# Fully turbulent flows of viscoplastic fluids in a rectangular duct

Rodrigo S. Mitishita 1<EMAIL_ADDRESS>Jordan A. MacKenzie 2 Gwynn
J. Elfring 1 Ian. A. Frigaard 1,3 1 Department of Mechanical Engineering,
University of British Columbia, 2324 Main Mall, Vancouver, BC, V6T 1Z4, Canada
2 Department of Chemical and Biological Engineering, University of British
Columbia, 2360 E Mall, Vancouver, BC, V6T 1Z3, Canada  3 Department of
Mathematics, University of British Columbia, 1984 Mathematics Rd, Vancouver,
BC, V6T 1Z2, Canada

###### Abstract

Turbulent flows of viscoplastic fluids at high Reynolds numbers have been
investigated recently with direct numerical simulations (DNS) but experimental
results have been limited. For this reason, we carry out an experimental study
of fully turbulent flows of a yield stress fluid in a rectangular aspect ratio
channel with a high-resolution laser doppler velocimetry (LDA) setup. We
employ aqueous Carbopol solutions, often considered to be a simple yield
stress fluid. We formulate different concentrations to address the effect of
the rheology of the fluid on the turbulence statistics at an approximately
constant Reynolds number. Additionally, we also perform experiments with a
single Carbopol formulation at different Reynolds numbers to study its effect.
The flow analysis is performed via rheology measurements, turbulence
statistics and power spectral densities of velocity fluctuations. The addition
of Carbopol to the flow increases turbulence anisotropy, with an enhancement
of streamwise velocity fluctuations and a decrease in wall normal velocity
fluctuations in comparison to water at the same mean velocity. This change is
reflected on the power spectral densities of streamwise velocity fluctuations,
where we observe a large increase in energy of large scale turbulent
structures. Conversely, the energy of smaller scales is decreased in
comparison to water, where the energy drops with a steeper scale than the
Newtonian power law of $k_{x}^{-5/3}$. As we increase the Reynolds number with
a Carbopol solution, the streamwise Reynolds stresses approach Newtonian
values in the core, which suggests diminishing effects of shear-thinning. The
power spectral densities reveal that the energy content at larger scales
decreases slightly with the Reynolds number. However, the shear thinning
effects do not disappear even as the Reynolds number approaches 50000.

###### keywords:

Turbulent flow, yield stress fluids, Carbopol.

††journal: J. non-Newt. Fluid Mech.

##  1 Introduction

Turbulent pipe flow of viscoplastic or yield stress fluids are common in a
large number of industrial processes, such as transport of slurries during
drilling and cementing of oil and gas wells [1, 2], particle transport
applications [3, 4], pulp suspension flows [5], and various other processing
flows (e.g. food and personal products). Although fluids used in industrial
applications often exhibit more complex behaviours than simple viscoplasticity
and shear-thinning, e.g. viscoelasticity [6] and time dependence/thixotropy
[7], these characteristics are often neglected in flows where viscous effects
are dominant [8]. By doing so, the only difference from the Newtonian
behaviour is that the viscosity is a function of the strain rate [9], such as
inelastic, non-thixotropic yield stress liquids. These fluids are aptly named
generalized Newtonian (GN) fluids, and often provide a good approximation for
predicting the flow behaviour of viscoplastic fluids in practical
applications. While turbulence is still an active research topic, and there
are a limited number of studies with yield stress fluids.

Early studies with viscoplastic fluid flows from the 1950s have been performed
via friction factor correlations. Metzner and Reed [10], Dodge and Metzner
[11] developed a methodology for predicting the friction factor of power law
fluids, together with the argument that the same method should be applicable
to all generalised Newtonian fluids and for different duct shapes. Application
to yield stress fluids can be found in [12, 13, 14]. Other authors have
developed their own correlations [15, 16, 17, 18, 19], some of which enjoy
popularity in different process industries. Anbarlooei et al. [20] and
Anbarlooei et al. [21] have developed friction factor correlations via a
phenomenological approach based on Richardson’s energy cascade and
Kolmogorov’s second similarity hypothesis for fully turbulent Newtonian flows,
which states that in the inertial range of intermediate length scales of the
energy cascade, the energy spectra of velocity fluctuations scales with a
power law of -5/3 in wavenumber space, while being uniquely determined by the
energy dissipation rate and independent of viscosity [22]. These arguments
were combined with the friction factor method of Gioia and Chakraborty [23].
To formulate their theory, the authors assume that Kolmogorov’s second
similarity hypothesis also holds for a viscoplastic fluid. Their studies
showed good agreement with Dodge & Metzner expressions for the turbulent
friction factor. However, there are discrepancies between predictions and
experiments, even when the correlations have been experimentally validated
[11, 7, 24, 25]. Many of the phenomenological theories also are used to
predict turbulent transition, again with noted differences [26, 27].

The discrepancies between experiments with viscoplastic fluids and friction
factor correlations may come from limited rheological measurements in early
years and from the materials used experimentally, which include slurries [28]
and pulp suspensions [5]. Viscoelasticity or thixotropy effects are not
usually accounted for, but to some extent are always present in non-Newtonian
flows. For example, when flexible polymers are added to water, the resulting
viscoelasticity can significantly reduce frictional drag in turbulent flow,
often near 70 % [29, 30, 31, 32, 33]. It is perhaps only in recent decades
that researchers have settled on specific fluids that are better suited as a
relatively simple yield stress fluid. By far the most widely used polymer
additive is Carbopol [34]. When correctly mixed, it provides a clear solution
which is ideal for visualization and laser velocimetry experiments [35, 25,
36, 8], while exhibiting little viscoelastic behaviour beyond the yielding
point and negligible thixotropy [35, 34, 37], with only the yield stress and
shear-thinning as dominant characteristics. Thus, Carbopol solutions have been
considered to be very close to an ideal viscoplastic fluid and are also used
in our study.

Non-intrusive flow velocimetry techniques such as particle image velocimetry
(PIV), laser doppler anemometry (LDA) and ultrasound doppler velocimetry (UDV)
advanced the field significantly. These techniques have been widely used over
the years in turbulence measurements to provide high-resolution measurements
of velocity fluctuations, combined with pressure and flow rate measurements
and rheological characterization of the working fluids. Notably, velocimetry
experiments provided valuable information on the transition to turbulence,
which is still an active research topic with both Newtonian and non-Newtonian
fluids. Peixinho et al. [25] observed that turbulent puffs develop more
frequently with an increase in Reynolds number, eventually becoming turbulent
once the puff frequency is high enough, i.e. a saturation phenomenon. Güzel et
al. [8] concluded that fully turbulent flows in viscoplastic fluids are
achieved only when the yield stress is exceeded by the fluctuating Reynolds
stresses, across the entirety of the pipe. Using data from [8], Güzel et al.
[27] concluded that an estimation of transitional Reynolds numbers can be
predicted from a radial averaged Reynolds number from the time-averaged
velocity profile. The average turbulence intensity across the pipe must be
sufficiently high to overcome the yield stress. An interesting rheological
effect is the asymmetry of the average velocity profile during transition, as
seen in experiments with viscoplastic fluids [25, 7, 8] and in viscoelastic
drag-reducing polymer solutions [38, 8].

LDA experiments by Peixinho et al. [25] showed results of turbulent pipe flow
in Carbopol solutions and found an upturn in the velocity profile scaled in
inner units, and an increase in the streamwise Reynolds stresses in comparison
to water, similar to drag-reducing solutions. The Carbopol used by Peixinho et
al. [25] showed strong drag reduction behaviour, more than a CMC solution also
used in the study. Whether the drag reduction in their Carbopol is evidence of
significant elasticity or shear-thinning is not clear [39, 40]. Presti [24]
also observed streamwise turbulent intensities being larger than the Newtonian
counterpart (similar to Peixinho et al. [25]) near the wall and radial
turbulent intensities being similar to the Newtonian case. As the Reynolds
number increases, streamwise and wall normal velocity fluctuations also seem
approach the Newtonian values, likely due to the increased viscous response to
high frequency turbulence. A similar Reynolds number dependence was also
observed during the turbulent flow of a Laponite solution, a thixotropic,
viscoplastic fluid in the experiments by Escudier and Presti [7]. More details
about the high frequency dynamics can be studied through the frequency spectra
of velocity fluctuations of viscoplastic fluids, which has not been
investigated experimentally, to the best of our knowledge.

