12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

DEVELOPMENT OF DRAG-REDUCED TURBULENT BOUNDARY LAYERS
ON SMOOTH WALLS WITH DIFFERENT POLYMER CONCENTRATIONSMohamed G. AbdElKader, John R. Elsnab, Nicholas Hutchins, and Jason P. Monty
Department of Mechanical Engineering
University of Melbourne
Parkville Victoria 3010 Australia
mabdelkaderg@student.unimelb.edu.au

ABSTRACT

stantaneous velocity fields of polymer drag-reduced turbulent
channel flow. He noticed damping in small-scale vortices in
the near-wall region accompanied by a reduction in the innerscaled wall-normal turbulent intensity. These modifications
reveal a measurable interaction between polymers and turbulence, especially the interaction with near-wall vortices and velocity streaks, which represents the turbulence regeneration cycle (Jiménez & Pinelli, 1999). Further research by White et al.
(2004) using wall-parallel-plane PIV detected a coarsening
and organization of the low-speed velocity streaks at 50% dragreduction. Dubief et al. (2004) used numerical simulation todescribe the role of polymer stretching and relaxing in disturbing the near-wall vortices and energizing the streamwise velocity fluctuation. However, the detailed polymer-turbulenceinteraction is still a subject of study, especially for the highdrag reduction regime (White & Mungal, 2008).
Despite the intensive research on the drag-reduced turbulent boundary layer characteristics, only a few studies investigated external flow applications (e.g., ship hulls, submarine
surface). Most of the external flow experiments dealt withthe behaviour of the turbulent boundary layer after polymer
injection, with a consequent variation of polymer concentration across the boundary layer, e.g., Fontaine et al. (1992);
Winkel et al. (2009). The current research investigates the effect of uniform polyacrylamide concentration (polymer ocean)on the development of turbulent boundary layers over a flatplate. This study gives clearer insight into drag-reduced turbulent boundary layer structure and the effect of different uniform polymer concentrations, namely 25, 50, and 100 ppm, onsuch flow structure in the outer region.

A range of polyacrylamide concentrations was used to investigate the development of drag-reduced turbulent boundary
layers over a towed flat-plate. A unique experiment was built
where a 5 m-long plate was towed through a large volumeof uniform concentration polymer solution. Time-resolved
particle image velocimetry data of instantaneous streamwise
wall-normal velocity fields were obtained owing to the uniqueframe of reference of the tow tank facility. Results reveal thatthe presence of polymer significantly affects the outer region
coherent structure of the turbulent boundary layer, via damping of small-scale vortices and the formation of high-strength
long shear layers.

INTRODUCTION
The turbulent flow of polymer solutions at minute concentrations displays a significant skin friction drag reduction ofnearly 80% relative to the turbulent flow of the solvent (usuallywater) alone (White & Mungal, 2008). This phenomenon is ofgreat importance as it potentially leads to substantial energy
savings in marine and liquid transportation applications. Since
the discovery of the effect of polymers on the turbulent boundary layer by Toms (1948), extensive research has been aimed
to describe the limitations and cause of the phenomenon. A variety of experimental, (Virk et al., 1967; Warholic et al., 1999;
White et al., 2004; Winkel et al., 2009; Elsnab et al., 2019) andnumerical (Sureshkumar et al., 1997; De Angelis et al., 2002;
Dubief et al., 2004) investigations in internal and external flow
systems, reveal the main characteristics which describe the turbulent boundary layer modifications due to polymer presence.
The reduction of skin friction drag is accompanied bymodifications to the turbulent boundary layer statistics. One
such modification is the slope change of the mean streamwise
velocity profile. Virk (1975) reported an upward shift in the
inner-scaled mean velocity profile logarithmic region at lower
drag reduction rates (Newtonian plug) and a slope increase inthis region with further drag reduction rise until an asymptotic
drag reduction value is attained. Additionally, the mean innerscaled streamwise turbulence intensity increases with polymer
concentration with its inner-scaled peak moving further awayfrom the wall, which was found to have a linear trend with thepercentage of drag reduction in the low drag-reduction regime(Elsnab et al., 2019).
The effect of polymer on the turbulent boundary layer
coherent structure has been investigated using experimental
flow visualization and simulation techniques. Warholic et al.
(2001) used particle image velocimetry (PIV) to obtain in-

