity of the ultraviolet source and the
5.1
Measurement conditions
Measurements with laser Doppler velocimetry were carried
out for four Reynolds numbers, Re=6000, 11,000, 20,700
and 48,000, based on the bulk velocity, the half-height of
the channel and the viscosity of the water. The parameters
are the concentration of the injected polymer solution, ci,
the concentration of the polymer in the test section
(so-called mixed concentration), cm, and the Reynolds
number. These are listed in Table 1 for the different
experiments. More than sixty pressure drop measurements
were obtained in order to establish the dependence of the
689
Table 1. Summary of velocity
measurements. Homogeneous
drag-reduction: Re=20,700,
48,000. Heterogeneous dragreduction: Re=6000, 11,000
690
Re
Cm
Ci
6000
6000
11,000
11,000
11,000
11,000
11,000
20,700
20,700
20,700
20,700
20,700
20,700
48,000
48,000
48,000
48,000
48,000
48,000
–
35.4
–
5.4
8.6
15
20
–
0.28
1
0.62
1.5
7.7
–
1.05
5
2.9
10.2
14.5
–
500
–
1000
1000
1000
1000
–
100
100
500
500
500
–
500
500
500
500
500
drag reduction on the above-defined parameters. A range
of drag reductions of 10 to 85% and a range of concentrations in the test section of 0.2 to 25 ppm were covered.
Visual observations, which were performed for all the
Reynolds numbers, indicated when sheets or filaments of
polymer molecules were present. If these structures were
not observed, drag reduction was not realized at Re=6000
and Re=11,000. Therefore, the work described in this paper is separated into three parts: Homogeneous drag
reduction studies are described for high Reynolds numbers in section 5.2. The structures of heterogeneous
polymer solutions are discussed for a range of Reynolds
numbers in section 5.3. The turbulence properties of
polymer solutions at low Reynolds numbers, when structures are present, is examined in section 5.4.
u*2
1.40
1.36
2.35
2.12
2.06
1.95
1.87
4.51
4.21
4.16
3.96
3.78
1.90
10.19
9.24
6.32
8.58
5.44
4.51
sw2
DP/Dx
ymax/h
%DR1
%DR2
1.95
1.84
5.52
4.47
4.25
3.81
3.48
20.36
17.72
17.29
15.67
14.30
3.61
103.98
85.26
39.93
73.5
29.50
20.32
0.74
0.70
1.96
1.63
1.58
1.46
1.35
6.92
6.09
5.95
5.45
4.99
1.42
32.74
29
15
25
11.5
8
1.04
1.04
1.11
1.08
1.06
1.03
1.01
1.16
1.15
1.14
1.13
1.13
1.00
1.25
1.157
1.048
1.15
1.01
1
–
5
–
15
16
20
25
–
11
13
19
25
76
–
0.45
41.82
14.19
53.62
67.41
–
6
–
19
23
31
37
–
13
15
23
30
82
–
18.00
61.60
29.31
71.63
80.46
2
k ¼ 8 u =Ub
ð4Þ
Here u* is the friction velocity and Ub is the bulk-averaged
velocity.
The maximum drag reduction was reached at
cm=7.7 ppm. A comparison with data obtained by
5.2
Measurements for high Reynolds numbers—homogeneous
solutions
Results for Reynolds numbers Re=20,700 and Re=48,000,
for which a venturi device was used in the mixing process,
are presented in this section. The dimensionless sand
roughness, kþ
s was estimated by Nakagawa et al. (2003a)
from the roughness function proposed by Nikuradse
(Schlichting 1960) for a fully rough surface. It had values
of 46.6 and 108, for water flows. The transition to a fully
rough region occurs at kþ
s ffi 70.
The dependence of the drag reduction on the mixed
concentration is depicted in Fig. 4a, for a Reynolds
number of 20,700. The average wall shear stresses for the
wavy and the flat wall were determined from pressure
drop measurements by using Eqs. 1,2. The estimation of
the location of zero Reynolds shear stress was established
from LDV measurements. Increases in the mixed concentration are accompanied by increases in drag reduction for both flat and wavy walls. However, the percent
drag reduction for the wavy wall is larger than for the flat
wall. This is consistent with the observation that the
friction factor for a water flow, k=0.0297, is larger for the Fig. 4. Percentage drag reduction as a function of the mixed
wavy wall than for the flat wall, k=0.01917, where k is
concentration for flat and wavy surfaces. a Re=20,700.
defined as
b Re=48,000
Warholic et al. (1999) shows that the maximum drag
reduction occurs at a larger mixed concentration,
cm=13 ppm, when both walls of the channel were flat.