With all the above complexities, it is interesting to reflect that the most
accurate way to study the fully turbulent flow of a GN fluid is via direct
numerical simulation (DNS). These studies have become more common in the past
decade but still lag behind those performed for Newtonian and viscoelastic
fluids. Resolution of models such as the Bingham or Herschel-Bulkley fluid
models is prohibitively expensive for DNS because of the singular viscosity at
zero shear rate [40]. Instead most of high Reynolds number DNS results that we
know have been computed using regularized versions of these models. Friction
factor results from the DNS of a power-law fluid in [41] agree well with the
measurements of Dodge and Metzner [11]. DNS can also provide turbulence
measurements that are challenging to obtain experimentally such as the energy
budgets, which require data on all three components of velocity, pressure
fluctuations and also viscosity fluctuations in non-Newtonian fluids
specifically. In the work of Singh et al. [42], the turbulent kinetic energy
budgets are presented for pipe flow of power-law fluids, where the amount of
shear-thinning is quantified by the decrease in the power-law index $n$, with
with $0<n<1$. The dependence of Reynolds stresses on the power law index $n$
is found mainly near the wall, where the velocity gradients and also
fluctuations of shear rate are larger. Additionally, the authors observed a
decrease in the turbulent production term, and an increase of the energy
dissipation term as $n$ decreases. Similarly, Singh et al. [41] analyze
Reynolds number effects in turbulent flows of shear thinning fluids. They
conclude that the effects of Reynolds number on velocity fluctuations are more
apparent only in the buffer layer and viscous sublayer, with increased
turbulence production and viscous dissipation as $Re$ increases.

Shear-thinning effects can play a large role in turbulence dynamics, as shown
by DNS studies [40, 43, 42]. The decrease in power-law index in a Herschel-
Bulkley fluid can increase the root mean square of velocity fluctuations
$u_{rms}$ and give a small decrease in Reynolds stress. These changes in
turbulent intensities are a consequence of the enhancement of more elongated
turbulent structures and the attenuation of smaller scale vortices [42]. a
similar effect to that observed in turbulent drag reduction with viscoelastic
fluids [30]. The investigation by Rosti et al. [44] is notable for including
viscoelasticity in the laminar regime via a Kelvin-Voigt modification of the
Bingham constitutive equation without use of regularization, but the turbulent
regime had negligible viscoelasticity with a relatively low Reynolds number.
Their results highlight that the yield stress can effectively dampen
turbulence structures, with larger unyielded volumes of fluid near the core as
the yield stress increases. As the yield stress becomes larger, streamwise
turbulent structures are also enhanced, and small scale structures are
suppressed [43, 44].

While these DNS studies are breaking new ground, the experimental results
available are limited and often have used fluids with a more complex rheology
than Carbopol, e.g. dilute viscoelastic fluids. A quantitative analysis of the
effect of Reynolds number and the most dominant rheological features of
viscoplastic fluids (the yield stress and shear-thinning viscosity) has not
been performed in detail with experiments. This is likely due to the
difficulties of achieving a turbulent flow by pumping a viscoplastic fluid in
experiments. To account for this gap, we present an experimental investigation
of turbulent flows with Carbopol solutions at three different concentrations.
We present velocity and Reynolds stresses measured with a two component LDA
setup, with high spatial and temporal resolution adequate to resolve the small
scales in the inertial range. We also provide a comparison of spectral
analysis between water and Carbopol solutions, to assess the changes in energy
content of different eddy length scales. Such an analysis has not yet been
performed with yield stress fluids experimentally or via DNS, to the best of
our knowledge. In the turbulent regime, Carbopol solutions are often
considered inelastic, and thus the viscous or shear-thinning contribution to
the flow is considered dominant [8]. Therefore, we focus our analysis on the
effect of the shear thinning behaviour, in addition to the possible influence
of the yield stress.

The paper is organized as follows. In Section 2 we outline the experimental
methods and procedures with both LDA and also rheometry. The results are
organized in two sections: In Section 3, the effect of the Carbopol
concentration in the turbulent flow is investigated via rheological
characterization of the test fluids, statistical methods and spectral analysis
at a similar Reynolds number. Then, Section 4 presents an analysis of the
effect of the Reynolds number in the flow statistics and spectra, while
keeping the Carbopol concentration the same. The major contributions of the
paper are summarized in Section 5.

##  2 Experiments

###  2.1 Flow loop

We carry out our turbulent flow investigation in a horizontal, pump driven
flow loop through a transparent $7.5\,m$ test section with a rectangular cross
section, connected to the pump discharge pipeline by directional valves. A
schematic of the flow loop is presented in Figure 1. The rectangular test
section (“Channel” in Figure 1) is made of three 2.5 $m$ long, clear acrylic
channels connected together by flanges. The inner dimensions of the channel
are width $W=50.8$ mm x height $H=25.4$ mm $=2h$, where $h$ is the channel
half-height, with hydraulic diameter $D_{h}=2WH/(W+H)$ of 33.8 $mm$. The flow
in the storage tank is driven by a Netzsch NEMO progressing cavity pump,
capable of a maximum flow rate of 20 $l/s$. This type of pump minimizes
shearing of the fluid, thus decreasing the degradation rate of polymer
solutions. A bypass pipe connects the the pump discharge to the tank. A Parker
pulsation damper was installed after the pump discharge to prevent pressure
fluctuations. An Omega FMG 606 magnetic flow meter, with accuracy of 0.2 % of
full scale is used to measure the average flow rate, and the average velocity
can be calculated by dividing the flow rate by the cross sectional area of
each test section. The pressure drop along 2.5 m of the rectangular test
section is measured by an Omega DPG 409 differential pressure transducer (P1
in Figure 1), with 0.08 % accuracy and data acquisition rate of 3 hz. The
temperature of the fluid in the tank is measured by an Omega TC-NPT pipe
thermocouple, of 0.5 % accuracy. We do not control the fluid temperature in
the loop, but the temperature increases by less than 2∘C during a full
velocity profile measurement. Remote control of the pump speed, as well as the
signal acquisition from the thermocouple, pressure transducers and flow meter
are provided by the National Instruments LabVIEW software and compact data
acquisition modules. We perform laser doppler velocimetry (LDA) experiments at
over 5 m (or approximately 150$D_{h}$) downstream of the test section inlet,
where we consider the flow to be fully developed.

Figure 1: Flow loop schematic.

###  2.2 Laser doppler anemometer setup

To acquire turbulent velocity data in the Carbopol solutions, we perform
instantaneous, point-wise velocity measurements with a two-component Dantec
Dynamics FlowExplorer LDA setup in back-scatter mode. The receiving optics and
laser system are contained in a probe. The laser source supplies a pair of 532
nm wavelength (green) laser beams, separated horizontally by a distance of 60
mm for velocity measurements in the streamwise direction (or x-direction), and
a pair of 561 nm wavelength (yellow) laser beams, separated vertically by a
distance of 60 mm for velocity measurements in the wall normal direction
(y-direction). For clarity, the LDA measurement schematic is shown in Figure 2
(a), along with main flow coordinates. The x-direction corresponds to the main
flow direction of positive streamwise velocity $U$. The y coordinate
represents the direction of $V$ or wall normal velocity, with the reference
zero point on the bottom wall. Therefore the positive y direction corresponds
to fluid moving away from the bottom wall at $y/h=0$ to the centre $y/h=1$.
The z coordinate corresponds to spanwise directions with the reference zero
point on the front wall. Therefore the measurement plane, which is the channel
centre plane, is located at $z/W=0.5$ or $z=25.4$ mm. These details
illustrated in Figure 2 (b). The optical setup allows for an ellipsoidal
measurement volume of 0.1 mm diameter and 0.3 mm length with a 150 mm focal
length lens. A frequency shift of 80 Mhz is applied to each laser by a Bragg
cell. The probe is connected by to the Burst Spectrum Analyser (BSA) signal
processor for data acquisition with the Dantec BSA software. Seeding particles
manufactured by Dantec Dynamics, of 5 $\mu$m average size, are used for the
experiments. The uncertainty of the LDA system measurements, according to the
factory calibration certificate, is 0.1 %.

Figure 2: Three-dimensional (a) and front view (b) schematics of the LDA
measurement setup in the channel. The channel centre plane, or the measurement
plane, is shown in dotted lines.