EXPERIMENTAL SETUP
Experiments were conducted in the Michell Hydrodynamics Laboratory at the University of Melbourne using the
tow tank facility (see figure 1). The facility consists of a
60 m × 2 m × 2 m (L × W × H) tank equipped with overheadrails on both tank sides. A carriage travels back and forth withminimal vibration along 30 m of the tank length at a speed
up to 1.5 m/s. A horizontally mounted flat-plate of plan area,
5 m × 1.2 m (L × W), is suspended from the carriage inside
the tank. The tank side is equipped with a 1.0 m × 1.5 m viewing glass window for under water optical access located at
21.6 m from the plate start position. The total length of thetank was divided into three sections separated by two barriers
1.0 m × 1.8 m to isolate the water between the barriers (middle
section) from the rest of the tank water. The total distance travelled by the plate is contained within the middle tank section

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1.

Schematic of tow tank facility at the University of Melbourne

30 m × 2 m. Such a setup was chosen to reduce the amount ofpolymer required to obtain the desired concentration in sucha large facility. All the experiments presented here were conducted in the middle tank section of the facility, referred to as
the ’test-section’, while the plate traversed between the barriers. The volume of the water in the test-section is approximately 41000 L. The flow at the upstream end of the plate wastripped using ≈ 1 mm diameter wire, which slightly overstimulates the turbulent boundary layer based on the study by Erm& Joubert (1991).
The polymer used in the current experiment is watersoluble anionic polyacrylamide (APAM) manufactured by
Gongyi Xinqi Chemical plant (anionic polyacrylamide 9260),and has an average molecular weight, Mw (g/mol), of 6 million according to manufacturer specifications. The preparation process of the polymer solution was challenging due tothe large volume of the test-section. For example, a 100 ppm
concentration requires 4.1 kg of APAM. The preparation process of the aqueous polymer solution was carefully made inseveral steps. Firstly, the necessary amount of polymer wasmeasured using a precision scale (Sartorius Entris224-1s) witha maximum weight capacity of 220 g and resolution of 0.1 mg.
Secondly, the weighed powder was divided into batches to produce a highly concentrated polymer solution in a 100 L external tank using water from the test-section. The concentrated
solutions were stirred using a spiral mixer at 30 rpm to reduce the polymer mechanical degradation during the solution
preparation process. Thirdly, the final polymer solution wasmade by introducing these batches to the test-section acrossthe water surface area. Finally, the solution was left in the tankfor two days to ensure complete homogeneous polymer mixing across the test-section. The final polymer concentration
was checked using an Ubbelohde viscometer. The viscometer is manufactured and calibrated by CANNON Instrument
Company. The concentration check process starts by carefully preparing small samples with concentrations equivalent
to the required test-section concentrations (25, 50, 100 ppm)
and then comparing the viscosity of such samples with theones collected from the test-section at different locations andtimes. The test-section samples were collected before, during
and after each experiment to ensure a constant polymer concentration throughout the experiment. The collected and reference samples were kept at a constant temperature of 16◦ C
during viscosity measurement. The change of solution viscosity with polymer concentration for the reference samples
is presented in figure 2(a). The linear fit to the data was used
to estimate the polymer concentration based on the solution

viscosity for the collected samples. The data presented in figure 2(a) indicates that the change in concentration is less than
13% for all the experiments conducted in the current study.
Furthermore, the variation of solution kinematic viscosity withtime is presented in figure 2(b). The figure displays that a constant viscosity is maintained throughout each experiment. Six
samples were collected every measurement day, and the error
bars represent the 95% confidence interval for such samples.
For data analysis, the solution viscosity for all cases was taken
as that for the solvent due to the small concentrations used in
the current study.
Time-resolved x-z instantaneous velocity fields were acquired where x, y and z are the streamwise, spanwise, andwall-normal coordinates, respectively, and u, v and w indicate the corresponding velocity components. Each data set
was obtained by towing the plate through the field of viewat a constant speed U∞ ≈ 0.9 m/s (Rex = 9.5 × 105 – 4.5 × 106
for the Newtonian case). The PIV setup used in the current
study consists of two PCO dimax HS4 CMOS cameras with
32GB buffer. The field of view illumination is supplied bya Photonics DM20-527 dual head Nd:YLF laser that delivers
100 mJ/pulse at 1000 Hz. Each time the plate is towed past
the stationary PIV system, we acquire ≈ 5000 instantaneous
velocity fields capturing the streamwise development of thedrag-reduced turbulent boundary layer. The acquired images
are processed using an in-house PIV processing. The image
processing algorithm uses a multigrid interrogation analysis
with a final interrogation window size of 32 × 32 pixels, equivalent to 1.5 mm × 1.5 mm in real space with 50% overlap. The
number of passes acquired for the pure water (Newtonian) andpolymer solution (non-Newtonian) cases is 100 and 85, respectively, which requires five days for the non-Newtonian experiments. The total number of passes used produces a statisticalconvergence for the mean streamwise velocity, U. However,
we need at least 700 passes to reach turbulent intensity convergence as discussed by Lee (2017), which was not possible
in the current study due to time limitations.

