3.1. Fluid rheology

The results of the shear viscosity measurements using the torsional rheometer are shown in figure 6(a). The demonstrated shear viscosities are the average of the thrice repeated measurements for each sample. Error bars are the range in the measurements at each $\dot{\gamma }$. Within the presented values of $\dot{\gamma }$, the measurements of μ show good repeatability and low random error; the range in the measurements are less than 5.7 %. Based on figure 6(a), the measured μ of domestic tap water at 25 °C is 0.861 ± 0.049 mPa s. The results for water can be contrasted with shear viscosity measurements of Nagashima (Reference Nagashima1977) and Collings & Bajenov (Reference Collings and Bajenov1983). They measured the viscosity of distilled water at 25 °C; finding it to be 0.891 mPa s. The discrepancy between the results of figure 6(a) for water and the measurements of Nagashima (Reference Nagashima1977) and Collings & Bajenov (Reference Collings and Bajenov1983) is within the estimated uncertainty based on the three repeated measurements, and is attributed to systematic uncertainties inherent with the torsional rheometer.

Figure 6. Rheology of aqueous solutions of drag-reduced additives including (a) shear viscosity as a function of shear rate, and (b) mid-point filament diameter with respect to time from uniaxial filament extension.

From visual inspection of figure 6(a), it is apparent that the XG solution is shear thinning. The viscosity of the XG solution reduces by 80.4 % between $\dot{\gamma }$ of 5 and 400 s⁻¹. For $\dot{\gamma }\; \gt 400\;{\textrm{s}^{ - 1}}$, Taylor instabilities produce a sudden increase in μ and the results were discarded. The values of $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ for the drag-reduced, turbulent flows being investigated are beyond 2000 s⁻¹, much greater than the maximum achievable $\dot{\gamma }$ of 400 s⁻¹ using this rheometer. Therefore, a predictive model is used to extrapolate the data and estimate μ _w of the drag-reduced turbulent flows. For the XG solution, the Carreau–Yasuda (CY) model (Carreau Reference Carreau1972; Yasuda, Armstrong & Cohen Reference Yasuda, Armstrong and Cohen1981) fit the measurements appropriately and is shown by the solid line in figure 6(a). The CY model is represented by the following equation,

(3.1)\begin{equation}\frac{{\mu - {\mu _\infty }}}{{{\mu _0} - {\mu _\infty }}} = \frac{1}{{{{(1 + {{({\lambda _t}\dot{\gamma })}^a})}^{n/a}}}},\end{equation}

where μ ₀ is the zero-shear-rate viscosity, μ _∞ is the infinite-shear-rate viscosity, λ_t is a fitting constant with a dimension of time, n is a dimensionless exponent and a is an additional fitting parameter introduced by Yasuda et al. (Reference Yasuda, Armstrong and Cohen1981). For XG, μ ₀ is 0.019 Pa s, μ _∞ is 0.937 mPa s, λ_t is 0.517 s, n is 0.466 and a is 1.935. The uncertainty in the extrapolated shear viscosity for XG is taken to be the maximum range in the thrice-repeated measurements of μ. Using the above (3.1), the μ _w of XG at HDR, which corresponds to the value of $\dot{\gamma }$ that was equal to $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, is 1.576 mPa s. Extrapolating the CY model may be subject to errors that can influence the variables derived for inner scaling, including τ_{w ,2}, u_τ and λ (Singh et al. Reference Singh, Rudman, Blackburn, Chryss, Pullum and Graham2016). We will go on to demonstrate that the DR ₂ derived using these rheology measurements is within 5 % of the DR ₁ determined from measurements of the streamwise pressure gradient. Propagation of uncertainty accounts for additional errors in the inner-scaling variables that can be seen by error bars in plots of the mean velocity profile and Reynolds stresses.

Solutions of PAM also demonstrate shear-thinning qualities, but to a much lesser extent than XG. The viscosity of PAM at MDR reduced by 7.4 % between $\dot{\gamma }$ of 10 and 180 s⁻¹. The viscosity of PAM at HDR reduces by 6.1 % across the same range in $\dot{\gamma }$. Below $\dot{\gamma }$ of 10 s⁻¹, measurements of μ were noisy and ambiguous. In either scenario, measurements of μ are approximately constant for $\dot{\gamma }\; \gt 180\;{\textrm{s}^{ - 1}}$, which is the maximum measurable $\dot{\gamma }$ of both PAM solutions (HDR and MDR) before Taylor instabilities impair the measurements. The Sisko (SI) model (Sisko Reference Sisko1958) was used to represent μ of the PAM solutions at moderate and large values of $\dot{\gamma }$. This model is typically used when measurements close to the zero-shear-rate viscosity are lacking (Barnes et al. Reference Barnes, Hutton and Walters1989). The fitted SI model is shown in figure 6(a) using a dashed line and is represented by the following equation,

(3.2)\begin{equation}\mu = {\mu _\infty } + K{\dot{\gamma }^{n - 1}},\end{equation}

where K and n are constants used to describe the power law decay in μ. The infinite-shear-rate viscosity, μ _∞, for PAM at HDR and MDR are estimated to be 1.072 and 1.087 mPa s, respectively. The fitting parameter n and K are 0.349 and 0.455 mPa sⁿ for PAM at HDR and 0.101 and 0.985 mPa sⁿ for PAM at MDR. Using the above (3.2), the μ _w of PAM at HDR and MDR is 1.074 and 1.088 mPa s respectively, not much greater than the corresponding values of μ _∞.

