3.1 Parameter space

The parameter space investigated in the present experiments is described in table 1. The friction Reynolds numbers, $\unicode[STIX]{x1D6FF}^{+}$ , ranges from 580 to 1690. Note that $\unicode[STIX]{x1D6FF}^{+}$ for the Newtonian flows were $\approxeq 650$ , $975$ and $1800$ . The ratio $Q_{i}/Q_{s}$ refers to the relative flow rate of the injected fluid to the wall layer flow rate, as discussed above. Here $C_{i}$ is the concentration of the injected fluid, and $C_{h}$ is the bulk concentration in the channel. The bulk concentration assumes a homogeneous mixture and is determined by using the measured volumetric flow rate obtained from a turbine style flow meter. The estimated drag reduction for each scenario is also listed. The friction Weissenberg number, $We_{\unicode[STIX]{x1D70F}}$ , defined as the ratio of the polymer time scale to the flow time scale is given by

(3.1) $$\begin{eqnarray}\displaystyle We_{\unicode[STIX]{x1D70F}}=\frac{\unicode[STIX]{x1D706}u_{\unicode[STIX]{x1D70F}}^{2}}{\unicode[STIX]{x1D708}_{0}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D706}$ is the polymer relaxation time and $\unicode[STIX]{x1D708}_{0}$ is the zero shear rate kinematic viscosity of the solution. In table 1, $\unicode[STIX]{x1D706}$ is determined using the Rouse–Zimm theory and is approximated as

(3.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}_{z}=0.422\frac{[\unicode[STIX]{x1D702}]M_{w}\unicode[STIX]{x1D707}_{s}}{N_{A}k_{B}T}, & & \displaystyle\end{eqnarray}$$

where $[\unicode[STIX]{x1D702}]=0.072M_{w}^{0.65}$ is the intrinsic viscosity of the PEO solution, $\unicode[STIX]{x1D707}_{s}$ is the solvent dynamic viscosity, $N_{A}$ is Avogadro’s number, $k_{B}$ is the Boltzmann constant, and $T$ is the temperature (Zimm Reference Zimm1956).

In the present study, depending on $C_{i}$ , the solution concentration from the point of injection to the MTV measurement station will range from semidilute $(C^{\ast }<C<C_{e})$ to dilute $(C<C^{\ast })$ , where $C^{\ast }$ is the critical overlap concentration and $C_{e}$ is the entanglement concentration beyond which topological interactions play a role. Correlations correcting for the affect of concentration on $\unicode[STIX]{x1D706}$ in the dilute regime can be found in Tirtaatmadja, McKinley & Cooper-White (Reference Tirtaatmadja, McKinley and Cooper-White2006). Note that at the MTV measurement station $C\approx C_{h}$ and the Zimm relaxation time provides a suitable measure of the polymer relation time. White, Somandepalli & Mungal (Reference White, Somandepalli and Mungal2004) reported a polydispersity of 1.45 for the same polymer molecular weight as used in the present study. It is therefore likely that the polymer in the present study has a similar polydispersity, which means there exists a distribution of molecular weights and, consequently, a single $We_{\unicode[STIX]{x1D70F}}$ is effectively a representative measure of the viscoelastic effects of the solution.

Table 1.

Experimental parameters for the polymer drag-reduced flows investigated.

3.2 Potential effects associated with the injection process

Significant care was taken to ensure that (i) the flow properties at the measurement location were not influenced by disturbances caused by the polymer injection, and that (ii) the flow properties were not influenced by an accumulation of polymer concentration in the facility over time. This section documents three tests that examined these potential effects. The tests quantify profiles of the inner-normalized mean velocity, r.m.s. streamwise velocity, and Reynolds shear stress, as computed via measurement of the mean velocity gradient and (1.4). In figures 7 and 8, the first test $(T1)$ refers to the results measured before polymer was injected, and thus quantifies the Newtonian fluid flow. Once polymer is injected, there is the possibility of a temporal increase in the test section polymer concentration, depending on how effectively the polymer chains are broken by passing through the pump. To explore this effect, the second test $(T2)$ involved measurements in the test section following a polymer injection experiment, but after the polymer had cycled at least once through the pump. Lastly, the third test $(T3)$ investigated the effect of the injection process itself by acquiring measurements for an injection of pure water. The $T1$ results provide clear evidence that the Newtonian fluid measurements in the test section closely adhere to DNS results at a comparable Reynolds number. It is also apparent that the drag-reducing ability of the injected polymer is, to within measurement uncertainty, completely destroyed by the pump, as the $T2$ mean velocity, r.m.s. $u^{\prime }$ , and Reynolds stress profiles all show close agreement with the DNS results. Similarly, the $T3$ results indicate that statistical properties of the flow were not altered by the injection itself. The results of figures 7 and 8 provide clear evidence that changes observed in the data presented below are due to the effect of the injected polymer. After each experiment, the MTV solution in the channel was filtered for 24 h using dual-stage $20~\unicode[STIX]{x03BC}\text{m}$ filters to remove any residual polymer.

Figure 7.

