Fully turbulent flows of viscoplastic fluids in a rectangular duct

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.

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 %.

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.

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:

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.

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.

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

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

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.

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.

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.

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 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.

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].

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.

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.