There is a negligible difference in measured values of μ for the 150 ppm C14 solution at HDR and the 200 ppm C14 solution at MDR. Unlike PAM and XG, solutions of C14 exhibit a Newtonian trend with constant μ for $10\;{\textrm{s}^{ - 1}} \lt \dot{\gamma }\; \lt 100\;{\textrm{s}^{ - 1}}$. Therefore, their viscosities were assumed constant for $\dot{\gamma }\; \gt 100\;{\textrm{s}^{ - 1}}$. The estimated μ _w of C14 at HDR is 0.911 ± 0.036 mPa s and C14 at MDR is 0.912 ± 0.024 mPa s. No SISs are observed for C14; however, that does not rule out the possibility of their presence at higher values of $\dot{\gamma }$.

Using the CaBER system, it was not feasible to measure λ_E of XG and C14 solutions, since the filament immediately ruptured upon moving the endplates. Similar findings for rigid polymer and surfactant solutions have been reported by previous investigations (Lin Reference Lin2000; Escudier et al. Reference Escudier, Nickson and Poole2009; Mohammadtabar et al. Reference Mohammadtabar, Sanders and Ghaemi2020). The two PAM solutions were the only fluids that showed a measurable λ_E using the CaBER apparatus. Figure 6(b) demonstrates the filament diameter, D, as a function of time, t. Here t = 0 indicates the end of the top plate displacement. Similar to the shear viscosity measurements, the thrice-repeated measurements of D(t) were averaged for each sample and the error bars show the range of the measurements. The solid black line represents the exponential fit of D(t) = Ae^−Bt − Ct + E. The resulting λ_E for PAM at HDR and PAM at MDR were 4.3 and 11.0 ms, respectively. For the purposes of our analysis, a comprehension that solutions of PAM have significantly larger extensional characteristics than those of XG and C14, will suffice.

Despite producing similar DR at HDR or MDR (see table 2), each drag-reducing solution exhibits a different shear viscosity and extensional characteristics. Of the additive solutions, XG has the largest overall μ and a strong shear-thinning behaviour. PAM has the next largest distribution in μ; however, only approximately 20 % larger than the average μ of water. C14, on the other hand, has a water-like distribution in μ. Although we were unable to measure λ_E for C14 and XG using the CaBER system, the fact that λ_E for PAM solutions could be measured implies that PAM has a larger λ_E than C14 and XG. Rodd et al. (Reference Rodd, Scott, Cooper-White and Mckinley2005) specified that the operable range of the CaBER is constrained to fluids with λ_E larger than approximately 1 ms when μ is smaller than 70 mPa s. Given the measured shear viscosities of XG and C14 are less than 70 mPa s, it is possible that their λ_E are less than 1 ms. However, further measurements of the extensional rheology are needed to confirm this hypothesis, one possible method being the dripping-onto-substrate technique detailed in Dinic, Jimenez & Sharma (Reference Dinic, Jimenez and Sharma2017). Such a method was capable of measuring the pinch-off dynamics of fluids with μ less than 20 mPa s and λ _E less than 1 ms, according to Dinic et al. (Reference Dinic, Jimenez and Sharma2017). Nonetheless, a correlation relating DR to λ_E, similar to that proposed by Owolabi et al. (Reference Owolabi, Dennis and Poole2017) for flexible polymers, may not apply to solutions of XG or C14. The above analysis using conventional torsional and extensional rheometers highlights that the drag-reduced solutions demonstrate different rheological characteristics.

Other authors have demonstrated that flows obtained from DNS and using the FENE-P (finitely extensible non-linear elastic spring, with a Peterlin approximation) model with large Weissenberg number, $Wi = {\lambda _E}\,\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, have an effective viscosity that increases with distance from the wall (Procaccia et al. Reference Procaccia, L'vov and Benzi2008). A viscosity that increases monotonically with distance from the wall is achieved inherently by shear-thinning fluids. We find it intriguing that DR exists for both XG with relatively small λ_E and large shear-thinning behaviour, and PAM with large λ_E and minimal shear-thinning characteristics. This could suggest that polymers achieve DR using a viscosity that increases monotonically with y. Flexible polymers achieve this viscosity gradient using polymer elasticity (i.e. Wi), while rigid polymers are naturally shear thinning. Such a hypothesis is only speculative. Measurements connecting the role of shear-thinning characteristics to DR are warranted.

3.2. Newtonian turbulent channel flow

The following section seeks to evaluate the 3D-PTV measurements for water by comparing them with DNS of Iwamoto, Suzuki & Kasagi (Reference Iwamoto, Suzuki and Kasagi2002) at Re_τ = 300, Moser, Kim & Mansour (Reference Moser, Kim and Mansour1999) at Re_τ = 395, and Lee & Moser (Reference Lee and Moser2015) at Re_τ = 550. The previously listed DNS data, in that order, are compared with the experimental water data at Re_τ = 307, 425 and 511, respectively, in figures 7 and 8. The comparison involves an evaluation of ${\langle U\rangle ^ + }$ in figure 7 and the Reynolds stress distributions in figure 8. The error bars in figures 7 and 8 originate from a propagation of uncertainty stemming from errors in velocity and shear viscosity measurements. For clarity of the figures, the error bars are down sampled in figures 7 and 8.

Figure 7. Inner-normalized mean streamwise velocity from 3D-PTV measurement for water in comparison with DNS and the law of the wall. The three profiles are shifted upward along the vertical axis by 10. 3D-PTV measurements at Re_τ = [307, 425, 511] are compared with DNS from Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) with Re_τ = 300; Moser et al. (Reference Moser, Kim and Mansour1999) with Re_τ = 395; and Lee & Moser (Reference Lee and Moser2015) with Re_τ = 550.