Noteworthy, is the observation of a drag reduction of 20%
with a mixed concentration of only 0.25 ppm. Similar results are found for a Reynolds number of 48,000, as shown
in Fig. 4b. However, a lower drag reduction of 12% was
observed at cm=0.25 ppm. Also, the mixed concentration
that is required for maximum drag reduction increased to
cm=14.5 ppm when the Reynolds number was increased
from 20,700 to 48,000.
The variation of the average streamwise velocity with
the distance from the averaged location of the wave surface, made dimensionless with the half-height of the
channel, is depicted in Fig. 5 for maximum drag reduction.
The profiles over the crest and the trough of the waves
were averaged. Despite the difference in the roughnesses of
the top and bottom wall, the velocity profile was symmetric for both Reynolds numbers, 20,700 and 48,000.
Furthermore, if a scale of 160 to 200 cm/s, rather than 0 to
210 cm/s, were used in Fig. 5 one could see that the
maximum is located at the center of the channel. Therefore, the mean flow is not dependent on the structure of
the wall at maximum drag reduction.
The profiles are more diffuse, close to the wall, at
maximum drag reduction than is observed for water, so
the shear rate at the wall, (dU/dy)wall, could be directly
determined. For a Reynolds number of 20,700, the shear
rate was 117.7 s)1; for Re=48,000, it was 582 s)1. The
dimensionless mean velocities, U+, versus the distance
from the wall, y+, normalized with the friction velocity and
the viscosity of the polymer solution near the wall, are
close to Virk’s profile for maximum drag reduction, only
for y/h £ 0.5.
Figure 6a presents selected measurements of the rootmean-square (RMS) of the streamwise velocity fluctuations, u¢, for Re=20,700 over a structured surface. The
peaks in u¢ are displaced outward with increasing drag
reduction. The peak values are roughly the same for drag
reductions of 30% or less. However, they increase when
normalized with the friction velocity, because of the
decrease in u* with increasing drag reduction. The magnitude of u¢ decreases in the outer flow with increasing
drag reduction. A plateau region which starts at y/h>0.17
for water flow over a wavy wall and at y/h>0.2 for a flat
wall becomes less prominent with increasing drag reduction. For a flow over a wavy wall at Re=48,000, the
dimensional u¢ decreases, over the whole flow, and the
peaks are displaced outward with increasing drag reduction, as depicted in Fig. 6b. For conditions close to maximum drag reduction, the magnitudes of u¢ are greatly
decreased and broad maxima occur in the neighborhoods
of y/h=0.1 for Re=48,000 and at 0.19 for Re=20,700.
Measurements of the dimensional root-mean-square of
the normal velocity fluctuations are plotted in Fig. 7. They
decrease systematically with increasing drag reduction for
Re=20,700 and for Re=48,000. The influence of wave-induced flows for Re=48,000 is clearly seen in Fig. 7b,
by comparing the v¢ for y/h<0.2 with that obtained for
1.8<y/h<2.0 (i.e., with a flat wall). This influence extends
over a distance of about two wavelengths for water flows.
Fig. 6. Streamwise turbulence intensity, u’, versus the normalized
distance from the wall, y/h, for two different Reynolds numbers:
a Re=20,700, ci=500 ppm; n Water, u2*=45.1 mm/s; h
cm= 0.62 ppm, %DR=23, u2*=39.6 mm/s; M cm=1.5 ppm,
%DR=30, u2*=37.8 mm/s; · cm=7.7 ppm. %DR=82, u2*=19 mm/s.
Fig. 5. The average of profiles of the mean velocity measured over b Re= 48,000, ci=500 ppm; n Water, u2*=101.9 mm/s; h
cm= 2.9 ppm, %DR=29, u2*=85.7 mm/s; M cm=10.2 ppm, %DR=72,
the trough and over the crest, at maximum drag reduction, for
u2*=54.3 mm/s; · cm=14.5 ppm. %DR=82, u2*=45 mm/s
two different Reynolds numbers
691
692
Of particular interest is the observation that the influence
of wave induced-flows remains qualitatively the same both
for low drag reductions and for conditions close to maximum drag reduction. These wave induced-flows are not
seen at Re=20,700, probably because of the smaller
dimensionless wave amplitude, a+.