RESULTS
The parameters for the different experimental cases are
summarised in table 1. The development of the momentum
thickness, θ , is shown in figure 3(a) along the streamwise
plate length, x plate . From figure 3(a), it is shown that the momentum thickness at a given x location reduces as the polymer concentration increases, indicative of an increasing dragreduction. The friction velocity, uτ , was calculated using the
momentum integral equation for the zero pressure gradient tur-2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 2. (a) Solution kinamtic viscosity vs. polymer concentration as measured using the Ubbelohde viscometer at 16◦ C. • reference
samples and the displayed equation represent the trendline equation fitting the reference samples. • 25 ppm, • 50 ppm, and • 100 ppm.
The error bars represent the 95% confidence interval for each case (13% viscosity variation for 25 ppm). (b) Kinamtic viscosity
variation with time during the different concentration experiments. The error bars represent the 95% confidence interval per day.
Table 1. Different exprimental cases in the current study at
x plate = 2.4 m

bulent boundary layer
Cf
dθ
=
=
dx
2



uτ
U∞

2
,

(1)

where C f is the coefficient of friction and U∞ is the free stream
velocity. The boundary layer thickness, δ99 , used for computing θ is identified as the location where the streamwise velocity reaches 99% of the free stream velocity. The streamwise
gradient of the momentum thickness, dθ /dx, was calculated
by numerical differentiation using the presented θ vs x plate
profile. Since the output is noisy, as expected from numerical differentiation, a fitting equation was used to smooth theresults based on the assumption that Reθ has a log-linear relationship with the coefficient of friction C f as presented byÖsterlund et al. (2000); Jiménez & Pinelli (1999). The following fit was assumed, C f = A log(Reθ ) + B, where A and B are
the fitting constants. The fitting to the developing C f obtained
for the Newtonian case is presented in figure 3(b). Note thatthe fitting points start at x plate & 1 m from the trip wire to avoid
any interference from the trip wire on the obtained C f .

[η]Mw µs,
NA kB T

δ99 (mm)

Reτ

%DR

Weτ

Symbols

Newtonian

42

1561

0

0



25 ppm

32

935

35.4

1.08

∗

50 ppm

22

544

56.4

0.732

x

100 ppm

19.9

422

65.5

0.575

O

determined using C∗ = 1/[η]. In the present study, thecritical overlap concentration C∗ is 763 ppm, which indicates that all the current concentration is in the dilute
regime.
The non-Newtonian friction velocity estimation was
performed using the same procedure as the Newtonian
one. Although the logarithmic fitting procedure has no
physical basis for the non-Newtonian cases, we find agood fit to the measured C f for x plate & 1 m. The percentage drag reduction is calculated using

The coefficient of friction, C f , variation along theplate length as a function of the streamwise Reynolds
number Rex for the Newtonian case is plotted and compared with the computed predictions in figure 3(c). The
solid lines represent the fittings by Schlichting, White
and Nikuradse as modified by Nagib et al. (2007) for thecanonical turbulent boundary layer. The calculated C f
is in good agreement with the predictions. The discrepancy at the lower Rex is attributed to the proximity of themeasurement location to the trip wire. Table 1 also includes the friction Weissenberg number, Weτ = λ u2τ /ν,which is the ratio of the polymer relaxation time, λ ,to the viscous time scale and represents the viscoelastic effect of the solution. λ , was calculated using Zimm
(1956) theory provided in
λ = 0.422