Figure 8. Reynolds stresses from 3D-PTV of water compared with DNS. (a) ${\langle {u^2}\rangle ^ + }$, where each data set is shifted upward along the vertical axis by 5, (b) ${\langle {v^2}\rangle ^ + }$ where each data set is shifted by 1, (c) ${\langle {w^2}\rangle ^ + }$ where each data set is shifted by 1 and lastly (d) ${\langle uv\rangle ^ + }$where each data set is shifted by −1. The legends are similar to figure 7. The 3D-PTV results with Re_τ = [307, 425, 511] are compared with DNS from Iwamoto et al. (Reference Iwamoto, Suzuki and Kasagi2002) with Re_τ = 300; Moser et al. (Reference Moser, Kim and Mansour1999) with Re_τ = 395; and Lee & Moser (Reference Lee and Moser2015) with Re_τ = 550.

As demonstrated in figure 7, the 3D-PTV measurements of mean velocity at the three Re_τ agree with the distributions established using DNS and the law of the wall. Rather remarkable is the spatial resolution at which these measurements can be attained. For the lowest velocity case of Re_τ = 307, the spacing of data points along y⁺ is 0.4λ and the velocity measurements are obtained for y⁺ as low as 2 (~60 μm from the wall). The spatial resolution of the velocity measurements with respect to inner scaling decreases with increasing Re_τ. For Re_τ = 511, the spatial resolution is 0.7λ and a minimum y⁺ of 4 (~60 μm from the wall). The closest data point to the wall is limited by the size of the tracer particles and glare spots that formed due to a reflection of the laser sheet from imperfections on the surface (small scratches and particles stuck to the wall). As shown in figure 7, there is no observable noise in the velocity distributions obtained from 3D-PTV based on STB.

The 3D-PTV measurements of the Reynolds stress profiles are compared with those of DNS in figure 8. The results from 3D-PTV and DNS agree well with one another, although there are some minor deviations. The maximum discrepancy in the peak streamwise Reynolds stress, ${\langle {u^2}\rangle ^ + }$, shown in figure 8(a), is approximately $0.4u_\tau ^2$. The maximum deviation in the y⁺ location of the peak in ${\langle {u^2}\rangle ^ + }$ is 2.6λ. The wall-normal Reynolds stress profile, ${\langle {v^2}\rangle ^ + }$, overlaps well with DNS for Re_τ of 425 and 511, as shown in figure 8(b). The ${\langle {v^2}\rangle ^ + }$ profile for data with a Re_τ of 307 has a constant deviation, relative to the DNS profile, approximately equal to $0.1u_\tau ^2$ for all y⁺. Generally, the spanwise Reynolds stress distributions, ${\langle {w^2}\rangle ^ + }$, for all 3D-PTV results, are in good agreement with DNS, as seen in figure 8(c). The 3D-PTV results and DNS also show good agreement in their Reynolds shear stress profiles, ${\langle uv\rangle ^ + }$, shown in figure 8(d). One minor exception might be that the 3D-PTV profile of ${\langle uv\rangle ^ + }$ at Re_τ of 425 has a marginally larger peak by approximately $0.1u_\tau ^2$ with respect to the DNS profile.

The profiles of ${\langle {v^2}\rangle ^ + }$ and ${\langle uv\rangle ^ + }$, shown in figure 8(b,d) both have visible low-amplitude noise. This is associated with the larger particle positioning error of 3D-PTV in the out-of-plane direction and the smaller flow motions in this direction (v component). The largest peak-to-peak noise oscillation in figure 8(b) is approximately $0.03u_\tau ^2$, occurring between y⁺ = 230 and 250 for the case of Re_τ = 425. This peak-to-peak noise corresponds roughly to a pixel disparity of 0.1 pixel, given the digital resolution of 27.9 μm pixel⁻¹ and the image acquisition rate of 2.9 kHz. Since 0.1 pixel is less than the assumed error of 0.2 pixel for v, the visible low-amplitude noise in figure 8(b,d) is within the assumed margin of uncertainty discussed in § 2.3, and is captured by the error bars.

3.3. Mean velocity profile

The mean velocity profiles normalized using outer scaling are compared for drag-reduced flows at HDR and MDR in figures 9(a) and 9(b), respectively. Here, h is the half-channel height. Error bars are excluded from this figure, as the estimated 3D-PTV uncertainty is equivalent to the line thickness used here. In these figures, the mean velocity profile for water at the same U_b as the drag-reduced flows is also presented. For water, this flow rate results in Re_τ of 793, which is larger than Re_τ of the drag-reduced flows. The magnitudes of mean velocity in the near-wall region for the drag-reduced solutions is smaller than mean velocity of water. Although not fully captured within the wall-normal extent of the 3D-PTV domain, farther away from the wall, mean velocity of the drag-reduced flows is expected to become larger than that of water to maintain a similar U_b.

Figure 9. Outer-normalized mean streamwise velocity profile for drag-reduced flows at (a) HDR and (b) MDR.

Based on the shape of velocity profiles in figure 9(a), we can also see that the wall-normal gradient of mean velocity at the wall, $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, for all three drag-reduced cases is smaller than $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ of water. The profiles of C14 and PAM at HDR appear to approximately overlap in figure 9(a). The XG solution, on the other hand, starts with a lower $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, and its $\langle U\rangle /{U_b}$ profile is smaller up until y/h of 0.42. The greater μ _w of XG compensates for its smaller $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, resulting in a similar wall shear stress as PAM and C14. Within the region of y/h < 0.4 shown in figure 9(b), mean velocity for the two MDR cases of PAM and C14 are significantly lower than water. The profiles also demonstrate that $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ of PAM and C14 are smaller than $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ of water. PAM at MDR has a marginally lower velocity for y/h < 0.5 when compared to C14.