Measurements of u¢ and v¢ at Re=20,700 and at drag
reductions less than 35% are similar to what has been
observed by Warholic et al. (1999, Figs. 4 and 5) for a flat
surface. The apparent differences in their Fig. 4 arise because u¢+ was plotted against y+. Taking this into account,
the locations and magnitudes of the peaks, and the decreases in u¢ in the outer flow (say, at y/[email protected]) are
approximately the same for flat and structured surfaces.
Similar observations can be made about the decreases in v¢
with the addition of polymer. Comparisons at large drag
reductions are best done in the system used in this research because they will be made with solutions that have
had the same history. Thus, the approximate symmetry
observed in the measurements of u¢ and v¢ at maximum
drag reduction indicates that u¢ and v¢ are the same for flat
and structured surfaces. Rough agreement is also found
for turbulence profiles determined in this research for
conditions other than maximum drag reduction. This can
be seen if the data in Fig. 7b for a drag reduction of 29%
are replotted as v¢/u* versus y/hi where i=1, 2 and hi is the
distance from a wall to the location of zero Reynolds shear
stress.
Reynolds shear stresses are presented in Fig. 8a
(Re=20,700) and 8b (Re=48,000). For water flows the
Reynolds shear stress equals zero at y=1.158 for Re=20,700
and at y=1.25 h for Re=48,000. The extrapolation of the
Reynolds shear stresses to the wavy wall gives a wall shear
stress, sw2, which is the same as determined from the
pressure drop for situations in which low drag reductions
were realized. As depicted in Fig. 8a, the Reynolds shear
stresses decrease for all values of y with increases in drag
reduction. They are found to be zero over the whole
channel cross section for 82% drag reduction. Similar
behavior is observed for studies at Re=48,000, as shown in
Fig. 8b. The Reynolds shear stresses are zero over the
whole channel cross section when cm=14.5 ppm and the
percent drag reduction equals 82.
We agree with the finding of Nakagawa and Hanratty
(2003a), for this system, that the location of zero Reynolds
shear stress is the same as the location of the maximum in
the velocity profile, ym; it depends on the amount of drag
reduction. For a Reynolds number of 20,700, ymax=1.158 h
for water flows, ymax=1.132 h for DR2=23% and
Fig. 7. Normal turbulence intensity, v’, versus the normalized
distance from the wall, y/h, for two different Reynold numbers:
a Re=20,700, ci=500 ppm; n Water, u2*=45.1 mm/s; h
cm=0.62 ppm, %DR=23, u2*=39.6 mm/s; M cm=1.5 ppm, %DR= 30,
u2*=37.8 mm/s; · cm=7.7 ppm. %DR=82, u2*=19 mm/s.
Fig. 8. Mean Reynolds shear stress,uv, versus the normalized
b Re=48,000, ci=500 ppm; n Water, u2*=101.9 mm/s; h
cm= 2.9 ppm, %DR=29, u2*=85.7 mm/s; M cm=10.2 ppm, %DR=72, distance from the wall, y/h, for two different Reynolds numbers. a,
b as in Fig. 6
u2*=54.3 mm/s; · cm=14.5 ppm. %DR=82, u2*=45 mm/s
ymax=1.127 h for DR2=30%. For Re=48,000 the location of
zero Reynolds shear stress gradually approaches the center
of the channel with increasing drag reduction in that
ymax=1.25 h for water, ymax=1.157 h for DR2=29% and
ymax=1.05 h for DR2=72%.
Frequency spectra of the streamwise velocity fluctuations are presented in Fig. 9 for Re=20,700 at y/h=0.3. The
important feature is that the addition of polymer decreases
the relative contribution of high frequency fluctuations to
the turbulence. The spectra are the same for runs with the
same drag reduction even if ci and cm are different. A
cutoff frequency of 100 to 200 Hz, depending on the
amount of drag reduction, is observed for low drag
reduction runs. For conditions close to maximum drag
reduction a drastic damping of high frequency fluctuations
is seen. The magnitude of the cutoff frequency (40 Hz) at
maximum drag reduction is the same as obtained by
Warholic et al. (1999), when both walls were flat, again
indicating that the flow is the same at maximum drag
reduction for flat and wavy walls. Similar results are obtained from the frequency spectra of the normal velocity
fluctuations.