Case

%Dr =

τN − τNon
× 100,
τN(3)

where τN and τNon are the Newtonian wall and nonNewtonian wall shear stress, respectively.
Inner-scaled mean streamwise velocity U + profiles
at x plate = 2.4 m from the trip wire are presented in figure 4(a). The dashed line represents data from theturbulent boundary layer simulation provided by EitelAmor et al. (2014) at a comparable Reynolds number,and the solid line represents the maximum drag reduction asymptote as provided by Virk (1975). For the
Newtonian case, a good collapse is observed beyondz+ ≈ 100 between the experimental streamwise mean
velocity profile and the simulation reference data at
comparable Reτ = 1561. For the non-Newtonian case,the errorbar represents ± 5% error in the uτ assumption as calculated using the error analysis criteria provided by Moffat (1988). Figure 4(b) presents the normalized mean streamwise u0 2 . The discrepancies foundin the figure near the wall are a result of the lower PIV
interrogation window spatial resolution and the lower(2)

where [η] = 0.01Mw 0.755 is the intrinsic viscosity for
APAM solution Kulicke et al. (1982), µs is the solvent(water) dynamic viscosity, NA is Avogadro’s number, kB
is the Boltzmann constant, and T is the solution temperature. The Zimm theory is valid for dilute polymer solutions below the overlap concentration, C∗ , which can be3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
10-3

8

8

6

6

6

4

4

4

2

2

2

0

0
0

1

2

3

4

5

2

4

0
105

6
10

10-3

106

107

3

Figure 3. (a) Development of the momentum thickness θ along the plate length x plate in meters (Rex = 9.5 × 105 - 4.5 × 106 ) for
different polymer concentrations add. Symbols description is presented in table 1, (b) Calculated coefficient of friction C f basedon the momentum intergral equation for the Newtonian case,the log-linear fitting, C f = A log(Reθ ) + B, with A = −8 × 10−4
−3
Schlichting fit
and B = 9.6 × 10 , (c) the comparison between the calculated C f from fitting for the Newtonian case and−7/3−2−0.1502(2 log10 (Rex ) − 0.65),
White 0.4177(log(0.06Rex )) and
Nikuradse 0.02666Rex.

Figure 4.

Mean statistics for the different expirmental results at x plate = 2.4 m. (a) Mean streamwise velocity, U + , (b) Mean streamwise

turbulence intensity, u0 2 . The blue line represents the Virk’s asymptote, 11.7 log(z+ ) − 17. The black dashed lines represent thestatistical reference data from Eitel-Amor et al. (2014). The dashed dot line is the under-resolved statistics using estimation tool fromLee et al. (2016). The error bar represents ≈ 5% error due to the mehtod used to calculate uτ . Symbols definition is presented in table 1.

number of samples used to calculate the mean statistics(only 100 realizations were used for every plate position
x plate ), which is not adequate for second order statistics
convergence in the current experiment (Lee, 2017).
The black dashed line in figure 4(b) shows the DNS
data of Eitel-Amor et al. (2014) at comparable Reτ tothe Newtonian case. Due to the effect of spatial resolution, the PIV resolved variance is below this curve.
The black dot-dashed line shows the predicted DNS result at a matched resolution to the PIV experiment, using the predictive tool of Lee (2017). This serves as afurther validation of these measurements, demonstratingthat any discrepancy between the PIV and DNS for the
Newtonian case is entirely attributable to spatial attenuation and the unconverged turbulence intensities. Additionally, figures 4(a,b) illustrate the typical effect of thedrag-reducing polymer on the turbulent boundary layer.
Such effect appears in the upward shift in the logarithmic layer of the inner-scaled mean velocity profile forlow drag reduction cases and the increasing of the innerscaled near-wall streamwise turbulence intensity withincreasing polymer concentration.

A snapshot of the acquired real space instantaneous spanwise vorticity fields in the streamwise wallnormal plan is shown in figures 5(a,b) for Newtonian
and 100 ppm non-Newtonian cases, respectively. The
colorbar represents the magnitude of the spanwise vorticity squared ωy 2 . The boundary layer thickness ofthe drag-reduced turbulent boundary layer at 100 ppm(δ99 ≈ 20 mm) is approximately half of the Newtonian
one (δ99 = 42 mm) as observed from figures 5(a,b). To
better compare the details of the coherent structures, theouter-scaled field of view for the same vorticity field ispresented in figures 5(c,d). The vorticity field of the
non-Newtonian case indicates a reduction in the number of small-scale vortices, and the formation of highstrength, more coherent and elongated inclined vorticalfissures or shear layers in the outer layer. Zadrazil et al.
(2012) also observed the presence of shear layers in thevorticity field of pipe drag reduced turbulent flow using
polymer injection. However, The features in the current
study extend to z ≈ δ in the outer region of the drag reduced turbulent boundary layer. These modifications become more observable with drag reduction increase. On
412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Shear layer