Figure 9 confirms that a similar DR does not ensure overlap of the mean velocity profile for different drag-reducing additives when the profiles are normalized using outer scaling. This was observed clearly for the XG solution in figure 9(a). The results also show that the difference in the mean velocity profiles of different drag-reducing additives at a similar DR is not associated with the difference in their Re _H. In both figures 9(a) and 9(b), the mean velocity profiles of PAM and C14 solutions are similar while their Re _H is different (see table 2). The properties of the solutions suggest that their shear viscosity plays an important role in setting the outer-normalized mean velocity profiles. At a similar DR, drag-reduced solution with larger μ _w have a lower $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ and $\langle U\rangle /{U_b}$ in the near-wall region. While solutions with a similar μ _w result in a similar $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ and $\langle U\rangle /{U_b}$ in the near-wall region.

The inner-normalized mean velocity profile, ${\langle U\rangle ^ + }$, in the immediate wall vicinity at y⁺ < 15 is demonstrated for all additives and for water in figure 10. The inner scales of the turbulent flows are estimated here by calculating $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ using a linear fit of the data at 2–4 < y⁺ < 5. The lower wall-normal limit corresponds to the first valid data point from the 3D-PTV system which is determined to be at y ≈ 60 μm. For consistency, we chose the upper bound to be the maximum limit of the linear viscous sublayer for a Newtonian flow. Figure 10 shows the linear fit used to calculate $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$, and confirms the presence of a linear region for all the flows. The estimated $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ values are presented in table 3 and are used to calculate the corresponding μ _w based on the shear viscosity models described in § 3.1. This results in μ _w and the other inner-scaling variables for the drag-reduced flows that are presented in the table 3. The comparison of the estimated DR ₂ (based on $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$) in table 3 with the DR ₁ (based on ΔP) in table 2 shows a reasonable agreement of the two methods. The difference between DR ₁ and DR ₂ is small and varies between 1.6 % to 4.8 %. The discrepancy is associated with several factors including the finite aspect ratio of the channel, deviation from the fully developed turbulence at the upstream pressure port, and the uncertainty in determining $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$.

Figure 10. Mean streamwise velocity profile in the immediate near-wall region for (a) PAM at HDR, (b) XG at HDR, (c) C14 at HDR, (d) PAM at MDR, (e) C14 at MDR and (f) water.

Table 3. The estimated inner scaling based on the wall-normal gradient of mean velocity at the wall for the drag-reduced flows.

The relatively good agreement amongst the wall statistics and DR using measurements of ΔP and 3D-PTV for XG, suggests the extrapolation of the CY model from § 3.1 can reasonably estimate μ _w. A further means of communicating the agreement of these measurements is by determining μ _w using $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ and τ_{w ,1}. Here, $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ is obtained from 3D-PTV measurements, and τ_{w ,1} is derived from measurements of ΔP. Such a validation has been done in experiments by Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) and Ptasinski et al. (Reference Ptasinski, Nieuwstadt and Hulsen2001). If we perform the same analysis, the viscosity of the XG solution at a shear rate of 2364 s⁻¹ ($\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ from table 3) is 1.44 mPa s (using τ_{w ,1} in table 1). This viscosity is approximately 0.14 mPa s lower than the μ _w listed in table 3, which is roughly 8 %. The majority of this uncertainty is reflected in the error bars that propagate from a random error in repeated viscosity measurements and are shown in figures of mean velocity profile and Reynolds stresses to follow.

As alluded to earlier in § 2.3, the method of multiplying $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ and μ _w to establish τ_{w ,2} for the non-Newtonian fluids is an approximation. Fluctuations in $\textrm{d}\langle U\rangle /\textrm{d}y{|_w}$ with respect to time can be significant and the instantaneous distribution of μ _w may not be simply determined by the mean shear rate. This is most significant for the XG solution, whose shear viscosity is described by the CY model. Gubian et al. (Reference Gubian, Stoker, Medvescek, Mydlarski and Baliga2019) demonstrated that τ_w can fluctuate by as much as 35 % of the nominal value of τ_w for a Newtonian turbulent channel flow with a Re_τ of approximately 300. Assuming such a variance in τ_w is applicable to XG, an uncertainty in μ _w of approximately 0.06 mPa s is expected. Such a fluctuation in μ _w is captured by the error bars in the mean flow statistics demonstrated in the figures to follow.

In addition to demonstrating the fit of the linear viscous sublayer, figure 10 presents some insight into the thickness of the viscous sublayer for drag-reduced flows. The elastic sublayer model of Virk (Reference Virk1971) proposed that all drag-reduced flows have a viscous sublayer thickness of y⁺ = 11.6 (corresponding to the tri-section point of the MDR asymptote, ${y^ + } = {\langle U\rangle ^ + }$, and the log law). However, figure 10 demonstrates that none of the drag-reduced flows, have a viscous sublayer thickness of y⁺ = 11.6 (represented by the maximum extent of the black line). However, there is still a considerable thickening of the linear viscous subregion relative to water for the drag-reduced flows. At y⁺ = 11.6, HDR flows of XG, C14 and PAM solutions deviate from the linear fit by 1.98u_τ, 1.44u_τ and 1.12u_τ, respectively. The largest deviation corresponds to the XG solution, which has the largest shear viscosity. Water has a deviation from the linear fit at y⁺ = 11.6 of 1.97u_τ, which is equivalent to the deviation of XG. For MDR flows of C14 and PAM, the relative deviation from the linear profile at y⁺ = 11.6 is smaller and equal to 0.6u_τ and 1.0u_τ, respectively.