A different behavior is noted for y £ 0.16, which is
approximately the distance over which wave-induced flows
exist (Nakagawa et al. 2001). Here, the spectra for water
and for polymer solutions that cause low drag reductions
are roughly the same.
The method of injection and the concentration ci, have
an influence on the drag reduction. The effect of ci is depicted in Fig. 10, where the percent drag reduction on the
wavy wall is plotted against the mixed concentration cm.
For the same cm, an increase of ci is associated with an
increase in the drag reduction. For example, with
cm=0.25 ppm, the drag reduction is decreased from 20% to
13% by reducing the injected concentration from 500 to
100 ppm. Also, the realization of a drag reduction of 74%
requires a larger mixed concentration (cm@13 ppm) for
ci=100 ppm than for ci=500 ppm. With ci=100 ppm the
Reynolds shear stress was not observed to be zero over the
whole channel cross section and maximum drag reduction
was not reached, even at higher mixed concentrations.
These results on the effect of ci have also been obtained by
Warholic et al. (1999), who suggest that they support the
notion that the polymer molecules exist as aggregates in
the flow channel (Cox and Dunlop 1974). However, an
increase in ci (at constant cm) is accompanied by a decrease in the injection velocity. One of the reviewers feels
that this could lead to an asymmetric distribution of
polymers and that this, in some way, is responsible for the
effects. We cannot prove this is wrong. However, mixing
studies carried out by Warholic and our measurements of
the spatial variation of turbulence quantities (particularly,
Reynolds shear stress) over the whole cross section of the
channel provide no evidence of an asymmetric distribution.
5.3
Formation of large scale gelatinous aggregates of polymer
molecules
This section describes results that were obtained for
solutions that were created by sprinkling polymer particles. Figure 11 presents observations of the presence of
polymer structures at different Reynolds numbers. The
wavy wall is clearly seen at the bottom of each image. The
field of view is in the spanwise direction. The arrows
indicate the streamwise and wall normal directions.
Fig. 9. Dimensional streamwise power spectral density function,
Wu(f), versus the dimensional frequency, f, for y/h=0.3 and
Re= 20,700; n Water; h cm=0.62 ppm, ci=500 ppm, %DR=23;
M cm= 1.5 ppm, ci=500 ppm, DR%=30; · cm=2.0 ppm,
ci=100 ppm, DR%=21; + cm=7.7 ppm, ci=500 ppm, %DR=82
Fig. 10. The percent drag reduction, %DR2, versus the mixed
concentration, cm, for different injection concentrations, ci
(Re= 20,700)
693
694
Fig. 12. Instantaneous fluorescence images from a top view of the
channel. The arrow indicates the flow direction: a Re=6000,
ci‡500 ppm. b Re=11,000, ci‡500 ppm. c Re=20,700, ci‡500 ppm.
d Re=48,000, ci‡500 ppm
Fig. 11. Visual observations of the initiation of the heterogeneous
drag reduction by using the fluorescence imaging method. The
arrow indicates the flow direction: a Re=6000, ci‡500 ppm. b
Re=11,000, ci‡500 ppm. c Re=20,700, ci‡500 ppm. d Re=31000,
ci‡500 ppm. e Re=48,000, ci‡500 ppm
Instantaneous fluorescence images captured from a top
view of the channel are given in Fig. 12 for the same
conditions as existed for Fig. 11. For all cases, the mixed
concentration is close to 5 ppm and the injected concentration is greater than or equal to 500 ppm. Threadlike
structures are clearly observed for all Reynolds numbers.
An increase of the Reynolds number, from 6000 to 48,000,
corresponds to an increase of the shear rate at the wall
from 150 s)1 to 2300 s)1 for water flows. Shear forces at or
near the wall break the structure into a larger number of
entangled filaments. By comparing Figs. 11a and 11b, it is
seen that the density and length of the filaments increase
and that the thickness decreases as the shear rate increases. Noteworthy, is the observation of the distribution
of the filaments in the x-y plane. For all the Reynolds
numbers, the filaments are most concentrated near the
center of the channel. A similar result has been obtained
by Usui et al. (1988) in their study of heterogeneous drag
reduction.