Inner-normalized velocity profiles in ascending order: $T1$ , before polymer injection; $T2$ , after polymer injection (with destroyed polymer); $T3$ , injection using water. The profiles are vertically shifted by 5.

Figure 8.

Streamwise r.m.s. velocity (a) and Reynolds stress (b) profiles in ascending order: $T1$ , before polymer injection; $T2$ , after polymer injection (with polymer that has passed through the pump); $T3$ , injection using pure water. See figure 7 for details; dotted line, uncorrected, and solid line, corrected r.m.s. $u^{\prime }$ . Profiles are shifted vertically by 1.

3.3 Redistribution of momentum and vorticity

Depending on the injected concentration, turbulent channel flow structure can be dramatically influenced by the addition of polymer, including the mean profiles of momentum and vorticity. In the present experiments the mass flow rate at a given initial $\unicode[STIX]{x1D6FF}^{+}$ was held constant for increasing polymer concentration. Accordingly, redistributions of momentum or vorticity only result when there is a net modification, i.e. reduction, in drag. This behaviour is made clear in figure 9(a,b), which plots mean velocity profiles normalized by the bulk (spatially averaged) velocity, $U_{b}$ , and centreline velocity, $U_{cl}$ , respectively. Here, the near-wall velocity profile falls below the Newtonian flow, but for $\unicode[STIX]{x1D702}=y/\unicode[STIX]{x1D6FF}>0.05$ the profile exceeds the Newtonian case, and then falls below in a region near the centreline. The mass flow rate was held constant, and thus the conservation of mass requires an area-preserving profile. The redistribution of momentum away from the wall leads to a reduction in the velocity gradient near the wall. Consistently, the profiles at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ also exhibit the same qualitative behaviour as the lower-Reynolds-number case.

Figure 9.

Mean velocity normalized with (a) the bulk velocity and (b) the centreline velocity. Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dash–dot line, $\unicode[STIX]{x1D6FF}^{+}=1463$ ; dashed line, $\unicode[STIX]{x1D6FF}^{+}=1691$ .

A number of measures can be used to quantify aspects of the mean profile features just noted. One such integral measure is the displacement thickness, $\unicode[STIX]{x1D6FF}^{\ast }$ , as it reflects properties associated with mass conservation as well as those associated with the mean vorticity distribution. For the present planar channel, the equation for $\unicode[STIX]{x1D6FF}^{\ast }$ is given by

(3.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{\ast }=\int _{0}^{\unicode[STIX]{x1D6FF}}\left(1-\frac{U}{U_{cl}}\right)\text{d}y. & & \displaystyle\end{eqnarray}$$

As first demonstrated mathematically by Lighthill (Reference Lighthill1958), $\unicode[STIX]{x1D6FF}^{\ast }$ has the precise physical interpretation as being the centroid of the mean vorticity distribution. This interpretation will be revisited in the discussion of (3.6) relative to the mean vorticity profile. The ratio of the displacement thickness in polymer flow, $\unicode[STIX]{x1D6FF}_{p}^{\ast }$ , to that in the experimental Newtonian flow, $\unicode[STIX]{x1D6FF}^{\ast }$ , is presented in figure 10. From this figure, it is apparent that the position of the mean vorticity centroid relative to that in Newtonian fluid flow decreases with increasing $DR$ . The curve fit of the data presented in figure 10 is given by

(3.4) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}_{p}^{\ast }}{\unicode[STIX]{x1D6FF}^{\ast }}=-0.00786(\%DR)+1. & & \displaystyle\end{eqnarray}$$

Figure 10.

Ratio of the measured displacement thickness in the polymer $DR$ flow to that in Newtonian fluid flow at a similar $\unicode[STIX]{x1D6FF}^{+}$ . The $\circ$ symbols are the experimental data, the solid line is the curve fit given by (3.4), and the dashed lines represent the bounds of the $95\,\%$ confidence interval.

Figure 11.

Inner-normalized mean velocity profiles. The solid line is for the $DR=0$ case at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines are for $DR>0$ .

Figure 12.

Inner-normalized velocity profiles for varying $\unicode[STIX]{x1D6FF}^{+}$ at two fixed injection concentrations: (a) $C_{i}=500$ wppm and (b) $C_{i}=1000~\text{wppm}$ . Solid lines correspond to $DR=0$ cases at $\unicode[STIX]{x1D6FF}^{+}\approxeq 650,975$ and $1800$ ; dashed lines are for $DR>0$ cases.