The results in figure 10 show that the thickness of the viscous sublayer is smaller for drag-reduced flows at HDR than MDR, suggesting that viscous sublayer thickens with increasing DR. We also see that the thickness of the viscous sublayer depends on the additive type, i.e. the thickness varies for different solutions at a similar DR. The results also suggest that in general the thickness of the viscous sublayer in inner scaling reduces with increasing shear viscosity. The XG solution has the highest shear viscosity and has an almost identical viscous sublayer thickness as water, while other HDR flows with lower shear viscosity have a thicker viscous sublayer.

The velocity profiles normalized by inner scaling and presented in a log–linear format are shown in figure 11. The inner-normalized mean velocity profiles are compared with both the Newtonian law of the wall and the ultimate profile for drag-reduced flows at MDR, ${\langle U\rangle ^ + } = 11.7\,\textrm{ln}\,{y^ + } - 17$ (Virk et al. Reference Virk, Mickley and Smith1970). The results for flows at HDR in figure 11(a) are discussed first, followed by the results for MDR in figure 11(b).

Figure 11. Inner-normalized mean streamwise velocity profile of drag-reduced flows at (a) HDR and (b) MDR.

The mean velocity profiles of the HDR flows in figure 11(a) are close to each other in the near-wall region. We also observe that with increasing y⁺, the HDR profiles of the three drag-reduced cases start to diverge and appear to have different slopes. Subject to the Virk (Reference Virk1971) elastic sublayer model for polymer flows at an intermediate DR, the ${\langle U\rangle ^ + }$ profile in the elastic sublayer (or buffer layer) is supposed to overlap with the ultimate profile, and for larger y⁺ a Newtonian plug layer with a logarithmic profile with a similar slope as the Newtonian log layer should propagate. As shown in figure 11(a), none of the HDR profiles overlap with the ultimate asymptote. Our observations for HDR flows show that in the HDR regime, DR does not uniquely define the inner-normalized mean velocity profile since the type of additive plays a role in shaping the profile. In comparing the mean velocity profiles of different experiments, White et al. (Reference White, Dubief and Klewicki2012) similarly observed variability in the outer layer of the mean velocity profile for polymer solutions with the same DR; albeit for cases of low DR, smaller than 40 %. Due to the differences amongst the data sets, White et al. (Reference White, Dubief and Klewicki2012) postulated that the velocity distribution in the outer layer depends on Re, properties of the additive, and the canonical flow type. It is important to note that the results in figure 11(a) do not exclude the effect of Re. In other words, the variations can be partly attributed to differences in the Re of the drag-reduced flows.

The mean velocity profile of the two drag-reduced flows at MDR are shown in figure 11(b). The profile of C14 has a higher ${\langle U\rangle ^ + }$ than PAM outside the viscous sublayer, which is consistent with its slightly higher DR ₂; 76.5 % for C14 versus 73.3 % for PAM solution. The C14 profile is also marginally greater than the MDR asymptote for y⁺ > 60. Both previous experimental and numerical simulations have observed a small overshoot of the MDR asymptote for velocity profiles of polymer solutions (Escudier et al. Reference Escudier, Nickson and Poole2009; White et al. Reference White, Dubief and Klewicki2012; Graham Reference Graham2014). Both profiles do not adhere to the MDR asymptote of Virk et al. (Reference Virk, Mickley and Smith1970) and intersect with it at different y⁺. In addition, the profile of C14 does not agree with the asymptote for drag-reducing surfactant solutions proposed by Zakin et al. (Reference Zakin, Myska and Chara1996); ${\langle U\rangle ^ + } = 23.4\,\textrm{ln}\,{y^ + } - 65$. This asymptote is not show in figure 11 for brevity. Considering the error bars and the slight difference in DR of C14 and PAM, the MDR asymptote seems to be unique and independent of the additive type and Re number. However, the drag-reduced flows of PAM and C14 at MDR do not follow the logarithmic trend proposed by Virk et al. (Reference Virk, Mickley and Smith1970); they share a similar S-shaped profile that straddles or at least intersects the asymptote of Virk et al. (Reference Virk, Mickley and Smith1970). To further evaluate the logarithmic behaviour, the indicator function, $\zeta = {y^ + }\,\textrm{d}{\langle U\rangle ^ + }/\textrm{d}{y^ + }$, is investigated next in figure 12. Using the indicator function to evaluate logarithmic dependency, White et al. (Reference White, Dubief and Klewicki2012) found that the inner-normalized mean velocity of polymer drag-reduced flows at MDR were not truly logarithmic functions of y⁺.

Figure 12. The indicator function for drag-reduced flows at (a) HDR and (b) MDR.

To establish $\textrm{d}{\langle U\rangle ^ + }/\textrm{d}{y^ + }$, and calculate ζ, a moving second-order polynomial filter, of length 10–15λ (250 μm), was applied to the distribution of ${\langle U\rangle ^ + }$ as a function of y⁺. The polynomials were then differentiated analytically. Figures 12(a) and 12(b) demonstrates ζ as a function of y⁺ for HDR and MDR flows, respectively. A region of y⁺ where ζ is constant is indicative of a layer where ${\langle U\rangle ^ + }$ varies logarithmically as a function of y⁺. For example, the distribution of ζ for water, shown in both figures 12(a) and 12(b), is approximately constant and equal to 2.5 for y⁺ > 30, which is indicative of a logarithmic layer for the Newtonian turbulent channel flows. White et al. (Reference White, Dubief and Klewicki2012), Elbing et al. (Reference Elbing, Perlin, Dowling and Ceccio2013) and White et al. (Reference White, Dubief and Klewicki2018) proposed that for a polymer drag-reduced flow, the shape of the mean velocity profile, and similarly ζ, depends on Re, polymeric properties and the canonical flow type. Figure 12(a,b) addresses the second postulate by comparing flows comprised of different additives at HDR and MDR.