The shape of the filaments is very close to a freeform
line. They display many twists and turns as they move in
the streamwise direction, approximately with the same
velocity as the surrounding fluid in the central region of
the channel. A filament was observed to occupy different
vertical locations at a given time and to change in shape
with time. An increase of the shear rate (the Reynolds
number) results in a complex fibrillar structure, similar to
the network structure discussed by Hagiwara et al. (1999),
which tends to stay away from the wall.
The presence of threadlike structures depends on the
concentration in the mixing tank (ci), the flow rate of the
injected solution and the mixing procedure. A critical injected concentration, ci, of 500 ppm is needed to form
filaments when the polymer solution is made by sprinkling
sieved particles onto the liquid in the mixing tank. By
increasing the injected concentration, the mixed concentration that is needed for the appearance of threadlike
structures in the test section is decreased. We should
emphasize (as discussed in section 4) that smaller aggregates of polymers probably exist when filamentous structures are not observed. However, we have no direct proof
for this.
Additional fluorescence measurements of structure
formation in the flow loop are depicted in Fig. 13 for injected polymer solutions with concentrations greater than
or equal to 1000 ppm. A large number of gel-like entities,
which circulate with the liquid during the agitation procedure, were formed in the mixing tank and a more
complicated network of filaments was observed in the flow
channel. An increase in the Reynolds number does not
change the size of the filaments, as is seen by comparing
Figs. 13b, and 13c. Again, the filaments are concentrated at
the center of the channel.
Figure 14 shows the threadlike structures from a top
view of the channel. Large filaments are present at a low
Reynolds number (Re=6000) over the whole channel
section. By increasing the Reynolds number (Re=11,000),
the filaments are stretched, and a network results. After
stopping the injection process the filaments disappeared,
695
Fig. 14. Visual observations of the top view of the channel for
high injected polymer concentrations: a Re=6000, ci‡1000 ppm,
cm= 35.4 ppm. b Re=11,000, ci‡1000 ppm, cm=5.4 ppm.
c Re= 20,700, ci‡1000 ppm, cm=6 ppm
because mechanical degradation occurred when the
solution circulated through the centrifugal pump.
Drag reduction is enhanced when large scale structures
are present. For Reynolds numbers 20,700 and 48,000,
larger drag reductions were realized when the polymers
were primarily in the form of filaments. For example, for
cm=5 ppm and Re=48,000, the amount of drag reduction
that is realized for ci=500 ppm is 58% when filaments are
present and 50% for homogenous drag reduction. Figure 14c corresponds to the condition of maximum drag
reduction. The most striking example of the beneficial
influence of polymer structures are the studies at Re=6000
and 11,000. No drag reduction was observed for homogeneous solutions over the range of cm covered in this
study. However, measurable amounts of drag reduction
were realized for non-homogeneous solutions.
5.4
Turbulence measurements for heterogeneous drag
reduction at low Reynolds numbers
Measurements of Reynolds shear stresses are presented for
Re=11,000 in Fig. 15a. The water flow was in the transition
region (kþ
s ¼ 22:4). A linear relation is observed for
y/h>0.2. Friction velocities obtained by extrapolating this
linear fit to the wavy wall agree with values obtained from
pressure drop measurements, even though the mixed
concentration is relatively high. The Reynolds shear stress
is observed to decrease for all values of y with increases in
drag reduction. For water flows, deviations from the linear
behavior for y/h<0.2 occur because of contributions to the
momentum flux by periodic variations of the mean velocity induced by the wavy wall and because of contributions
by molecular viscosity.
Figure 15b presents measurements of the average of the
Fig. 13. Demonstration of filaments for fully heterogeneous drag
root-mean-square
of the streamwise velocity fluctuations,
reduction cases: a Re=6000, ci‡1000 ppm, cm=35.4 ppm.
obtained over the crest and over the trough, against the
b Re= 11,000, ci‡1000 ppm, cm=5.4 ppm. c Re=20,700,
distance from the wall, y/h, made dimensionless with the
ci‡1000 ppm, cm=6 ppm
696
half channel height. The Reynolds number was Re=11,000. flat, and with results obtained by Den Toonder et al. (1997)
Close agreement is observed between the u¢ and results
for turbulent pipe flow if the streamwise velocity fluctuaobtained by Warholic et al. (1999), in which both walls are tions are made dimensionless with the friction velocity.