3.4 Inner-normalized mean velocity profiles

Inner-normalized mean velocity profiles are presented in figures 11 and 12. The effect of injection concentration for a fixed initial $\unicode[STIX]{x1D6FF}^{+}$ is clearly presented in figure 11 whereas the effect of initial $\unicode[STIX]{x1D6FF}^{+}$ is clearly shown in figure 12. For pure water flow ( $DR=0$ ) Elsnab et al. (Reference Elsnab, Monty, White, Koochesfahani and Klewicki2017) showed that the 1c-MTV results exhibit excellent agreement with the similar $\unicode[STIX]{x1D6FF}^{+}$ DNS profiles. At a fixed initial $\unicode[STIX]{x1D6FF}^{+}$ , increasing the polymer concentration decreases the drag force. For $\unicode[STIX]{x1D6FF}^{+}\approxeq 650$ , however, increasing the concentration has minimal effect (see figure 12 a,b). This trend is also evident for the $\unicode[STIX]{x1D6FF}^{+}\approxeq 975$ case. These results indicate that, for injected polymers, the capacity of the polymer to reduce drag apparently depends on the concentration and the intensity of the turbulence, as nominally quantified by the magnitude of the Reynolds number. This intensity connects to the magnitude of the vorticity stretching and its associated capacity to stretch the polymer. It is well known that polymers do not reduce drag in laminar flow. The observations just noted would seem to be consistent with this phenomenon since vortex stretching is not present in laminar flows. More specifically, this observation also appears to be in accord with the results from the DNS simulations of Dubief et al. (Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004), indicating that large drag reductions correlate with the polymer chains undergoing a high degree of stretching.

The classical logarithmic equation for the mean velocity profile in statistically stationary turbulent wall-bounded flow is given by (1.1). The constant $\unicode[STIX]{x1D705}$ has special physical significance in this equation, as this leading coefficient reflects the slope of the profile, and thus is an average measure of how the turbulence distributes mean momentum as a function of $y$ (Klewicki, Fife & Wei Reference Klewicki, Fife and Wei2009). Accordingly, the mean flow theory based upon the leading balance structure of the mean momentum equation described in § 1.1 analytically connects $\unicode[STIX]{x1D705}$ to an asymptotically constant flux of turbulent inertial force (Klewicki et al. Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014). More generally, since $\unicode[STIX]{x1D705}$ is a manifestation of the underlying dynamical behaviours of wall-bounded flow, understanding how polymers modify $\unicode[STIX]{x1D705}$ is important to understanding the dynamics of polymer drag-reduced flow and, consequently, the overall phenomena of polymer drag reduction (White et al. Reference White, Dubief and Klewicki2012, Reference White, Dubief and Klewicki2018).

Empirical estimation of the value of $\unicode[STIX]{x1D705}$ requires specification of the relevant flow domain. Herein, the slope of the logarithmic mean profile is determined by curve-fitting the inner-normalized velocity profiles. In accord with the leading balance structure of (1.3), the domain of this fit is beyond the peak in the combined Reynolds and polymer stress, $1.5y_{m}$ , to $y=0.2\unicode[STIX]{x1D6FF}$ . As revealed later in figures 19 and 20, this is where the profiles of the $RPS$ are essentially parallel to the corresponding $DR=0$ profile. Relative to the leading balance structure of figure 1, this corresponds to a position slightly beyond the start of layer $IV$ and extending into the lower portion of layer $IV$ (Klewicki et al. Reference Klewicki, Fife, Wei and McMurtry2007). Physically, the start of this domain is where the viscous force loses leading order. Consistent with the notion of an inertial sublayer in physical space, the end of the specified domain extends to only a small fraction of $\unicode[STIX]{x1D6FF}$ , and is well within the theoretically estimated bounds of the domain where (1.2) admits a self-similar invariant form owing to a constant wallward flux of turbulent inertial force (Klewicki Reference Klewicki2013b ). The specified domain of the logarithmic profile is consistent with the high-Reynolds-number data analysis of Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013), and has also been shown to reliably locate the inertial sublayer in rough-wall flows (Mehdi, Klewicki & White Reference Mehdi, Klewicki and White2013; Ebner, Mehdi & Klewicki Reference Ebner, Mehdi and Klewicki2016; Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016).

Examination of figure 13(a) suggests that the estimated value of $\unicode[STIX]{x1D705}$ begins changing for $DR$ values much less than the $40\,\%$ threshold found by Warholic et al. (Reference Warholic, Massah and Hanratty1999). Conversely, these results reinforce the observation of White et al. (Reference White, Dubief and Klewicki2012) that polymer effects do not adhere to the Virk model (Virk Reference Virk1975).

The indicator function ( $\unicode[STIX]{x1D701}=y^{+}\text{d}U/\text{d}y^{+}$ ) is commonly used to empirically assess the existence of logarithmic profile variation. The $\unicode[STIX]{x1D701}$ estimates from the present data are given in figure 13(b). These data show a region of local minima approaching a constant, $1/\unicode[STIX]{x1D705}$ , indicating logarithmic dependence. Although the mean velocity distribution in the buffer layer appears logarithmic in wall coordinates, the indicator function reveals that it is clearly not logarithmic. This test further indicates that for non-zero $DR$ , $\unicode[STIX]{x1D705}$ exhibits convincing deviations from the Newtonian flow value, and that there is no dip in $\unicode[STIX]{x1D701}$ below $1/\unicode[STIX]{x1D705}$ for $DR\gtrsim 10\,\%$ . This second feature is due to the manner by which $\text{d}U^{+}/\text{d}y^{+}$ decays with distance from the wall, as exemplified by where the profiles in figure 15 take on their outer region approach towards zero. Note that with increasing $DR$ this occurs increasingly farther from the wall than in the Newtonian case. Consistently, the log-layer (as defined) is also farther from the wall, and at $DR=26\,\%$ there is scant evidence of an inertially dominated region. Similar behaviours have also been observed in the HDR regime ( $DR>40\,\%$ ) (Warholic et al. Reference Warholic, Massah and Hanratty1999; White et al. Reference White, Dubief and Klewicki2018), but, to the authors’ knowledge, have not been previously reported relative to the LDR values studied herein. Overall, the present mean profile behaviours support the description given in White et al. (Reference White, Dubief and Klewicki2018).