Considering that LDA measures fluid velocity at a point, the probe requires a
traverse system for velocity profile measurements. Our custom traverse system
consists of a linear motorized actuator by Zaber Motion Control of a total
travel distance of 100 mm with 25 $\mu$m accuracy for vertical motion, to
allow traverse in the y-direction. Attached to the vertical traverse is
another linear actuator by Zaber for horizontal movement in the z-direction in
Figure 2. This actuator has a total travel distance of 250 mm with 63 $\mu$m
accuracy. We estimate the position of the walls ($y=0$ and $z=0$ in Figure 2)
with the LDA probe by monitoring the photomultiplier anode current value while
traversing from a position within the channel towards the wall. We record the
zero point where the anode current reading shows very high values due to
reflection of the laser beams, as the measurement volume approaches the wall
location. The accuracy of the bottom (via y-direction traverse) and front wall
(via z-direction traverse) location measurement is assumed to be 0.1 mm and
0.3 mm, which is the diameter and length of the measurement volume
respectively. The effect of refraction of the laser beams while traversing in
the z-direction is calculated with the refractive indices of both Carbopol or
water ($n_{r}=1.33$) and the clear plexiglass ($n_{r}=1.49$). This results in
a measurement volume traverse of 0.691 mm within the channel as the probe
moves 0.5 mm.

###  2.3 Rheometry

Rheological measurements are performed to characterize Carbopol solutions
prepared for the flow loop experiments. A high-resolution Malvern Kinexus
Ultra+ rheometer with a set of roughened parallel plates with 40 mm diameter
is employed for all rheology experiments to prevent wall slip, which is common
in yield stress fluids. The samples used in the rheometer were taken from the
flow loop tank after a full traverse with the LDA at a constant bulk velocity.
The rheometry is performed at the same temperature as the average recorded
from the flow loop during the experiments. Prior to every rheolometry test, we
perform a pre-shear at $\dot{\gamma}=100s^{-1}$ for 30 s followed by a 30 s
controlled stress rest at 0 Pa. We perform shear-rate controlled ramp-up/down
tests with a gap height of 1 mm to assess thixotropy and measure the viscosity
of the Carbopol solutions at low shear rates. Additionally, these experiments
are useful to estimate the dynamic yield stress of the fluids via a fit to the
Herschel Bulkley model for viscoplastic fluids:

$\tau=\tau_{y}+K\dot{\gamma}^{n},$ (1)

where $\tau$ is the shear stress [Pa], $\tau_{y}$ is the yield stress [Pa],
$K$ is the consistency index [Pa sn] and $n$ is the power law index [-]. Note
that the Equation 1 is valid only if $\tau>\tau_{y}$, because if
$\tau<\tau_{y}$ then $\dot{\gamma}=0$. To measure the viscosity at high shear
rates, a steady state flow curve test is performed with a gap of 0.3 mm to
minimize inertial effects. Next, stress-controlled amplitude sweeps at 0.5 hz
were carried out to measure the values of the storage ($G^{\prime}$ [Pa]) and
loss ($G^{\prime\prime}$ [Pa]) moduli. Finally, we perform frequency sweeps
from 1 to 0.01 Hz in the linear viscoelastic regime.

###  2.4 Carbopol mixing and flow loop experiment procedure

Before showing the main turbulence results, we present the procedure for
mixing the Carbopol solutions, followed by the flow loop data acquisition
procedure with the LDA, pressure, temperature and flow rate sensors. The
mixing procedure is similar to Bizhani et al. [2]. The preparation of the
Carbopol solution consists in mixing concentrated batches of 30 l of water at
a specific concentration of the Lubrizol Carbopol EZ-2 powder. The
concentrated solutions were mixed in a bucket with a three blade impeller at
150 rpm for a whole day. Each specific concentration of Carbopol is measured
relative to the total desired volume of 220 l of water. Next, the concentrated
solution at a given concentration and 190 l of water are added to the flow
loop tank. This solution is acidic and needs to be neutralized with NaOH to
build viscosity and yield stress. With the fluid mixing at a constant pump
speed, an aqueous solution of sodium hydroxide of 1/3.5 NaOH to Carbopol
weight is added to the tank, and mixed for 45 minutes for homogenization.
Three concentrations of Carbopol were used in our experiments: 0.06%, 0.08%
and 0.10%.

The main LDA velocity measurements were performed at bulk velocity $U_{b}=3.8$
m/s for 0.06% Carbopol, $U_{b}=5.8$ m/s for 0.08% and $U_{b}=7.0$ m/s for
0.10%, with $U_{b}$ calculated from the flow meter measurements. As we show in
the next section, these velocity parameters allow us to investigate the effect
of changes in rheology in the turbulent flows at somewhat similar Reynolds
numbers of the order of 20000. Higher velocities were needed for the more
concentrated solutions to achieve a fully turbulent flow. Additional
experiments with the 0.06% Carbopol were performed at $U_{b}=2.8$ and $5.8$
m/s, to analyze the influence of the Reynolds number with a constant Carbopol
concentration. For comparison, we also performed LDA experiments with water at
bulk velocities $U_{b}=2.8,3.8$ and $5.8$ m/s and only pressure, flow rate and
temperature measurements at $U_{b}=7.0$ m/s.

We consider the velocity profile to be symmetric along the centre plane of the
channel during fully turbulent flow, since reports of asymmetry in the
velocity profile have been reported during transition to turbulence in
Carbopol solutions [25, 8]. Thus, a traverse to measure half of the total
velocity profile corresponds to measurements on the centre plane of the
channel (Figure 2), from the bottom wall at $y/h=0$ to the centreline. Taking
this into account with the Carbopol solutions, we measure the velocity at each
position in the y-direction until at least 50000 measurements are acquired for
the $U$ component, and at least 30000 points for the $V$ component in non-
coincidence mode. Coincidence mode velocity measurements with Carbopol were
limited to 30000 data points due to the lower data rate. At least 50000 data
points were measured for all water measurements. A verification of statistical
convergence is reported in the supplementary material.

The temperature, flow rate and pressure drop are recorded at the same time as
the LDA experiments. The pressure drop measurements $\Delta P$ are used to
calculate the mean wall shear stress $\tau_{w}=\Delta PD_{h}/4L$, where $L$ is
the length between each pressure tap. With $\tau_{w}$ we define the friction
velocity $u_{\tau}=\sqrt{\tau_{w}/\rho}$. The density of Carbopol is the same
as water ($\rho$ = 999 kg/m3). With the bulk velocity and the friction
velocity we define the generalized Reynolds number and the frictional Reynolds
number, $Re_{G}=\rho U_{b}D_{h}/\eta_{w}$ and $Re_{\tau}=\rho
u_{\tau}h/\eta_{w}$ respectively, where $\eta_{w}=\tau_{w}/\dot{\gamma}_{w}$
is the viscosity of the fluid at the wall and $h$ is the channel half-height,
which is also the boundary layer height in fully developed channel flow. The
mean shear rate at the wall $\dot{\gamma}_{w}$ can be obtained with the wall
shear stress and rheometer flow curve data fit to a constitutive equation.
This definition of $\eta_{w}$ has been employed in a few other experimental
works such as Escudier et al. [31] and Owolabi et al. [45] given the
difficulty to establish the Reynolds number with shear-thinning fluids, where
the viscosity varies across the channel. The viscosity at the wall is also
used to define the wall unit $y^{+}_{0}=\eta_{w}/\rho u_{\tau}$, a viscous
length scaled used to normalize the wall normal coordinate $y$, measured from
the bottom wall of the channel.

The scope of our experimental work is to analyze the effect of the rheological
changes on the statistics of the turbulent channel flow in Section 3. We also
study the effects of a relatively wide range of Reynolds numbers with the same
Carbopol concentration in Section 4. We present the main flow loop parameters
in Table 1, for the experiments with water and all concentrations of Carbopol
solutions.

Ub [$m/s$] |  $Re$ [-] |  $u_{\tau}$ [$m/s$] |  $Re_{\tau}$ [-] | T [∘C] |  $\tau_{w}$ [$Pa$]
---|---|---|---|---|---
Water
2.86 | 105480 | 0.134 | 1850 | 23.6 | 17.9
3.85 | 147660 | 0.173 | 2490 | 25.3 | 29.8
5.86 | 232810 | 0.249 | 3700 | 26.8 | 61.7
7.14 | 256290 | 0.304 | 4090 | 22.4 | 92.1
Carbopol 0.06%
2.79 | 9430 | 0.161 | 200 | 25.4 | 25.9
3.80 | 19370 | 0.210 | 400 | 25.8 | 44.1
5.72 | 50670 | 0.296 | 970 | 26.6 | 87.8
Carbopol 0.08%
5.82 | 24740 | 0.306 | 490 | 26.9 | 93.4
Carbopol 0.10%
7.08 | 21000 | 0.373 | 420 | 26.7 | 139.0
Table 1: Parameters for water and Carbopol flow loop experiments.