Figure 5. Moving plate frame of reference x-z plane of the spanwise vorticity strength ωy2 at x plate = 2.4 m (Rex ≈ 2.25 × 106 ). (a) and(c) pure water (Newtonian) %DR = 0 and (b) and (d) 100 ppm polymer concentration (non-Newtonian) %DR = 65 of the developing
turbulent boundary layer on smooth wall. The plate is moving from right to left. (a) and (b) dimensional coordinates, (c) and (d)outer-normalized coordinates.

𝜃𝑟

Figure 6. Mean 2-pt correlation for the streamwise velocity fluctuation Ruu for x plate =2.4 m and zre f = 0.2δ . (a) Newtonian, (b) 25
ppm, (c) 50 ppm, and (d) 100ppm. The dashed dot line shows the larger pricipal axis as determined from the principal component
analysis. (e) the change in the correlation feature inclination angle, θr , with polymer concentration.

the other hand, The Newtonian turbulent boundary layershows scatterd features in the vorticity field. The dragreduced turbulent boundary layer coherent structure canalso be assessed from the normalized two-point correlation for the streamwise velocity fluctuation field Ru0 u0 as
shown in figure 6 for x plate = 2.4 m, and reference position zre f = 0.2 δ . The normlized cross correlation coefficient was calculated usingRuu =u0 (x, z)u0 (x + ∆x, z + ∆z),
σu0x σu0

where σ is the standard deviation of the corresponding signal. It shows that the mean coherent structure is gradually squeezed in the wall-normal direction and elongated in the streamwise direction with increasing polymer concentration. The black dashed lines
in figures 6(a–d) represent the contour for Ru0 u0 = 0.1.
The contour extends to ≈ 4δ for Ru0 u0 = 0.1 in thestreamwise direction for the 100 ppm case, which is thehighest polymer concentration investigated in the current study. Furthermore, the coherent inclination angle
was determined using the principal component analysis(4)x+∆x,z+∆z

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
(PCA). The angle of the major axes obtained from PCA
is used as the feature inclination angle, θr , an example
of which is given by the solid line on figure 6(d). Modification to this computed angle with polymer concentration is shown in Figure 6(e). As presented, the angle
becomes smaller while increasing polymer concentration. Fontaine et al. (1992) also calculated the principal stresses using the scatter plot of the streamwise andwall-normal velocity fluctuations and found a decreasein the major principal stress axis angle with drag reduction.

ological behaviour of polyacrylamide. Prog. Polym.
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Approach to an asymptotic state for zero pressure gradient turbulent boundary layers. Philos. T. R. Soc. A.
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Österlund, J. M., Johansson, A. V., Nagib, H. M. &Hites, M. H. 2000 A note on the overlap region in
turbulent boundary layers. Phys Fluids 12 (1), 1–4.
Sureshkumar, R., Beris, A. N. & Handler, R. A. 1997
Direct numerical simulation of the turbulent channel
flow of a polymer solution. Phys Fluids 9 (3), 743–
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Toms, B. A. 1948 Some observations on the flow of linear polymer solutions through straight tubes at large
Reynolds numbers. Proc. of In. Cong. On Rheology,
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CONCLUSION
It is well known that polymers significantly affectthe near-wall region of the turbulent boundary layer. In
this study, an investigation for the boundary layer regionz > 0.2δ is presented using a uniform polyacrylamide
concentration (polymer ocean). A homogenous concentration eliminates the effect of polymer mixing accompanied by the injection process on the turbulence features. The polymer has a noticeable effect on the outerregion vortical structure. Such effect is observed in thepresence of high-strength and elongated features that extend to z ≈ δ . Two-point correlations of streamwise velocity fluctuations show an extended feature in the orderof 4δ in the streamwise direction at high drag reduction, approximately 2–3 times the length of the structurein the Newtonian case. The inclination angles of thesefeatures were determined using the principal component
analysis. Such angles decrease with increasing polymer
concentration. These findings indicate that the polymer
effect extends to the outer region of the turbulent boundary layer.
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