The peaks in u¢ are displaced outward with increasing drag
reduction. For Re=11,000, when filaments are present, the
peak value of u¢ is located at y/h=0.08 (y+=43) for
DR%=31. This is the same as observed for Re=20,700
(y+=42, DR%=29) when the polymer solution may be
considered to be homogeneous.
Measurements of the root-mean-square of the normal
velocity fluctuations are presented in Fig. 15c. They decrease systematically with increasing drag reduction for y/
h<0.25 (y+<150). Approximate agreement is observed with
results obtained in a channel with flat walls with approximately the same amount of drag reduction (Warholic
et al. 1999). Therefore, the presence of filaments does not
dramatically affect the turbulence statistics (at both large
and small Reynolds numbers), when comparisons are
made for the same amounts of drag reduction.
The first indication of drag reduction for Re=6000 was
found at a high mixed concentration (cm=35.4 ppm) when
threadlike structures were present. The non-dimensional
mean streamwise velocity, U+, is plotted in Fig. 16 against
the non-dimensional distance from the wall, y+, for
Re= 6000 and ci=1000 ppm. For water flow, the presence
of wall roughness is associated with a downward
Fig. 15. Measurements of the Reynolds shear stress: a The
streamwise turbulence intensity. b The normal turbulence
intensity. c For Reynold number Re=11,000. Here the injection
concentration is: ci=1000 ppm: n Water, u2*=23.5.1 mm/s;
h cm=5.4 ppm, DR%=19, u2*=21.15 mm/s; M cm=8.6 ppm,
DR%=23, u2*=20.62 mm/s; + cm=15 ppm, DR%=31,
u2*=19,52 mm/s; · cm=20 ppm, DR%=37, u2*=18.65 mm/s
Fig. 16. Non-dimensional mean streamwise velocity, U+, versus
the non-dimensional distance from the wall, y+ for Re=6000 and
ci=1000 ppm
displacement of the profile by an amount equal to DU/
u*=1.36. The downward displacement is reduced when
drag reduction is realized.
Frequency spectra at y/h=0.3 are given in Fig. 17a, b for
cases in which filaments existed. The frequency and the
spectral density function are scaled with the root-mean1=2
square of the streamwise velocity fluctuations, u2
¼ u00 .
The amount of drag reduction increases with increasing
mixed concentration, from 19% with cm= 5.4 ppm to 37%
with cm=20 ppm at Re=11,000. The polymer solution had
threadlike structures of the type seen in Figs. 13 and 14.
Figure 17a presents frequency spectra of the streamwise
velocity fluctuations for Re=11,000 and ci=1000 ppm at
y/h=0.3, for four different mixed concentrations. These
suggest a cut-off frequency, fc/u¢, which decreases from
fc/u¢=0.2 m)1 to 0.1 m)1 as the amount of drag reduction
increases from 19% to 37%. This behavior occurs for
0.3<y/h<1.0.
Frequency spectra are a sensitive indicator of drag
reduction, in that the cut-off frequency changes with small
changes on the amount of drag reduction. This can be seen
in Fig. 17b, where the spectral density functions, Wu(f),
normalized with the root-mean-square of the streamwise
velocity fluctuations, are plotted for Re=6000 and y/h=0.3.
The drag reduction was only 6% for cm=35.4 ppm. A
decrease in the cut-off frequency is seen for the polymer
solution.
6
Concluding remarks
6.1
Effect of wavy wall on turbulence and drag reduction
Larger drag reductions are realized with a wavy wall than
with a flat wall because the drag for a water flow is larger.
However, the fluid turbulence is roughly the same for flat
and wavy surfaces. The region close to the wall, where
wave-induced flows are important, is excluded from consideration. Particularly noteworthy is the finding that the
spectral density function of the velocity fluctuations and
the mean velocity profiles are the same, and the Reynolds
shear stresses are zero for flat and wavy surfaces at maximum drag reduction.