##  3 Results and discussion: effect of Carbopol rheology

###  3.1 Rheology and pressure measurements

To better contextualize the turbulence measurements, we first present the
rheological characterization of the Carbopol solutions. Figure 3 shows the
ramp-up and ramp-down flow curves with samples from the three Carbopol
solutions, collected from the tank after each LDA traverse. The ramp time was
3 minutes from a shear rate of $0.001$ to $200$ s-1. All three Carbopol
solutions exhibit yield stress and shear-thinning behaviour. The ramp down
flow curves were then fitted to the Hershel-Bulkley model to estimate the
dynamic yield stress, consistency index and power law index. There is a small
deviation between the model fit and the steady shear experiments only at very
high shear rates, over 1000 s-1. No effects of inertia were observed in the
rheological measurements due to the low gap between the parallel-plate
geometry. The fitting parameters are also reported in Figure 3. As usual with
Carbopol solutions, we observe hysteresis at low shear rates which is likely a
consequence of the elasticity of the fluids near the yielding point [46]. The
steady-shear-rate viscosity curves match the ramp-up/down curves quite well,
with no discernible hysteresis at the high shear rates relevant for turbulent
flows; thus, we consider that all Carbopol solutions shown in this work are
non-thixotropic.

Figure 3: Flow curves of the Carbopol solutions used in flow loop. Empty
symbols correspond to the ramp experiments, and filled symbols correspond to
the steady state flow curves. The dashed lines represent fits to the Herschel-
Bulkley model Figure 4: Stress amplitude sweep at $\omega=3.14$ rad/s (a) and
small amplitude oscillatory shear measurements (b) at $\gamma=0.1$ % for
Carbopol solutions.

We characterize the elastic properties of Carbopol by performing stress-
controlled amplitude sweeps for each of the Carbopol solutions. The results
are reported in Figure 4 (a). The amplitude sweep results show that the
elasticity is important in the linear viscoelastic region of constant
$G^{\prime}$ and $G^{\prime\prime}$ where the fluid is unyielded. The linear
viscoelastic region falls below the yield stress which is of the same order of
magnitude of the stress value at the crossover point between $G^{\prime}$ and
$G^{\prime\prime}$ [47]. Within the linear viscoelastic regime obtained from
the amplitude sweeps, we perform strain-controlled, small amplitude
oscillatory shear experiments (SAOS) with both 0.08% and 0.10% Carbopol
solutions. SAOS measurements with the 0.06% solution were not possible due to
inertial effects. The results from Figure 4 (b) show the usual gel-like
behaviour of the solutions, which are characterized by a $G^{\prime}$ plateau
approximately independent of the frequency of oscillation $\omega$.
$G^{\prime\prime}$ is also independent of the frequency below 1 rad/s. We note
here that we assume that the viscoelasticity in the non-linear regime with
Carbopol solutions is small, according to other experimental works [35, 48].
We can confirm this statement by estimating the relaxation time of the
Carbopol in the non-linear viscoelastic regime with first normal stress
differences ($N_{1}$) measurements at steady shear with a cone-plate geometry.
However, these experiments are often very challenging or impossible at the
high shear rates in the boundary layer of turbulent flows, which can be over
1000 s-1. Because of this reason, we did not perform measurements of $N_{1}$
in this work.

Friction factor measurements are useful to determine whether or not the
experiments have reached the fully turbulent regime. From the pressure drop
measurements, we compute the mean wall shear stress and the Fanning friction
factor from the pipe flow of water and the three Carbopol solutions, according
to the equation

$f=\frac{2\tau_{w}}{\rho U_{b}^{2}}.$ (2)

For reference, we compare our experimental friction factor results to the
empirical Colebrook equation for the Fanning friction factor given by

$\frac{1}{\sqrt{f}}=-4.0\,log\,\left(\frac{\epsilon/D_{h}}{3.7}+\frac{1.255}{Re_{G}\sqrt{f}}\right),$
(3)

which estimates the friction factor of turbulent Newtonian flows of
$Re>10^{5}$ with smooth inner surfaces (roughness $\epsilon$ = 0). We also
show the asymptote of Virk [49],

$\frac{1}{\sqrt{f}}=19.0\,log\,\left(Re_{G}\,\sqrt{f}\right)-32.4,$ (4)

which predicts the state of maximum drag reduction in turbulent flows with
viscoelastic polymer solutions. The friction factor results are plotted in
Figure 5 against the generalized Reynolds number.

Figure 5: Measurements of the Fanning friction factor of the Carbopol
solutions.

The friction factor results for water nicely match the Colebrook equation. The
Carbopol results show a limited drag reduction even at fully turbulent
Reynolds numbers depicted here, when compared to the friction factor given by
the Colebrook equation at the same generalized Reynolds number. The small
difference suggests that there is little viscoelasticity effect in the
experiments shown in Figure 5. However, if we plot the friction factor of
Carbopol as a function of the solvent (water) Reynolds numbers, we observe a
drag increase with the addition of Carbopol at the same bulk velocities
because of the increase in viscosity. This is due to the fact that a Carbopol
solution is not a viscoelastic drag reducing fluid and can be considered to be
an ideal viscoplastic fluid in turbulent flow conditions. DNS simulations with
generalized Newtonian fluids such as those by Singh et al. [43, 42] are
qualitatively similar.

###  3.2 Turbulence statistics

Now we present the turbulence measurements of average velocities and Reynolds
stresses of the flow of Carbopol from the LDA velocity measurements and
compare those to Newtonian flows, both from our experiments with water and
with DNS. For the analysis of LDA measurements, we employ the Reynolds
decomposition of the Navier-Stokes equations [22]. The instantaneous
streamwise velocity component $U$ is given by the sum of the mean streamwise
velocity $\langle U\rangle$ and the fluctuations $u$: $U=\langle U\rangle+u$.
The same decomposition is applied to the wall normal velocity component:
$V=\langle V\rangle+v$. The profiles of the local time-averaged streamwise
velocities normalized with the friction velocity ($U^{+}=\langle
U\rangle/u_{\tau}$) are plotted against the wall normal position normalized by
the wall unit ($y^{+}=y/y^{+}_{0}$) in Figure 6. Note that the wall unit
changes for each fluid formulation due to different effective viscosities at
the wall. The velocity profiles, which were measured along the channel centre
plane, are shown in Figure 6.

Figure 6: Velocity profiles of the turbulent flow of water and Carbopol
solutions in wall units. The dot-dashed lines represent the viscous sublayer,
the full lines represent the log law of the wall for turbulent flow of
Newtonian fluids, and the dashed line shows Virk’s asymptotic velocity profile
for drag reducing polymer solutions. The error bar represents the variation of
$u_{\tau}$ along the walls of the channel cross-section.

We observe good agreement between the experimental data set of water at
$U_{b}=5.8$ m/s ($Re$ = 232810) and the log law profile. The velocity profiles
of Carbopol show a slight upward shift in the log-layer when compared to water
flow, consistent with experimental [25] results with Carbopol and numerical
[43, 44] results with inelastic generalized Newtonian (GN) models for
viscoplastic fluids. Therefore, the upward shift in our velocity profiles
could be a consequence of shear thinning of the fluid, and not
viscoelasticity, which appears to be significant only in the linear
viscoelastic regime (or before the yielding transition). We also observe a
very slight concentration dependence in the velocity profiles of each
solution. The velocity profile of the 0.10% solution is slightly steeper than
the 0.06% solution. This is likely due to the decrease of the power-law index
$n$ with Carbopol concentration; in other words, an increase in shear-thinning
effects. A similar result was been observed in the DNS results of [40] for
turbulent pipe flow of power-law fluids with decreasing $n$ values. The
increase in the slope of the velocity profile agrees with the decrease in
friction factor with concentration as shown in Figure 5.

Regarding the aspect ratio of our channel, we did not perform experiments with
the Carbopol in positions away form the channel centre plane because of fluid
degradation concerns with the additional time required to perform these
experiments. Nevertheless, we note that our flow cannot be approximated to a
channel flow because of the 2:1 aspect ratio of the rectangular duct. To
support our data, we show that the turbulence statistics of water are quite
similar to what is calculated from Newtonian flow DNS at similar Reynolds
number. Therefore, we believe this limitation in our experiment does not
affect our main conclusions of this study. Another effect that is difficult to
quantify is that only the mean wall shear stress is evaluated from the
pressure measurements. Our channel has a 2:1 aspect ratio and we expect
therefore to have significant variations in wall shear stress along the wall
and into the corners, especially for a shear-thinning yield stress fluid. This
results in $u_{\tau}$ varying $\sim 15\%$ across the width of channel. To
clarify this effect of the geometry of our setup, the 15% variation of
$u_{\tau}$ across the channel is represented by an error bar in Figure 6.
However, even with the differences in wall shear stress across the channel,
the mean velocity profile of water shows good agreement with the Newtonian
log-law profile, and the computation of $u_{\tau}$ with the mean wall shear
stress is adequate. Note that if we do not consider the variation of mean wall
shear stress in the channel, the estimated uncertainty in $U^{+}$ is near
$3\%$, following the procedure of Whalley et al. [50].