Figure 13.

Curve fits for the mean velocity used to estimate $\unicode[STIX]{x1D705}$ (a) and the associated indicator function $\unicode[STIX]{x1D701}$ (b). Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines are $DR>0$ ; dotted lines are the curve fits over $1.5y_{m}\lesssim y\lesssim 0.2\unicode[STIX]{x1D6FF}$ ; the thick line denotes the slope of $1/\unicode[STIX]{x1D705}=1/0.4$ .

Figure 14.

Normalized slope of the log region, where $\unicode[STIX]{x1D705}^{\ast }=\unicode[STIX]{x1D705}_{p}/\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}=0.4$ . The $\circ$ symbols are the experimental data for $\unicode[STIX]{x1D702}=0.2$ ; solid lines are linear curve fits for $\unicode[STIX]{x1D702}=0.15,0.2,0.25,0.3,0.4,0.5,0.6$ and $1.5y_{m}^{+}$ ; the dashed lines denote the $95\,\%$ confidence interval for $\unicode[STIX]{x1D702}=0.2$ .

Figure 15.

Mean velocity gradient profiles. Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines denote $DR>0$ cases; dash–dot line, DNS at $\unicode[STIX]{x1D6FF}^{+}=2004$ .

Indeed, in the direct comparisons now described, the data indicate continuous variations in $\unicode[STIX]{x1D705}$ that are in excess of the $95\,\%$ uncertainty bound for $DR$ values greater than approximately $15\,\%$ . Furthermore, based upon the data trend, this variation would seem to begin for $DR\lesssim 15\,\%$ . The relevant results are depicted in figure 14, where the end point of the logarithmic region used to determine $\unicode[STIX]{x1D705}$ via a linear curve fit is varied from $\unicode[STIX]{x1D702}=0.15,0.2,0.25,0.3,0.4,0.5,0.6$ (see figure 13(a) for an example of the curve fit). The start point for the curve fit is $1.5\times$ the peak location in the combined Reynolds and polymer stress. This moves the start point to within layer $IV$ (i.e. well within the inertial layer) and ensures that the curve-fit region is not biased by the transition regime. A conservative minimum end point of $\unicode[STIX]{x1D702}=0.15$ was used by Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013), while Pope (Reference Pope2000) reported $\unicode[STIX]{x1D702}=0.3$ . These two values are the lower and upper bounds reported in the literature for the end point of the logarithmic region. Note that using values of $\unicode[STIX]{x1D702}>0.3$ violates the notion that the log-layer is an inertial sublayer, so the results presented in figure 14 for $\unicode[STIX]{x1D702}>0.3$ should be interpreted with caution, as the end point is likely beyond the logarithmic region. Here it is shown that the slope of the log-layer profile normalized by the Newtonian value ( $\unicode[STIX]{x1D705}=0.4$ ) decreases approximately linearly with increasing $DR$ . It is clear that there is an effect on the slope that is dependent upon the end point of the log-layer, but all of the fits lie within the $95\,\%$ confidence interval for an end point of $\unicode[STIX]{x1D702}=0.2$ and start point of $1.5y_{m}^{+}$ . As just noted, it is also evident that a change in slope is nearly a statistical surety for $DR\gtrsim 15\,\%$ , and that these changes are much larger than those one might attribute to variations in Reynolds number. A curve fit of the data in figure 13 for $\unicode[STIX]{x1D702}=0.2$ is given by

(3.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}^{\ast }=\frac{\unicode[STIX]{x1D705}_{p}}{\unicode[STIX]{x1D705}}=-0.0086(\%DR)+1. & & \displaystyle\end{eqnarray}$$

Note that under the condition of $DR=0$ , one would expect the data to reflect the Newtonian flow value, and thus an additive coefficient of $1.0$ . This condition is enforced for (3.5). If a floating parameter fit is performed, the data fit is within approximately $3\,\%$ of this value.

3.5 Mean velocity gradient

The mean velocity gradient ( $\text{d}U^{+}/\text{d}y^{+}$ ) was estimated using a second-order forward finite difference scheme. As is apparent, the $DR=0$ profile of figure 15 exhibits very good agreement with the DNS data for $y^{+}\gtrsim 20$ . As expected, the profiles for $DR>0$ indicate a redistribution of mean vorticity, $\text{d}U^{+}/\text{d}y^{+}=-\unicode[STIX]{x1D6FA}_{z}$ . Here it is relevant to note that the displacement thickness is equivalently given by

(3.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{\ast }=\frac{\displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}}y\unicode[STIX]{x1D6FA}_{z}\,\text{d}y}{\displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}}\unicode[STIX]{x1D6FA}_{z}\,\text{d}y}, & & \displaystyle\end{eqnarray}$$

indicating that $\unicode[STIX]{x1D6FF}^{\ast }$ is the centroid of the mean vorticity distribution. The trend in the profiles of figure 15 is seen to be consistent with the data of figure 10 by recognizing that the integrand contributions to (3.3) over a range of relatively large $y$ , or equivalently, the large $y$ contributions to the numerator in (3.6), are reduced under the influence of the polymer. (Also see the changes to the mean profile shape in figure 9.)