A common feature for both flat and structured surfaces
is a decrease in the relative importance of high frequency
(or high wavenumber) velocity fluctuations. Measurements of the cutoff frequencies are presented in Fig. 18 for
streamwise velocity fluctuations. The darkened symbols are
for Re=20,700, the open for Re=48,000. The squares are for
homogeneous solutions and the triangles for heterogeneous solutions. This cutoff is seen to decrease with
increasing cm (or increasing drag reduction) and to be
smaller for heterogeneous solutions. The differences be-
Fig. 17. Streamwise velocity spectra with the spectral density
function and frequency normalized by the root-mean-square of
the velocity fluctuations: a Re=11,000, ci=1000 ppm, y/h=0.3.
b Re=6000, ci=1000 ppm, y/h=0.3
Fig. 18. The streamwise cut-off frequency versus the mixed
concentration for y/h=0.3. n Re=20,700, ci=500 ppm; N Re=
20,700, ci= 1000 ppm; h Re=48,000, ci=500 ppm; M Re=48,000,
ci=1000 ppm
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698
tween the data for Re=20,700 and Re=48,000 are smaller if
a wavenumber, defined as the quotient of the frequency
and the root-mean-square of the streamwise velocity
fluctuation, is used.
The turbulence measurements, reported in this paper,
mainly reflect the behavior of the outer flow. The observation that the influence of polymers is similar to what is
observed for a flat wall is consistent with recent findings.
Liu et al. (2001) carried out PIV measurements in our
channel under conditions that both walls were flat. These
reveal that a characteristic of flow outside the viscous wall
region is the appearance of turbulence structures that can
extend over a large portion of the cross section. Two-point
spatial correlation functions were used to determine the
proper orthogonal modes.
These show that these large scale structures can be
represented by a small set of low-order eigenmodes that
contain a large fraction of the kinetic energy of the
streamwise velocity fluctuations and a small fraction of the
kinetic energy of the wall-normal velocities. Surprisingly,
the set of large-scale modes that contain half of the total
kinetic energy also contain two-thirds to three-quarters of
the total Reynolds shear stress in the outer region. From a
DNS, Na et al. (2001) called these structures ‘superbursts’
and argued that they may be pictured as plumes that
emerge from the wall.
Warholic et al. (2001) used PIV at Re@20,000 to study the
effects of drag reducing polymers on the turbulence in the
same system used by Liu et al. (2001). The most noticeable
effect was a damping of large wave number eddies. The
resulting decrease in wall-normal velocity fluctuations was
manifested by the diminishing of small swirls and by the
appearance of large regions in which velocity vectors in the
x-y plane are almost unidirectional. The decrease in the
small scale turbulence is accompanied by a decreased
activity of the wall in creating turbulence at high drag
reductions. This is demonstrated by an observed decrease
(or, even, elimination) of large scale ejections from the wall.
The study of Nakagawa et al. (2003a) with the structured
surface used in this research at Re=46,000 provides
approximate support to the suggestion of Raupach et al.
(1991) that the outer flow has universal characteristics. For
example measurements of u¢ and v¢, normalized with the
friction velocity, of the Reynolds shear stress coefficient,
and of the von Karman constant for the wavy and flat surfaces agree. (Similar results were obtained in the study of
Hudson et al. (1996).) Furthermore the PIV measurements
by Nakagawa and Hanratty (2001) at Re=46,000 with the
wavy wall used in this research show large scale structures
in the outer flow similar to what was observed by Liu et al.
(2001) for a smooth surface. The influence of polymers on
the large scale turbulence could, therefore, be expected to
have effects similar to what was observed for a flat wall in
that a greatly reduced role of the wall in creating turbulence
would be manifested by a decrease in large scale ejections.
6.2
Preparation and delivery of the polymers/influence
of large scale filaments and aggregates
The results presented in this paper document the notion
that drag reduction can be greatly affected by how the
polymer solution is prepared and delivered. A striking
example is that for ci=500 ppm the injected solution can
be heterogeneous or homogeneous. The heterogeneous
solution contained large filaments of gelatinous polymer
aggregates that resembled those that have been observed
by a number of previous investigators when they injected
very concentrated polymer solutions into the field through
tubes located in the flow. As has already been noted in
previous studies, larger drag reductions are observed with
the same mixed concentration when large filaments of
polymers are present. This is particularly evident in the
experiments at Re=6000 and Re=1100 for which drag
reduction was not observed when large filaments were not
present.