Figure 7 (a) shows a diagnostic plot [51], where the root mean square of the
velocity fluctuations normalized with the mean velocity measured at the centre
of the channel $U_{CL}$, is plotted as a function of the mean local velocity
normalized by the mean centreline velocity $\langle U\rangle/U_{CL}$. This is
a useful representation of the turbulence intensities $u_{rms}$ (or
$\sqrt{\langle u^{2}\rangle}$) because it allows us to visualize the behaviour
of the fluctuations without the wall position. We compare our experimental
results with water to DNS results of Newtonian flows by Ahn et al. [52] at
$Re_{\tau}=3000$ (shown in dark blue lines) in a pipe. The reason we present
DNS pipe flow results instead of channel flow is to indicate that even though
our channel flow is three-dimensional due to the 2:1 aspect ratio, the
experimental results with water in the centre plane of the channel (see Figure
2) agree, at least qualitatively. The Carbopol results are quite different
from water. We notice a large increase in the turbulent intensities throughout
most of the $y$-direction. The peak in turbulence production moves away from
the wall, and is not dependent of the concentration of the Carbopol solution.
We observe a good collapse of the Carbopol $u_{rms}$ data in the diagnostic
plot, with no visible differences between concentrations. Although we cannot
measure velocity in the buffer layer ($5\leq y^{+}\leq 30$) and viscous sub-
layer ($y^{+}<5$) with water due to the very small boundary layer thickness at
the high Reynolds numbers of our experiments, the data show good qualitative
agreement with the Newtonian DNS at similar Reynolds number. Figure 7 (b)
indicates the streamwise Reynolds stresses are at least two orders of
magnitude larger than the yield stress during all experiments presented in the
velocity profiles of Figure 6. The calculation of the Bingham number
$Bi=\tau_{y}/[K(U_{b}/h)^{n}]$ [25] also results in small values. According to
[8], this is shows that the Carbopol is fully yielded during all measurements
performed here, and confirms that our experiments are indeed in the fully
turbulent regime. Therefore, the results seem to point towards a negligible
influence of the yield stress during fully turbulent flows of viscoplastic
fluids.

Figure 7: Streamwise turbulence intensities of the turbulent flow of water and
Carbopol solutions normalized by the centreline velocity (a) and streamwise
Reynolds stresses of the Carbopol solutions normalized by the yield stress
(b), plotted against the local average velocity normalized by the centreline
velocity.

Reynolds stresses of water and Carbopol solutions are shown in Figure 8 (a-c).
All turbulent quantities in Figure 8 (a-c) are normalized with $U_{b}$, which
corresponds to the bulk velocity calculated from the flow meter during the
experiments, except for $\langle u^{3}\rangle$ which is normalized by
$u_{rms}^{3}=\langle u^{2}\rangle^{3/2}$, resulting in the skewness in Figure
8 (d). The streamwise turbulence intensities show a similar behaviour as the
diagnostic plot except for the fact that we display it as a function of
position this time. The peak in $\langle u^{2}\rangle$ occurs near $y/h\sim
0.1$ for all Carbopol concentrations, and the fluctuations seem slightly
larger for the 0.10% Carbopol in comparison to the other concentrations.
Overall the streamwise turbulence intensities are larger than for Newtonian
fluids, and the decrease of the power-law index $n$ with Carbopol
concentration does not seem to play a very large role here. These experimental
observations agree qualitatively with the results from [25] and [44], although
the increase in $\langle u^{2}\rangle$ is rather small in their numerical
simulations, when compared to experiment. The measurements of the wall normal
turbulent intensities in Figure 8 (b) are quite similar to Newtonian fluids,
albeit at slightly smaller values overall. The effect of the parameter $n$ is
more apparent in Figure 8 (b) than (a), where the wall-normal fluctuations
decrease with $n$. We note here that due to limitations of our LDA system, it
is difficult to measure the $V$ component of velocity below $y/h\sim 0.28$.
Nevertheless, our measurements still highlight the increase in anisotropy of
the turbulent flow of Carbopol near the wall, indicated by the significant
increase in $u$ fluctuations and slight decrease in $v$ fluctuations when
compared to water. Turbulence anisotropy is also enhanced by larger
concentrations of Carbopol, with decreasing $n$ values.

Reynolds shear stresses -$\langle uv\rangle$ in Carbopol solutions, shown in
Figure 8 (c), seem to increase near the wall when compared to water, which is
likely due to the large $\langle u^{2}\rangle$ observed in Figure 8 (a). Due
to the impossibility of near wall measurements, we cannot be sure of the
behaviour of the Reynolds shear stresses at lower values of $y/h$. However,
from the $\langle u^{2}\rangle$ results, we can speculate that the peak in
Reynolds shear stresses might occur farther away from the wall than the
Newtonian case. The values of -$\langle uv\rangle$ are also expected to be
lower than water near the wall, similar to the trend seen in the $\langle
u^{2}\rangle$ results. The observation that all values converge near the core
is also interesting. The Reynolds and viscous shear stresses vanish in the
turbulent core, and we speculate that the yield stress effect might play a
role at suppressing turbulent fluctuations. Additionally, we note that in case
of water flow, all results displayed here agree with the available DNS results
for pipe flow for $Re_{\tau}=3000$ when scaled with $U_{b}$.

The skewness values shown in Figure 8 (d) depict changes in the near wall
turbulence dynamics in the flows of Carbopol when compared to Newtonian
fluids. An increase in positive skewness near the wall indicates more
intensity in sweeping motions, or high speed fluid moving towards the wall
($u>0,v<0$) [53] when compared to water, and further away from the wall we
observe an inflection point near $y/h=0.07$, after which the skewness of the
fluctuations become negative. The negative skewness indicates enhanced
ejection motions, characterized by low speed fluid moving away from the wall
($u<0,v>0$) [53], which supports the results from Figure 8 (a) where the
increase in streamwise fluctuations denote more production of turbulence [54].
Overall, Figure 8 (a-d) highlights changes in Reynolds stresses due to the
addition of Carbopol, but there is not a large difference between the three
concentrations, due to the rather narrow power law index $n$ range in the
formulated solutions.

Figure 8: Streamwise Reynolds stresses $\langle u^{2}\rangle$ (a), wall normal
Reynolds stresses $\langle v^{2}\rangle$ (b) and Reynolds shear stresses
$-\langle uv\rangle$ (c) normalized by the bulk velocity $U_{b}^{2}$ and
skewness $\langle u^{3}\rangle/u_{rms}^{3}$ (d) of water and Carbopol
solutions.

Figure 9 presents conditional averages of the Reynolds shear stresses with
respect of $\langle uv\rangle$ with respect to the sign of the velocity
fluctuations $u$ and $v$: $u>0,v>0$: Q1 events or outward interactions);
$u<0,v>0$: Q2 events or ejections; $u<0,v<0$: Q3 events or wallward
interactions and $u>0,v<0$: Q4 events or sweeps [53]. The largest
contributions towards the Reynolds shear stresses come from Q2 events or
ejection motions, and Q4 events or sweep motions [55] as seen from Figure 9
(b) and (d). In the particular case of Carbopol solutions, both Q1, Q3 and Q4
seem relatively unaltered when compared to water, whereas Q2 events in Figure
9 (b) and (c), are the most affected. Specifically for Figure 9 (b), the
rheological changes in the fluid impart a large increase in ejection motions
for all fluids studied here. This is connected to the large enhancement in
streamwise turbulence intensities seen previously.

We also show the probability density functions of the streamwise velocity
fluctuations in Figure 10 to highlight the difference in dynamics very close
to the wall and near the core. Near the buffer layer, at $y/h\sim 0.04$ in
Figure 10 (a), we observe larger probabilities of large $u$ velocity
fluctuations with Carbopol than water. The narrower pdf of water also
indicates a larger probability of smaller $u$ velocity fluctuations. Figure 10
(b) shows the probability density function (pdf) of $u$ velocity fluctuations
at $y/h\sim 0.5$, where the difference between Carbopol and water is small,
although this difference increases somewhat with the concentration. In this
case, the pdf of the Carbopol solutions is slightly more skewed towards
negative values, as previously shown by Figure 8 (d). Also, the probability of
small $u$ velocity fluctuations reduces slightly with the concentration of
Carbopol.