3.6 Streamwise velocity RMS profiles

The present inner-normalized streamwise velocity r.m.s. profiles are shown in figures 16 and 17. With increasing $DR$ , the peak r.m.s. $u^{\prime }$ values increase, as does the $y^{+}$ position, $y_{p}^{+}$ , of this peak. Consistent with the previous results of Warholic et al. (Reference Warholic, Massah and Hanratty1999), this trend reinforces their physical interpretation that increasing the Reynolds number increases the capacity of the polymer at a fixed injection flow rate/concentration to reduce skin friction. This trend is illustrated in figure 17. Here it is noted that an increase in $\sqrt{\overline{u^{\prime 2+}}}$ is primarily attributable to a decrease in $u_{\unicode[STIX]{x1D70F}}$ , with $\sqrt{\overline{u^{\prime 2}}}$ remaining nearly constant (White & Mungal Reference White and Mungal2008). Thus, the reduction in skin friction could either be associated with a decrease in the intensity of $\sqrt{\overline{v^{\prime 2+}}}$ , or by an off-setting decrease in the correlating interaction between $u^{\prime }$ and $v^{\prime }$ . A decrease in $v^{\prime }$ is consistent with the existing literature (Wei & Willmarth Reference Wei and Willmarth1992; Warholic et al. Reference Warholic, Massah and Hanratty1999), and with the notion that polymers modify the momentum mixing (homogenization) mechanisms associated with the near-wall vorticity field (White et al. Reference White, Dubief and Klewicki2018). The physics of the associated energetics are discussed extensively by Dubief et al. (Reference Dubief, White, Terrapon, Shaqfeh, Moin and Lele2004). Their physical premise is that the polymer extracts energy from the buffer-layer vortices and releases it in the near-wall streaks, leading to enhanced $u^{\prime }$ and reduced  $v^{\prime }$ .

Figure 16.

Streamwise velocity r.m.s. profiles. The solid line denotes the $DR=0$ case at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines denote $DR>0$ cases.

Figure 17.

Streamwise velocity r.m.s. profiles for (a) $C_{i}=500$ wppm and (b) $C_{i}=1000$ wppm. Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 650,975,1800$ ; dashed lines denote $DR>0$ cases.

The present data allow the rate at which $y_{p}^{+}$ increases to be quantified as a function of $DR$ . The position of these peaks is plotted in figure 18(a). According to these data, the outward movement of the peak is nearly linear with $DR$ , as exemplified by the curve fit

(3.7) $$\begin{eqnarray}\displaystyle \sqrt{\overline{u^{\prime 2+}}}|_{y_{p}^{+}}=1.165(\%DR)+14.1. & & \displaystyle\end{eqnarray}$$

In figure 18(b), the polymer flow peak r.m.s. $u^{\prime }$ data are normalized by the peak value in the Newtonian flow, with the normalization denoted by an asterisk. A curve fit of the data yields

(3.8) $$\begin{eqnarray}\displaystyle \sqrt{\overline{u^{\prime 2+}}}^{\ast }=0.011(\%DR)+1.0. & & \displaystyle\end{eqnarray}$$

To the authors’ knowledge the quantifications of figure 18 are the clearest elucidation of these trends to date.

Figure 18.

Position (a) and peak value (b) of r.m.s. $u^{\prime }$ normalized by the corresponding $DR=0$ value. Solid lines are the curve fits given by (3.7) and (3.8). The dashed lines indicate a $95\,\%$ confidence interval. The experimental data are given by  $\circ$ .

3.7 Combined Reynolds and polymer stress profiles

Reynolds and polymer stress $(RPS)$ profiles were obtained from the measured mean velocity gradient profiles, and the integrated mean momentum equations (1.4) and (1.5), respectively. These stress profiles are presented in figure 19. Under the influence of polymer, the inner-normalized peak in the combined stress decreases and moves outwards in viscous units with increasing $DRu^{\prime 2+}$ . This is consistent with the previous observation indicating that while $\sqrt{\overline{u^{\prime 2+}}}$ increases, $\sqrt{\overline{v^{\prime 2+}}}$ decreases (Warholic et al. Reference Warholic, Massah and Hanratty1999). Here it is useful to reiterate, however, that the polymer stress cannot be independently determined from the present experiments. After close inspection, it is also apparent from figure 20 that the capacity of the injected polymer to reduce skin friction is Reynolds number dependent. This is exemplified by the $RPS$ profile at $\unicode[STIX]{x1D6FF}^{+}\simeq 650$ , which shows minimal change even though the injected polymer concentration doubled. Upon increasing $\unicode[STIX]{x1D6FF}^{+}$ to $975$ , however, the $RPS$ shows an increasing deviation with increasing concentration. It is also clear from figure 20 that the modification of the combined stress profiles occurs in the near-wall region with a much smaller (essentially immeasurable) effect on the outer region when compared to the Newtonian flows at similar $\unicode[STIX]{x1D6FF}^{+}$ . This behaviour is consistent with the polymer modifying the vorticity stretching and reorientation processes that underlie the three-dimensionalization of the near-wall vorticity field (for example, White et al. Reference White, Dubief and Klewicki2018).