Figure 12(a) shows that the HDR flows of C14 and PAM have similar distributions of ζ. White et al. (Reference White, Dubief and Klewicki2012) stated that HDR flows are distinct in their lack of a Newtonian plug. By observation of figure 12(a) none of the HDR flows have a y⁺ range where ζ appears constant and a Newtonian plug does not exist within the measurement domain. However, this does not rule out the possibility of a Newtonian plug existing at larger y⁺. The profile of ζ for XG show relative similarity with the other HDR flows for y⁺ < 30; however, the peak in its profile, though subject to experimental noise, appears to be marginally higher and located at larger y⁺. The larger y ⁺ location of ζ peak for XG solution indicates that the centre of the elastic sublayer (buffer layer) is farther away for the wall. Therefore, the indicator function also provides further evidence that the shape of the velocity profile and the thickness of the sublayers is not uniquely defined by DR. Here, the thicker elastic sublayer of the XG solution is associated with its larger shear viscosity and lower Re number. The y ⁺ location of the peak in the distribution of ζ, shows that the elastic sublayer is thinner for drag-reduced solution with higher Re number (Re_H or Re_τ).

Figure 12(b) compares the plots of ζ for C14 and PAM at MDR. The two profiles appear similar for all y⁺. The y⁺ location and value in the peak of ζ is approximately (y⁺, ζ) = (70, 14) for both drag-reduced flows. The peak is larger and farther away from the wall relative to the HDR cases, indicating a thicker elastic sublayer. Due to the lack of a region with constant ζ, White et al. (Reference White, Dubief and Klewicki2012) concluded that the exact shape of the MDR profile was not logarithmic. Instead, MDR was achieved when the peak in ζ equals 11.7, corresponding to the slope in the MDR asymptote proposed by Virk et al. (Reference Virk, Mickley and Smith1970). Figure 12(b) demonstrates that the peak exceeds this limit for both PAM and C14 solutions. In plotting ζ for experimental data from Escudier et al. (Reference Escudier, Nickson and Poole2009) collected for a rigid polymer solution at MDR with DR of 67 %, White et al. (Reference White, Dubief and Klewicki2012) demonstrated a similar overshoot of ζ = 11.7. Elbing et al. (Reference Elbing, Perlin, Dowling and Ceccio2013) also shows a peak in ζ greater than 11.7 for a flexible polymer solution with DR = 65 %. Therefore, further doubt is cast on the exactness of the slope of the MDR profile of Virk et al. (Reference Virk, Mickley and Smith1970). Figure 12(b) also appends the conclusion of White et al. (Reference White, Dubief and Klewicki2012) to state that surfactant drag-reduced flows at MDR, in addition to polymer flows, also do not possess a logarithmic layer. Furthermore, while the shape of the two mean velocity profiles at MDR are not exactly logarithmic, they are similar. This implies that a universal distribution of ${\langle U\rangle ^ + }$, and ζ, for drag-reduced flows at MDR, that is irrespective of the additive type and Re number, may exist.

3.4. Reynolds stresses

The Reynolds stresses profiles for the HDR cases are compared in figure 13. In addition to the drag-reduced flows, the Reynolds stress profiles for water at four Re_τ that are similar to Re_τ of the drag-reduced cases are presented. For example, the Reynolds stress profiles of C14, PAM and XG, with Re_τ of 503, 424 and 287, are shown alongside those for water with a Re_τ of 511, 425 and 307. As expected, all of the Reynolds stress profiles of water show similar distributions, relative to one another, within the linear sublayer and buffer layer. Larger differences in the outer layer, amplify with increasing y⁺ as expected.

Figure 13. Inner-normalized mean Reynolds stress profiles of drag-reduced flows at HDR showing (a) streamwise Reynolds stress, (b) wall-normal Reynolds stress, (c) spanwise Reynolds stress profiles and (d) Reynolds shear stress.

Figure 13(a) shows that all HDR flows possess a large peak value of ${\langle {u^2}\rangle ^ + }$ that is also shifted away from the wall, relative to water at a similar Re_τ. The ${\langle {u^2}\rangle ^ + }$ profiles of C14 and PAM appear similar for y ⁺ < 70 although the ${\langle {u^2}\rangle ^ + }$ peak is smaller for PAM. The two profiles deviate with further increase of y ⁺. Compared to C14 and PAM, XG has a smaller peak value of ${\langle {u^2}\rangle ^ + }$, which is displaced farther from the wall. Therefore, ${\langle {u^2}\rangle ^ + }$ peak is smaller and farther away from the wall for solutions with higher shear viscosity. In addition, the notion that drag-reduced flows of different additives at the same DR have a similar ${\langle {u^2}\rangle ^ + }$ peak appears to be invalid. The shift in the peak of ${\langle {u^2}\rangle ^ + }$ away from the wall is an indication of a thicker buffer layer that is consistent with our previous observations.