Before moving on to the next section, we summarize the main findings from the
turbulence statistics of different Carbopol solutions. The main observation is
that, for high $Re_{G}$ turbulence, the yield stress is insignificant relative
to the Reynolds stresses. Therefore, similarly to previous studies such as
[11, 56, 40, 42], only the parameter $n$, or the amount the fluid shear thins,
seems to be of most influence here. Indeed, from the perspective of
dimensional analysis, where $\tau_{y}/\tau_{w}\ll 1$, only $n$ should affect
the flow. The effects of $n$ are observed as an increase in the velocity
profile when scaled with inner units, and a decrease in wall-normal and
Reynolds shear stresses , $\langle v^{2}\rangle$ and -$\langle uv\rangle$.
Conversely, the streamwise Reynolds stresses $\langle u^{2}\rangle$ increase
quite significantly, and we conclude that the main consequence of Carbopol
addition in the Reynolds stresses to enhance streamwise turbulent structures,
making the flow more anisotropic. This is likely due to the more pronounced
viscosity differences across the channel with larger shear-thinning effects -
the larger viscosities in the core dampen the transfer of momentum to the
wall, leading to a decrease in $\langle v^{2}\rangle$ and -$\langle uv\rangle$
[57, 42], and thus increased $\langle u^{2}\rangle$. However, our range of $n$
is quite narrow to observe large changes in statistics. Since we cover
$0.42\leq n\leq 0.56$, our results across all three Carbopol formulations are
quite similar. Numerical simulations allow coverage of a much wider range of
parameters. We note that the experimental conditions would not allow much
smaller values of $n$ to be achieved, since we need to increase the Carbopol
concentration to decrease $n$. The resultant increase in viscosity and
$\tau_{y}$ would make reaching turbulence more difficult.

Figure 9: Conditional averages of Reynolds stresses of the turbulent flow of
water and Carbopol solutions plotted in the four quadrants: $\langle
uv\rangle_{Q1}$ (a), $\langle uv\rangle_{Q2}$ (b), $\langle uv\rangle_{Q3}$
(c) and $\langle uv\rangle_{Q4}$ (d). Figure 10: Probability density functions
of $u/U_{b}$ of water and Carbopol solutions. Image (a) corresponds to
$y/h\sim 0.04$ while image (b) shows $y/h\sim 0.5$

###  3.3 Power spectral densities

We investigate the one-dimensional power spectral densities (PSDs),
represented by $E_{uu}$ for streamwise velocity fluctuations $u(t)$ and
$E_{vv}$ for wall normal velocity fluctuations $v(t)$. We compare energy
spectra results with water and Carbopol solutions in different channel
positions. The power spectral density is an estimate of the energy
distribution throughout the frequency range, i.e.
$E_{uu}\propto|u_{f}(f)|^{2}$, where $u_{f}(f)$ is the Fourier transform of
$u(t)$, for a given $y/h$ position. The frequency data $f$ is then converted
into wavenumber space by $k_{x}=2\pi f/\langle U\rangle$, and made
dimensionless multiplied by with the boundary layer thickness $h$, or channel
half-height, for fully developed channel flow. This is possible by using
Taylor’s hypothesis, which has been widely used for LDA experiments [22, 29].
We consider that the hypothesis is acceptable if the root mean square of the
velocity fluctuations $u$ and $v$ is less than 20% of the mean local velocity
$\langle U\rangle$. The PSD is made dimensionless dividing by $\nu_{s}U_{b}$,
where $\nu_{s}$ is the kinematic viscosity of the solvent. The normalization
is to aid comparison between different fluids.

We employ linear interpolation of the velocity-time signal to obtain equally
spaced data points. The interpolation frequency is the average data rate of
the experiments (similar to [58]) which varies between 3000 and 4000 Hz for
the $U$ component, and between 2000 and 4000 Hz for the $V$ component. The
limitation of this method is that the interpolation acts as a filter to the
data signal at high frequencies, and because of this, we cut off our spectral
results at a maximum frequency of approximately 1/4 of the total data rate,
which is a reasonable approximation as shown by Ramond and Millan [59], for
instance. Even without the high frequency results, we are able to estimate the
inertial range spectra in our water results.

As an initial benchmark, we show the PSDs of the $u$ fluctuations, given by
$E_{uu}$, in Figure 11 (a) and $v$ fluctuations given by $E_{vv}$ in Figure 11
(b) for water at three distinct Reynolds numbers at various $y/h$ positions
specified in the figures. The dotted lines represent the wavenumber scaling
for the inertial range of the turbulent energy cascade $k_{x}^{-5/3}$, where
the energy transfer from large to small eddies depends only on the energy
dissipation rate [22]. By means of scaling with the bulk velocity $U_{b}$ and
the kinematic viscosity $\nu_{s}$, there is a very good collapse, both for
$E_{uu}$ and $E_{vv}$. We observe that, for both positions shown in Figure 11
(a), energy decay from small wavenumbers (representative of large eddies) to
large wavenumbers (representative of small eddies) follow the $k_{x}^{-5/3}$
scaling quite closely for $k_{x}h>5$. With Figure 11, we can establish that
the experimental results are able to provide the energy spectra in the
inertial range at reasonable wavenumbers. A similar conclusion is drawn from
Figure 11 (b), where the energy content from $v$ fluctuations also falls with
the same $k_{x}^{-5/3}$ typically expected from Newtonian fluids at high $Re$.
Figure 11 (c) presents $E_{uu}$ and $E_{vv}$ at the centre of the channel. In
this figure we observe a typical behaviour of isotropic turbulence (at least
considering the measurement plane for our LDA results) in the centre of the
channel, in which both $E_{uu}$ and $E_{vv}$ collapse in the high wavenumber
range, also with approximately a $k_{x}^{-5/3}$ scaling [60].

Figure 11: Power spectral densities of velocity fluctuations during the flow
of water. (a) shows the PSDs of the streamwise velocity fluctuations at
$y/h\sim 0.12$ (thin lines) and at the centre plane at $y/h\sim 1$ (thick
lines). The $k_{x}^{-5/3}$ scaling is shown by dotted lines. (b) shows the
PSDs of the wall normal velocity fluctuations at $y/h\sim 0.28$ (thin lines)
and at the centreline at $y/h\sim 1$ (thick lines). Image (c) shows the PSDs
of streamwise ($E_{uu}$, thin lines) and wall normal velocity fluctuations
($E_{vv}$, thick lines) at the centreline $y/h\sim 1$.

The PSDs of streamwise velocity fluctuations with the addition of Carbopol are
shown in Figure 12, for similar $Re_{G}$ conditions. In Figure 12 (a), the
curves for different Carbopol concentrations are quite close to each other by
scaling with $U_{b}$ and $\nu_{s}$ at each $y/h$ position. Interestingly, the
scaling of energy for higher wavenumbers ($k_{x}h>5$) for the Carbopol
measurements is close to $k_{x}^{-7/2}$ instead of the usual $k_{x}^{-5/3}$
for the inertial range in Newtonian fluids, indicating a larger power decay
for higher wavenumbers (small scales). The energy drop off at large
wavenumbers also appears to be independent of Carbopol concentration and
position in the channel, at least for the range of parameters investigated
here. The energy spectra decay of $k_{x}^{-5/3}$ can be observed in the PSDs
of Carbopol, but in a lower range of $1\leq k_{x}h\leq 4$ approximately, which
is a consequence of low Reynolds numbers. The theoretical studies by
Anbarlooei et al. [20] and Anbarlooei et al. [21] have proposed that
Kolmogorov’s $k_{x}^{-5/3}$ scaling for the energy spectra in the inertial
range is valid for viscoplastic fluids, to enable scaling expressions for the
wall shear stress and thus to compute the friction factor in pipe flows. An
interesting observation from this section is that the PSDs appear to confirm
their statement because the $k_{x}^{-5/3}$ power law is still observed, albeit
for different $k_{x}h$ values than water.