Figure 19.

Reynolds plus polymer stress profiles. Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines denote $DR>0$ cases.

Figure 20.

Reynolds and polymer stress for two fixed injection concentrations, (a) $C_{i}=500$ wppm, and (b) $C_{i}=1000$ wppm. Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 650,975,$ and $1800$ ; dashed lines denote $DR>0$ cases.

Figure 21.

Normalized maximum position (a) and value (b) of the $RPS$ versus $DR$ . Newtonian baseline $\unicode[STIX]{x1D6FF}^{+}$ , $\circ \,\approxeq \,\unicode[STIX]{x1D6FF}_{1800}^{+}$ , $\diamond \,\approxeq \,\unicode[STIX]{x1D6FF}_{975}^{+}$ , $\triangle \,\approxeq \,\unicode[STIX]{x1D6FF}_{650}^{+}$ ; solid line, curve fit; dashed line, $95\,\%$ confidence interval.

The $y^{+}$ location of the maximum $RPS$ value, as normalized by its experimental Newtonian value, moves outwards approximately linearly with $DR$ , as shown in figure 21(a). The curve fit on this figure is given by

(3.9) $$\begin{eqnarray}\displaystyle RPS^{+}|_{y_{m}^{+}}^{\ast }=(-\overline{u^{\prime }v^{\prime }}^{+}+\overline{\unicode[STIX]{x1D70F}_{xy}^{p}}^{+})|_{y_{m}^{+}}^{\ast }=0.0174(\%DR)+1. & & \displaystyle\end{eqnarray}$$

Similarly, the normalized maximum value decays approximately linearly, as shown in figure 21(b), and decays according to

(3.10) $$\begin{eqnarray}\displaystyle RPS^{+\ast }(y^{+}=y_{m}^{+})=-0.00176(\%DR)+1. & & \displaystyle\end{eqnarray}$$

The trend of figure 21(a) directly correlates with the outward movement (in $y^{+}$ ) of the inertial domain with increasing $DR$ . This is because the peak in the $RPS$ coincides with an exchange in leading-order balance that culminates with the $VF$ term becoming subdominant. This and other effects on the structure of the mean dynamical balance are now investigated in greater detail.

3.8 Analyses based on the MMB

The above data presentation largely employed the traditional means by which to represent and interpret the statistical profiles of wall turbulence. As demonstrated by comparing smooth- and rough-wall flows (Mehdi et al. Reference Mehdi, Klewicki and White2013; Ebner et al. Reference Ebner, Mehdi and Klewicki2016; Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016), and most recently for polymer flow (White et al. Reference White, Dubief and Klewicki2018), data representations and interpretations in the context of the MMB provide new insights into the underlying dynamics. Accordingly, the present section first reveals and characterizes the modifications to the four-layer structure of the MMB under the influence of polymer, and then self-consistently uses this structure to revisit the behaviours of some of the statistical profiles just examined. In the following sections, data presentation focuses on the higher- $\unicode[STIX]{x1D6FF}^{+}$ data set. These data exhibit clear trends in the MMB structure.

The experimental version of figure 1 is presented in figure 22. Per (1.3), generating the profiles in figure 22 requires taking the ratio of two quantities that depend on the second $y$ derivative of the experimentally determined mean velocity profile. Assessment of the veracity of these profiles is reflected in the very good agreement between the measured ratio profiles for Newtonian flow, as compared to the DNS profiles at similar Reynolds number.

Figure 22.

Profiles of $VF/(TI+GPS)$ . Solid line, $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines denote $DR>0$ cases; dash–dot line, DNS at $\unicode[STIX]{x1D6FF}^{+}=2004$ .

For Newtonian flows, a subjectively yet self-consistently defined end point criterion for layer $II$ was incorporated in the analysis of Wei et al. (Reference Wei, Fife, Klewicki and McMurtry2005). Namely, this is where the ratio of $VF/TI=-2$ . Similarly, the end of layer $III$ is where $VF/TI=0.5$ . Using these criteria, the effect of the polymer on the layer bounds can be determined by finding where $VF/(TI+GPS)=-2$ and $0.5$ . For Newtonian flows, the $y^{+}$ end points of layers $II$ and $III$ are respectively well represented by $1.6\sqrt{(\unicode[STIX]{x1D6FF}^{+})}$ and $2.6\sqrt{(\unicode[STIX]{x1D6FF}^{+})}$ (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005; Klewicki Reference Klewicki2013b ). A curve fit of the present Newtonian data for layer $II$ yields $1.53(\unicode[STIX]{x1D6FF}^{+})^{0.49}$ and for layer $III$ yields $5.9(\unicode[STIX]{x1D6FF}^{+})^{0.38}$ . The deviation in the experimental result for layer $III$ is likely influenced by the uncertainty in our ability to precisely find where the profiles cross $0.5$ . For polymer-influenced flows, it is clear that the mean viscous force retains leading order farther from the wall, and accordingly the end point of layer $II$ moves to greater $y^{+}$ . Consistently, it is similarly apparent from figure 22 that the width and end point of layer $III$ also increases. Further evidence that the width of layer $III$ increases is given by (3.11) and (3.12). This is also due to the viscous force retaining significance to greater distances from the wall.