Figure 13(b,c) demonstrates significant attenuation in the profile of ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ of the drag-reduced flows relative to water. For ${\langle {v^2}\rangle ^ + }$, this agrees with the observations of Escudier et al. (Reference Escudier, Nickson and Poole2009) for polymers and also Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) for surfactants. Attenuation in the profile of ${\langle {w^2}\rangle ^ + }$ has been shown by White et al. (Reference White, Somandepalli and Mungal2004) for polymers. To the authors’ knowledge, ${\langle {w^2}\rangle ^ + }$ has never been demonstrated for surfactant drag-reduced flows. Similar to their ${\langle {u^2}\rangle ^ + }$ profiles, C14 and PAM display rather similar profiles for ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ with subtle discrepancies. The ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ profiles for XG, on the other hand, are noticeably more attenuated than the other HDR flows. The peak value in the ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ distributions of XG are approximately 50 % those of C14. Figure 13(d) demonstrates similar profiles in ${\langle uv\rangle ^ + }$ for C14 and PAM, but again a more attenuated distribution for XG. The larger attenuation in ${\langle uv\rangle ^ + }$ is likely attributed to a larger imposition of viscous stresses due to the larger overall shear viscosity of the XG solution. Therefore, different drag-reduced solutions at an identical DR do not exhibit identical distribution of Reynolds shear stresses, in particular when their shear viscosity is different. A lack of consistency in the shear viscosity of the drag-reduced solutions is also reflected by differences in the Re number of the solutions with similar DR (i.e. similar u_τ). Therefore, the discrepancy in the Reynolds stress distributions of the HDR flows can be similarly explained by differences in the Re of the drag-reduced solutions.

Figure 14 demonstrates the Reynolds stresses of C14 and PAM at MDR. Having observed that the Reynolds stresses of XG were much lower than the other HDR flows in figure 13, it was perceived to be prudent to include XG at HDR in the comparison with the MDR flows in figure 14. This was based on prior knowledge that the Reynolds stresses are more attenuated for flows with larger DR (Warholic et al. Reference Warholic, Massah and Hanratty1999a; Ptasinski et al. Reference Ptasinski, Nieuwstadt and Hulsen2001; Escudier et al. Reference Escudier, Nickson and Poole2009). Similar to figure 13, figure 14 presents the Reynolds stresses of the drag-reduced flows alongside the distributions of water that share a similar Re_τ. C14 and PAM at MDR, alongside XG at HDR, with Re_τ of 363, 324 and 307, are presented together with the distributions of water with Re_τ of 363 and 307.

Figure 14. Inner-normalized mean Reynolds stress profiles of drag-reduced flows at MDR and XG at HDR; (a) streamwise Reynolds stress, (b) wall-normal Reynolds stress, (c) spanwise Reynolds stress profiles and (d) Reynolds shear stress.

In figure 14(a), there is relatively good overlap in the distributions of ${\langle {u^2}\rangle ^ + }$ for the three solutions. Here the similarity in the XG profile with the other two profiles is striking, despite 18–21 % difference in DR ₂ of XG at HDR and the other two MDR flows. For polymer flows, Escudier et al. (Reference Escudier, Nickson and Poole2009) demonstrated that for DR > 40 %, ${\langle {u^2}\rangle ^ + }$ decreases as a function of DR; albeit, results appeared mixed for other authors (Warholic et al. Reference Warholic, Massah and Hanratty1999a). In the current investigation, the ${\langle {u^2}\rangle ^ + }$ peak of C14 and PAM at MDR decreased relative to their corresponding HDR cases. However, the peaks did not decrease to a point where they are lower than the peak measured for water. While Li et al. (Reference Li, Kawaguchi, Segawa and Hishida2005) and Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) demonstrate a lower peak in ${\langle {u^2}\rangle ^ + }$ for surfactant drag-reduced flows with large DR they have similarly shown that the peak in ${\langle {u^2}\rangle ^ + }$ largely depends on the Re_τ of the flow. Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) demonstrated this in their sweep of Re for different HDR flows, where the peak in ${\langle {u^2}\rangle ^ + }$ was larger than water for surfactant drag-reduced flows with large Re, but smaller than water for low Re. Thais, Gatski & Mompean (Reference Thais, Gatski and Mompean2012) showed the peak in ${\langle {u^2}\rangle ^ + }$ had a similar dependence on Re based on DNS using the FENE-P model. Figure 14(b,c) demonstrates that the distributions of ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ for C14 and PAM at MDR, and XG at HDR, have nearly identical profiles that are also significantly suppressed relative to water. Li et al. (Reference Li, Kawaguchi, Segawa and Hishida2005) and Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) also observed significant attenuation in profiles of ${\langle {v^2}\rangle ^ + }$ for surfactant drag-reduced flows near MDR. The overlap in ${\langle {u^2}\rangle ^ + }$, ${\langle {v^2}\rangle ^ + }$ and ${\langle {w^2}\rangle ^ + }$ implies that the mean turbulent kinetic energy is the same for the three drag-reduced flows.

Lastly, figure 14(d) demonstrates that ${\langle uv\rangle ^ + }$ profiles of C14 and XG are slightly larger than the ${\langle uv\rangle ^ + }$ profile of PAM at y ⁺ < 100. However, for all three flows, the ${\langle uv\rangle ^ + }$ magnitudes are small and have the same order of magnitude as the error bars. Therefore, the values should be considered negligible and differences are not tangible. Several authors have shown both finite and also negligible ${\langle uv\rangle ^ + }$ profiles for polymer drag-reduced flows near MDR (Warholic et al. Reference Warholic, Massah and Hanratty1999a; Ptasinski et al. Reference Ptasinski, Boersma, Nieuwstadt, Hulsen, van den Brule and Hunt2003; Escudier et al. Reference Escudier, Nickson and Poole2009). Similarly, Tamano et al. (Reference Tamano, Uchikawa, Ito and Morinishi2018) presented a finite ${\langle uv\rangle ^ + }$ distribution, while Warholic et al. (Reference Warholic, Schmidt and Hanratty1999b) demonstrated a ${\langle uv\rangle ^ + }$ profile approximately equal to zero for flows of surfactant drag-reducing additives at MDR. The discrepancies in the small residual values of ${\langle uv\rangle ^ + }$ is potentially associated with measurement uncertainties as they are also present in the current measurements.