The PSDs of wall-normal velocity fluctuations shown in Figure 12 (b) also
follow the $k_{x}^{-7/2}$ scaling at large wavenumbers approximately, but
unlike Figure 12 (a) the $k_{x}^{-5/3}$ scaling is not as apparent at
intermediate wavenumbers. The good collapse of the spectra results in Figure
12 (a) is also seen in Figure 12 (b). The energy content of $v$ fluctuations
of 0.1% Carbopol is slightly lower overall, which correlates to the lower
values of $\langle v^{2}\rangle$ observed previously. Figure 12 (c) shows that
the Carbopol solution is also isotropic in the centreline in the high
wavenumber range, albeit with the spectra scaling to the power law of
$k_{x}^{-7/2}$. It is interesting to observe isotropy at the centreline at
high $k_{x}$ range, even though there is a clear increase in turbulent
anisotropy near $y/h\sim 0.1$ due to the large peak in $\langle u^{2}\rangle$.
Summarizing the results from Figure 12, the PSDs for all three Carbopol
formulations investigated here are quite similar to each other. The similarity
in the PSDs are expected since the values of streamwise Reynolds stresses
$\langle u^{2}\rangle$ were quite similar across all concentrations of
Carbopol solutions (or values of the power-law index $n$) investigated. The
values of wall-normal Reynolds stresses $\langle v^{2}\rangle$ also did not
change significantly. This means that the PSDs are not very sensitive to
changes in the rheology of the fluid, not unlike the results observed in the
turbulence statistics. We can expect that, similarly to what was observed in
the wall-normal Reynolds stresses results, $\langle v^{2}\rangle$ decreases
with $n$, so it is likely that the energy content decreases as well. DNS
results show that $\langle u^{2}\rangle$ increases with $n$ [40, 42], so
following the same trend the energy spectra should also increase. However,
without a wider range for the power-law index $n$, it remains difficult to
experimentally investigate its effect.

Figure 12: Power spectral densities of velocity fluctuations during the flow
of Carbopol solutions in concentrations of 0.06 (black), 0.08 (blue) and 0.10%
(red). Image (a) shows the PSDs of the streamwise velocity fluctuations at
$y/h\sim 0.12$ (thin lines) and at the centre plane at $y/h\sim 1$ (thick
lines). The $k_{x}^{-5/3}$ scaling is shown by the dotted line, and the
$k_{x}^{-7/2}$ scaling is shown by the dashed line. Image (b) shows the PSDs
of the wall normal velocity fluctuations at $y/h\sim 0.28$ (thin, lines) and
at the centre plane at $y/h\sim 1$ (thick lines). The plot (c) shows the PSDs
of streamwise ($E_{uu}$, thin lines) and wall normal velocity fluctuations
($E_{vv}$, thick lines) at the centre plane ($y/h\sim 1$).

We compare the PSDs of both water and Carbopol at the same $U_{b}$ values in
Figure 13 in two positions in the channel centre plane. Closer to the wall in
Figure 13 (a), the addition of Carbopol to the flow enhances the energy
content in small wavenumbers, or larger eddy length scale, when compared to
water. This happens in both measurement positions shown. The consequence of
this effect is the increase of streamwise fluctuations in the velocimetry
measurements of Figure 8 (a). As noted previously, the energy content in
$k_{x}h>5$ decreases as $k_{x}^{-7/2}$ for the Carbopol solutions, and
$k_{x}^{-5/3}$ for water. The PSDs of the $v$ component fluctuations of
Carbopol in Figure 13 (b) are compared to the water results. Similarly to the
$E_{uu}$ results, the energy at low wavenumbers is enhanced with the addition
of Carbopol, while the energy content at high wavenumbers is lower. The
$k_{x}^{-7/2}$ power law scale is also observed in the Carbopol results, as a
consequence of less energy contained in $k_{x}h>4$.

To further investigate the steeper slope in the high wavenumber range in the
Carbopol solutions, we plot the dissipation spectra (which can be estimated by
$2\nu\int_{0}^{\infty}k^{2}E_{uu}(k)dk$), normalized by $U_{b}^{3}$, in Figure
14. As with the PSD results, we cut off the dissipation spectra plot at
$k_{x}h=10$ to avoid aliased results due to the uneven data rate of the LDA.
We observe that, for the same bulk velocities, the majority of the energy
dissipation in Carbopol occurs at larger scales than with water. There is a
peak at $k_{x}h=2$ and a decrease in dissipation at $k_{x}h>4$ in both Figure
14 (a) and (b). The decay in dissipation is not well-resolved in the water
results due to data rate limitations, but we see that a large amount of energy
dissipation happens at smaller scales than in the Carbopol solutions.

Figure 13: Power spectral densities of streamwise (a) and wall normal (b)
velocity fluctuations during the flow of water and Carbopol solutions.
Streamwise velocity fluctuations in (a) were measured at $y/h\sim 0.12$ (lines
with symbols) which is the closest position to the wall where the Taylor’s
hypothesis is valid and $y/h\sim 1$. Wall normal velocity fluctuations in (b)
were measured at $y/h\sim 1$. Figure 14: Estimation of the dissipation spectra
for water and the 0.06% Carbopol solution at $U_{b}=3.8$ m/s (a), and water
and the 0.08% Carbopol solution at $U_{b}=5.8$ m/s (b) from the respective
streamwise PSDs, measured at $y/h\sim 1$.

##  4 Results and discussion: effect of the Reynolds number

In this section we investigate the flow of Carbopol at 0.06% concentration at
different $Re_{G}$ values, which range from weakly turbulent at $Re_{G}=9430$
to strongly turbulent $Re_{G}=50670$. We remind the reader that the full
experimental parameters are listed in Table 1. We consider the rheology of the
fluid at all three Reynolds numbers investigated here to be the same as
presented in Figure 3. Indeed, rheological tests with samples taken after
experiments at different Reynolds numbers have shown little effect of
degradation (not shown). The velocity profiles normalized by the friction
velocity $U^{+}=\langle U\rangle/u_{\tau}$ plotted against the wall-normal
positions normalized by the wall unit $y^{+}=y/y_{0}^{+}$ are presented in
Figure 15 (a) for each $Re_{G}$ values, and the friction factor results,
calculated similarly to Figure 5, are shown in Figure 15 (b).

Figure 15: Mean velocity profiles normalized in wall units (a) and friction
factor (b) during turbulent flow of 0.06% Carbopol solutions. In plot (a), the
dot-dashed lines represent the viscous sublayer, the full lines represent the
log law of the wall for turbulent flow of Newtonian fluids, and the dashed
line shows Virk’s asymptotic velocity profile for drag reducing polymer
solutions.

The velocity profiles in Figure 15 (a) show an interesting dependence in
$Re_{G}$. At a low $Re_{G}$ value of 9430, the velocity profile is somewhat
similar to the turbulent flow of viscoelastic, drag reducing (DR) fluids [30,
31], where there is an upturn in the velocity profile near the buffer layer.
This is also seen in the friction factor plot as a function of $Re_{g}$ in
Figure 15 (b), where the Carbopol friction factor is lower than the Newtonian
case at $Re_{G}=9430$, and moves towards the Colebrook friction factor line as
$Re_{G}$ increases. Escudier and Presti [7] observed similar DR-like results
in pipe flow of a Laponite solution, which is a thixotropic viscoplastic
fluid. However, their flow was clearly transitional at $Re_{G}=3400$, whereas
our Reynolds number is quite a bit higher. As we increase the Reynolds number,
the flow seems to approach a Newtonian-like velocity profile, specifically at
$Re_{G}=50670$ where the velocity profile of Carbopol 0.06% matches the log-
law profile for Newtonian fluids. Therefore, the turbulent flow of Carbopol
appears to become increasingly Newtonian-like as $Re_{G}$ is increased.

Figure 16: Streamwise Reynolds stresses $\langle u^{2}\rangle$ (a), wall
normal Reynolds stresses $\langle v^{2}\rangle$ (b) normalized by the bulk
velocity $U_{b}^{2}$, and skewness $\langle u^{3}\rangle/u_{rms}^{3}$ (c) of
water and Carbopol solutions.

The flow statistics are presented in Figure 16. We note a significant change
in streamwise Reynolds stresses with $Re_{G}$ in Figure 16(a). As we increase
the Reynolds number, the peak in $\langle u^{2}\rangle$ moves closer to the
wall, and approaches the Newtonian values near the turbulent core as $Re_{G}$
is increased. It seems that for a very large Reynolds number, it is reasonable
to expect the $\langle u^{2}\rangle$ profile to further resemble the Newtonian
in a larger range of $y/h$ values. Considering the $\langle v^{2}\rangle$
results in Figure 16(b), a similar assessment can be made as the Carbopol
values approach the Newtonian profiles as $Re_{G}$ increases. The values of
$\langle v^{2}\rangle$ are also quite low at $Re_{G}=9430$, which correlates
to larger $U^{+}$ in the velocity profiles in Figure 15 (a). This effect of
$Re_{G}$ in $\langle u^{2}\rangle$ and $\langle v^{2}\rangle$ is qualitatively
similar to simulations by Singh et al. [42]. As $Re_{G}$ increases, the
effects of the polymer additive are observed ever closer to the wall. It is
important to note that the Carbopol used in the present work does not appear
to be viscoelastic according to data from Section 3, and the evidence provided
by the experiments and numerical simulations of GN fluids [40, 42] seem to
point out to shear-thinning effects only. The Reynolds number affects the
skewness of the $u$ fluctuations as seen in Figure 15 (c). As $Re_{G}$
increases, regions of positive skewness are observed closer to the wall. In
addition, the skewness profile for the $Re_{G}=50670$ case is quite similar to
the water measurements, from $y/h\sim 0.4$.