Equations (3.11) and (3.12) are curve-fit functions of $\unicode[STIX]{x1D6FF}^{+}$ and $DR$ . These describe additive corrections to the end points of layers $II$ and $III$ owing to the dynamical influence of the polymer and are presented in figure 23. The scaling for the end point of layer $II$ based upon viscoelastic DNS data is estimated as ${\sim}2.1\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ . A comparison between experimental data and the scaling for layer $II$ are in excellent agreement. For example, at $\unicode[STIX]{x1D6FF}^{+}=1463$ , equation (3.11) predicts the end point of layer $II$ to occur at $y^{+}=79$ , whereas the viscoelastic scaling predicts $y^{+}=81$ . Note that the correction for layer $III$ has increased uncertainty due to uncertainty in estimating the very small second derivative of $U^{+}$ at large $y^{+}$ . This effect is clearly reflected in the width of the associated confidence interval shown in figure 23(b). Despite this increased uncertainty, the trend indicated by (3.12) is in accord with recent estimates from lower $\unicode[STIX]{x1D6FF}^{+}$ DNS. Specifically, at $\unicode[STIX]{x1D6FF}^{+}=1463$ (3.12) prescribes a layer $III$ end point near $y^{+}=150$ . Consistently, White et al. (Reference White, Dubief and Klewicki2018) used viscoelastic DNS to estimate that layer $III$ terminates at $y^{+}=4\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ , which for $\unicode[STIX]{x1D6FF}^{+}=1463$ gives $y^{+}=153$ . Note that even though the corrections agree with viscoelastic DNS data, they are clearly not perfect since these results implicitly assume that the percentage drag reduction and Reynolds number effects are decoupled.

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle II_{end}^{+}=1.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}+1.796(\%DR)^{0.703} & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle III_{end}^{+}=2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}+13.2(\%DR)^{0.415}. & \displaystyle\end{eqnarray}$$

Figure 23.

Corrections to the Newtonian values for the ends of layer $II$ (a) and $III$ (b) as a function of increasing $DR$ at $\unicode[STIX]{x1D6FF}^{+}=1800$ , 1691, 1687, 1652, 1592, 1507, 1463. The data are given by $\circ$ ; the solid lines denote the curve fit; the dashed lines denote a $95\,\%$ confidence interval.

With the determination of the layer boundaries, one can then estimate the velocity increments across each layer. Beyond constituting a basic property of the mean dynamical structure, this information has theoretical value relative to, for example, educing the existence of a logarithmic versus power-law mean profile (Wei et al. Reference Wei, Fife, Klewicki and McMurtry2005). Newtonian flow analysis reveals that the mean velocity increments across layers $I$ and $III$ scale with $u_{\unicode[STIX]{x1D70F}}$ . Conversely, the increment across layer $II$ is very close to $0.5U_{cl}$ independent of $\unicode[STIX]{x1D6FF}^{+}$ , while the increment across layer $IV$ approaches $0.5U_{cl}$ as $\unicode[STIX]{x1D6FF}^{+}\rightarrow \infty$ . Table 2 provides velocity increment data from the present experiments. If the increment across layer $I$ is taken to be approximately $3u_{\unicode[STIX]{x1D70F}}$ , then consistent with previous Newtonian flow analysis, the increment across layer $II$ in the Newtonian flow is very close to $0.5U_{cl}$ . In agreement with previous quantifications, the measured increment across layer $III$ in the $DR=0$ case is approximately $1.2u_{\unicode[STIX]{x1D70F}}$ . On the other hand, it is apparent that with increasing $DR$ the velocity increment from the wall to the outer edges of layer $III$ steadily increases. (Interestingly the increment across layer $II$ remains nominally fixed.) Nevertheless, the overall observation is that the mean viscous force retains leading order increasingly farther from the wall with increasing drag reduction. Dynamically, this indicates that, on average, more of the total velocity increment (circulation per unit length) is attributable to the region where the $TI$ term in (1.3) behaves like a momentum source. Consistent with this, previous measurements of the layer velocity increments in rough-wall flows (Mehdi et al. Reference Mehdi, Klewicki and White2013) indicate that the increment across the momentum source region decreases with increasing roughness drag. Thus, as a general momentum field diagnostic it would seem that increases in the velocity increment from the wall to the start of layer $IV$ , relative to the smooth-wall case, are associated with drag reductions, while decreases in this increment are associated with increasing drag. This has apparent similarity with previous characterizations of drag-reducing flows as behaving like a negative roughness (Hoyt Reference Hoyt1991). Lastly, the relative concentration of velocity increment in layer $II$ with increasing $DR$ consistently correlates with the aforementioned reduction in $\unicode[STIX]{x1D6FF}^{\ast }$ , and apparently supports the notion (analogy) that polymer $DR$ promotes a flow structure resembling that found in reverse transition. Further evidence in support of this is that the low-Reynolds-number onset of the four-layer regime correlates with $\unicode[STIX]{x1D6FF}^{\ast }$ first exceeding the width of layer $II$ (Klewicki Reference Klewicki2013b ).