For homogeneous flows the drag reduction increases
with an increase in the concentration of polymers in the
injected solution. This supports the notion that drag
reduction is enhanced by the presence of small aggregates
of polymers, which increase in size and concentration with
increasing ci (See Cox and Dunlop 1974).
6.3
Maximum drag reduction
Maximum drag reduction is usually defined from experiments in which the pressure drop is measured as a function of the polymer concentration, cm. Some complications
arise because increases in cm can be associated with
changes in viscosity. For the experiments at Re=20,700,
represented by Fig. 4a, the cm is small and the rate of
shearing at the wall is large enough that the viscosity at the
wall is not changing appreciably with cm. A clear-cut leveling off of DR is observed at large cm. A maximum drag
reduction of 80% can be defined for the wavy wall at
cm=7.7 ppm. From Fig. 8a it is seen that the Reynolds
shear stress is zero at this condition. The leveling off is not
so clearly defined in Fig. 4b for Re=48,000. However, if
maximum drag reduction is defined as occurring at
cm=14.5 a value of 82% is obtained. From Fig. 8b it is seen
that the Reynolds shear stress is zero at this condition.
However, it is also seen that the Reynolds shear stress is
not equal to zero for cm=10.2 ppm at which %DR=72. It
would seem that a profile of zero Reynolds stresses could
provide a better-defined criterion for maximum drag
reduction.
Several investigators have found that the Reynolds
shear stress is zero at large drag reduction for polymer
(Warholic et al. 1999a) and for surfactant flows (Warholic
et al. 1999b; Kawaguchi et al. 2002; Gyr and Bewersdorff
1995). Gampert and Yong (1990) present results of a study
with a polyacrylamide copolymer in a channel flow. They
obtained values of the Reynolds shear stress correlation
coefficient about 0.1 for Re=10700 when DR=57%, and for
Re=16000 when DR=70%. Measurements of zero Reynolds
shear stress have been observed in our laboratory in several experiments other than those reported in this paper
and in the work of Warholic et al. (1999a). These include
studies in which the polymer was premixed and circulated
through the system (Vlachogiannis et al. 2002) and studies
in which mixing was accomplished by injecting a concentrated solution into the flow channel (Baik et al. 2003)
under conditions that filaments were produced. In the
latter experiments a completely different technique, PIV,
was used to measure the Reynolds shear stress.
On the other hand, Ptasinski et al. (2001) have recently
reported non-zero values of the Reynolds shear stress in a
pipe flow for which a drag reduction of 70 percent was
realized. The reason for this difference is not known.
However, it should be noted that these experiments differ
from the ones reported in this paper. A pipe flow, low
Reynolds numbers, polymer concentrations as high as
435 ppm and a developing length of roughly 850 diameters
were used. No assurance is given that maximum drag
reduction was achieved. Furthermore, it should be recognized that at large drag reductions the measured Reynolds
shear stresses are sensitive to changes in drag reduction.
This is illustrated in Fig. 8b for drag reductions of 72%
and 82%.
6.4
Theoretical consideration
The results presented in this paper could have implications
in developing a general understanding of polymer drag
reduction. Most applications will involve the injection of
concentrated polymers into a flow system. The interpretation of studies in such systems could be different if large
polymer filaments are present. For example, we now have
evidence that these structures probably did not exist in most
of the experiments of Warholic et al. (1999).
There are reasons to suspect that the presence of small
polymer aggregates enhanced drag reductions observed by
us and by Warholic et al. (1999). This could affect the type
models used to describe the interaction between the
polymers and the turbulence (at least for systems similar to
the one used in this research). Thus, it would be productive
to examine theoretically the behavior of entangled, in
addition to single, polymer molecules in a turbulent field.
The development of a theory that will encompass both
smooth and rough walls would seem to require the
development of constructs which are not tied too strongly
to the details of the interaction of the turbulent flow with
the wall. In this sense, a consideration of changes in the
spectral density function of the velocity fluctuations could
be useful. We find that the damping of high frequency (or
high wavenumber) fluctuations is very sensitive to the
amount of drag reduction and that spectra for smooth and
rough walls are similar when compared at the same percent drag reduction.
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