Considering PAM and C14 at MDR, the measurements presented in figure 14 show that Reynolds stress profiles of drag-reduced flows at MDR overlap. We observed a perfect overlap for all components except Reynolds shear stress. For the latter component, there are subtle differences with the same magnitude as the measurement uncertainties. Therefore, we can conclude that at MDR, the Reynolds stress profiles are not a function of additive type and Reynolds number. At MDR, the Reynolds stress profiles converge to a common set of distributions for polymer and surfactant drag-reduced flows with different Re.

The C_f values presented based on ΔP in figure 4, and mean velocity profiles of figure 11(a), suggest that XG is not at MDR. In contrast, the results of figure 14 demonstrate that Reynolds stress profiles of XG are similar to those of PAM and C14 at MDR. The measurements of DR ₁ (based on ΔP) for XG in figure 3(c) also show that a higher level of DR was not achievable for XG with increasing its concentration; DR ₁ plateaus to a constant 58.5 % for c in excess of 300 ppm. Why XG has a lower asymptotic DR ₁, relative to C14 and PAM at MDR, is likely attributed to the imposition of larger viscous stresses. To summarize, it is evident that the DR ₁ of XG has attained an asymptotic state, according to figure 3(c). The Reynolds stresses also demonstrate that XG shares dynamical similarities with other MDR flows (see figure 14). Therefore, with respect to the turbulent flow and production of turbulent kinetic energy, XG is at an MDR state. The discrepancies in DR and mean velocity profile of XG with respect to the MDR state of the other drag-reduced flows is associated with larger inherent viscous stresses of this polymer solution.

3.5. Low- and high-speed streaks

The following analysis evaluates the length scale of the dominant flow structures at HDR and MDR using two-point correlation of streamwise velocity fluctuations. The spatial, two-point correlation is computed as

(3.3)\begin{equation}{R_{uu}}(\Delta z) = \frac{{\langle \; {u_{({x_0},{y_0},{z_0})}}\; {u_{({x_0},{y_0},{z_0} + \Delta z)}}\; \rangle }}{{\sqrt {\langle \; u_{({x_0},{y_0},{z_0})}^2\rangle } \; \sqrt {\langle \; u_{({x_0},{y_0},{z_0} + \Delta z)}^2\rangle } }}.\end{equation}

Here, (x ₀, y ₀, z ₀) is the coordinate of the reference point selected at (0, 0.4 h, 0), which is positioned within the logarithmic layer for Newtonian flows. The dominant coherent structures at this location are low and high-speed streaks that have also been observed in drag-reduced flows (White et al. Reference White, Somandepalli and Mungal2004; Mohammadtabar et al. Reference Mohammadtabar, Sanders and Ghaemi2017). At higher Re and in Newtonian flows, these streaks form the very large-scale motions (Hutchins & Marusic Reference Hutchins and Marusic2007). The incremental displacement along the spanwise direction is indicated as Δz, relative to the z ₀ reference point. As a result, R_uu characterizes the spanwise scale of the low and high-speed streaks in the drag-reduced flows.

Figure 15(a) presents R_uu along Δz/h for the HDR flows. The R_uu functions for water are shown alongside the drag-reduced flows. The overlap in the R_uu profiles indicate that the width of the streaks for the Newtonian cases are similar. The R_uu profiles for C14 and PAM at HDR are also approximately similar, indicating a similar streak spacing. This suggests that the R_uu distribution for drag-reduced flow may not be a strong function of Re as PAM and C14 flows have different Re. The XG demonstrates a rather larger R_uu relative to C14 and PAM, which indicates even wider streaks. Therefore, the turbulent streaks of drag-reduced flows of PAM and C14 with similar shear viscosities appear to be more alike, while XG – a solution with a much larger overall shear viscosity – is distinct.

Figure 15. Two-point correlation of streamwise velocity fluctuations in the spanwise direction for drag-reduced flows at (a) HDR and (b) MDR. The reference location for the two-point correlations is at (x ₀, y ₀, z ₀) = (0, 0.4 h, 0).

Figure 15(b) presents R_uu of drag-reduced flows of PAM and C14 at MDR, and XG at HDR. The profiles approximately overlap, and therefore streak spacing is expected to be similar for the three drag-reduced flows. Using a similar two-point correlation analysis, Li, Sureshkumar & Khomami (Reference Li, Sureshkumar and Khomami2006), White et al. (Reference White, Somandepalli and Mungal2004) and Tamano et al. (Reference Tamano, Uchikawa, Ito and Morinishi2018) demonstrated a monotonic increase in the spanwise width of the low- and high-speed streaks for polymer and surfactant drag-reduced flows with increasing DR. Comparing figure 15(a), with figure 15(b), both C14 and PAM exhibit growth in the average streak spacing with respect to DR. The XG profile appears to show more similarities in the width of its streaks with respect to solutions of C14 and PAM at MDR. This reinforces the notion that XG has attained a state of MDR regarding turbulent dynamics.