It is difficult to arrive to a conclusive explanation for the $Re_{G}$
dependence of turbulence statistics. A similar result has been observed in
simulations of turbulent channel flows of Bingham fluids in Rosti et al. [44].
They state that the fluid becomes Newtonian-like as the Bingham number, or the
ratio of yield stress to viscous stress, decreases. By increasing $Re_{G}$ we
effectively decrease the Bingham number of our flows as well. However, the
range of $Re_{G}$ values in our experiments are already quite high so this
Reynolds number dependence is more likely an effect of diminishing shear
thinning effects due to the very large shear rates near the wall. As $Re_{G}$
increases, so do the shear rates, which may lead to the approach to an
asymptotic viscosity plateau $\eta_{\infty}$ value for very high shear rates,
similar to a Carreau-Yasuda fluid. At such high shear rates, the fluid’s
viscosity might not change as much compared to lower shear rates. However,
rheometry limitations prevent us to estimate the viscosity at very high shear
rates due to inertia and low-gap errors with the parallel-plate geometry.

We investigate the effects of the Reynolds number on the PSDs of 0.06%
Carbopol solutions in Figure 17. Considering the PSDs of streamwise velocity
fluctuations $u$ in Figure 17 (a), we notice that the spectra seem to better
resemble $k_{x}^{-5/3}$ decay as $Re_{G}$ increases, and is especially
noticeable at $Re_{G}=50670$ at $y/h\sim 0.12$, but changes are somewhat
subtle in the high wavenumber range. The $E_{vv}$ energy content appears to
increase with $Re_{G}$ at the centreline for $k_{x}>1$. Then, larger $Re_{G}$
suggests an approach towards a Newtonian-like energy spectra in $E_{uu}$, with
lower energy at small wavenumbers and an extended wavenumber range for
$k_{x}^{-5/3}$, more evident for $y/h\sim 0.12$. These results suggest a
diminished importance of the non-Newtonian rheology in the turbulence dynamics
with large $Re_{G}$, although the effects of Carbopol such as increased
anisotropy and large $\langle u^{2}\rangle$ persist even at a very high
$Re_{G}$ near the wall. Regarding the $k_{x}^{-7/2}$ drop in power,
considering that the energy cascade from large to small eddies is supposed to
be independent of viscosity in the inertial range [22], we can hypothesise
that the lower inertial effects in the turbulent flow of Carbopol might narrow
the $k_{x}^{-5/3}$ wavenumber range. Then, the $k_{x}^{-7/2}$ may be a
consequence of the energy decay due to dissipation at larger scales in the
Carbopol solutions than water. Another reason could be increased elastic
effects in the small (high frequency) length scales in the energy cascade,
specifically. This explanation was proposed in Presti [24] to justify the
Reynolds number dependence of turbulent flows of Carbopol. In summary, the
effect of Reynolds number in the turbulent flow of Carbopol appears to be the
decrease of shear-thinning effects, both in the mean velocity profile and in
the Reynolds stresses. This is more evident approach of the wall-normal
Reynolds stresses $\langle v^{2}\rangle$ to the Newtonian values. The
$k_{x}^{-5/3}$ range in $E_{uu}$ is extended to smaller scales as $Re_{G}$
increases, again further resembling the water measurements. We can assume that
with increasing shear rates due to the increase in $Re_{G}$, the effect of
shear-thinning is diminished as an viscosity plateau $\eta_{\infty}$ is
approached.

Figure 17: Power spectral densities of streamwise velocity fluctuations (a)
and a comparison of streamwise and wall normal velocity fluctuations during
the flow of Carbopol at 0.06% concentration during two experimental
conditions: $Re_{G}=9430$ and $Re_{G}=50670$. Streamwise velocity fluctuations
in (a) were measured at $y/h\sim 0.12$ (thin lines) which is the closest
position to the wall where the Taylor’s hypothesis is valid, and the
centreline of the channel $y/h\sim 1$ (thick lines). Measurements in (b) were
performed at $y/h\sim 1$.

##  5 Conclusion

We performed an experimental investigation of the turbulent flows of Carbopol
solutions with LDA measurements in a recirculating flow loop setup, with the
main focus on both the rheology effect with similar $Re_{G}$, and the effect
of different $Re_{G}$ with a constant fluid formulation. Three concentrations
of Carbopol were investigated: 0.06%, 0.08% and 0.10%. For comparison and
benchmarking puruposes, we also presented turbulence data for the flow of
water. All measurements are performed in the fully turbulent regime within a
2:1 channel, and the Carbopol is assumed to be fully yielded in all
experiments.

We investigated the effect of shear-thinning of Carbopol solutions in the flow
field and Reynolds stresses. The addition of Carbopol to water provides an
slightly increased slope of the velocity profile $U^{+}(y^{+})$ in comparison
to the Newtonian (log law) case. The streamwise Reynolds stresses $\langle
u^{2}\rangle$ in Carbopol increase by a large amount when compared to water,
but the effect of rheological changes between each Carbopol formulation was
negligible in $\langle u^{2}\rangle$. The wall-normal Reynolds stress
component $\langle v^{2}\rangle$ was affected more noticeably by changes in
rheology, with the values of $\langle v^{2}\rangle$ decreasing with
concentration. This result suggests large anisotropy in turbulence structures
near the wall due to an enhancement in streamwise turbulent structures.
Overall, to observe a more significant effect of rheological parameters in
Reynolds stresses, a wider range of formulations have to be studied.

The increase in $\langle u^{2}\rangle$ with the Carbopol addition in
comparison to water is reflected in the $E_{uu}$ by an enhanced energy content
at low wavenumber, when compared to the flow of water at the same $U_{b}$. The
power spectral densities reveal that the energy scales with a $k_{x}^{-7/2}$
power law for large wavenumbers during Carbopol flows. This is in contrast to
the $k_{x}^{-5/3}$ power law encountered in water in the same wavenumber range
of $k_{x}h>5$. The $k_{x}^{-5/3}$ scaling does not disappear with the addition
of Carbopol, but happens in a lower range than water, near $1\leq k_{x}h\leq
4$. The $k_{x}^{-7/2}$ slope is also seen in the $E_{vv}$ spectra. We can
speculate that the $k_{x}^{-7/2}$ scaling in the high wavenumber range is due
to decreased inertial effects in the turbulent flow of Carbopol solutions,
since the Reynolds numbers are much lower than in water flows, with a narrower
inertial range wavenumbers and with dissipation at larger length scales than
water. Another possible explanation could be elasticity effects of the
Carbopol, which could be significant only at small scales and high
frequencies. Richardson’s energy cascade theory predicts that the energy
transfer in the inertial range is independent of viscosity [22], and this
statement supports our interpretation. Nevertheless, the turbulent spectra
could be investigated in more detail, and we plan to carry out experiments in
a circular cross-section pipe in a future study.

The effect of the Reynolds number during the flow of Carbopol 0.06% reveal
interesting turbulence dynamics. At $Re_{G}=9430$ the velocity profile and
turbulent statistics reveal a large increase in the slope of $U^{+}(y^{+})$,
and smaller $\langle v^{2}\rangle$ quantities. Conversely, at the high
Reynolds number case $Re_{G}=50670$, the flow becomes somewhat similar to a
Newtonian flow in aspects such as the velocity profile $U^{+}(y^{+})$ matching
the Newtonian log law, and streamwise Reynolds stresses also approaching
Newtonian values near the core. We believe that this is due to the decrease of
shear thinning effects in the turbulence statistics near the wall, where the
shear rates are very high. However, the turbulence spectra showed little
change in $k_{x}$ power-law scaling at the three $Re_{G}$ studied, but the
energy content was slightly altered as $Re_{G}$ increased. We expect that the
results presented in this work can aid in comparisons to turbulent flow
experiments with other fluids and DNS simulations, especially when
investigating the effects of shear-thinning rheology to the flow field.

## Acknowledgments

This research was made possible by research funding from Schlumberger and
NSERC under the CRD program, project 505549-16. Experimental infrastructure
was funded by the Canada Foundation for Innovation and the BC Knowledge Fund,
grant number CFI JELF 36069. This funding is gratefully acknowledged. We also
acknowledge support from the University of British Columbia 4 Year Fellowship
PhD scholarship program (R.S.M.).

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