Table 2.

Velocity increments for layers $III$ and $IV$ associated with the layer structure of figure 1 at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ and for $DR>0$ at $\unicode[STIX]{x1D6FF}^{+}=1691,1687,1652,1592,1507,1463$ . The velocity at the end points of layers $II$ and $III$ is also presented.

3.9 Profiles of the momentum transport length, $W$

The mean momentum equations for the canonical turbulent wall flows, including (1.2), admit a self-similar structure on two domains (Klewicki Reference Klewicki2013b ). The mathematically simpler of these two is operative on an interior inertial domain. According to the leading-order structure of (1.2) this inertial domain begins near the onset of layer $IV$ . Here, analytical treatment of (1.2) reveals that the average size of the momentum transporting motions varies linearly with distance from the wall, $y$ (Fife, Klewicki & Wei Reference Fife, Klewicki and Wei2009). On this domain, direct integration of (1.2) yields a similarity solution that asymptotically conforms with the universal logarithmic mean velocity profile (Klewicki & Oberlack Reference Klewicki and Oberlack2015). In accord with the existence of this similarity solution to (1.2), there is a coordinate stretching function that ensures solution invariance for varying $\unicode[STIX]{x1D6FF}^{+}$ . This coordinate stretching function is found analytically, denoted by $\unicode[STIX]{x1D719}$ , and defined by

(3.13) $$\begin{eqnarray}\displaystyle \frac{\text{d}W^{+}}{\text{d}y^{+}}=\unicode[STIX]{x1D719}^{-1}. & & \displaystyle\end{eqnarray}$$

In this expression, $W^{+}(y^{+})$ is the continuous distribution of the widths of a self-similar hierarchy of scaling layers termed the $L_{\unicode[STIX]{x1D6FD}}$ hierarchy (for example, Klewicki Reference Klewicki2013b ). Formally, $W^{+}(y^{+})$ is related to the decay rate of the $TI$ term in (1.2). Practically, however, $W^{+}$ is most accurately computed using $W^{+}=-(\text{d}^{2}U^{+}/\text{d}y^{+2})^{-1/2}$ . Physically, at any location on the $L_{\unicode[STIX]{x1D6FD}}$ hierarchy $W^{+}$ is the mean length scale of the inertial turbulent motions responsible for the wallward transport of momentum (Klewicki et al. Reference Klewicki, Philip, Marusic, Chauhan and Morrill-Winter2014). Conceptually, $W^{+}$ is similar to von Kármán’s representation of Prandtl’s mixing length. His formulation, however, stems from an intuitive similarity hypothesis, while the properties of $W(y)$ are determined from the analysis that determines an invariant representation of the mean dynamical equation that holds on each of the hierarchy layers. Specifically, normalization of the mean momentum equation using $u_{\unicode[STIX]{x1D70F}}$ and $W$ formally yields a single universal form that holds over the domain where the $TI$ term is a decreasing function of $y$ , i.e. on the domain of the so-called $L_{\unicode[STIX]{x1D6FD}}$ hierarchy. The $L_{\unicode[STIX]{x1D6FD}}$ hierarchy domain extends from the inner peak of the $TI$ (denoted as $y_{pi}^{+}$ ) to the outer peak (denoted as $y_{po}$ ). In Newtonian flow this corresponds to $y^{+}=y_{pi}^{+}\approxeq 7$ to $y/\unicode[STIX]{x1D6FF}=y_{po}/\unicode[STIX]{x1D6FF}\approxeq 0.5$ , or equivalently between the peak momentum source, $y_{pi}^{+}$ , and sink, $y_{po}$ , locations (Klewicki, Ebner & Wu Reference Klewicki, Ebner and Wu2011). The portion of the layer hierarchy where the logarithmic similarity solution holds corresponds to the inner part of layer $IV$ , where the mean dynamics become inertially dominated and the $W^{+}$ profile becomes a linear function of $y^{+}$ .

Estimates of the $W^{+}$ profiles from the present measurements are displayed in figure 24. It is clear from figure 24 that with increasing $DR$ the mean length scale responsible for the transport of momentum is reduced for $y^{+}>20$ . Viscoelastic DNS data indicate that for $y^{+}<20$ , the mean length scale increases (White et al. Reference White, Dubief and Klewicki2018). This correlates with a reduction in the intensity of near-wall vorticity stretching. The present data also suggest an increase in $W^{+}$ for $y^{+}<20$ , but uncertainties in these estimates become significant as the wall is approached.

Figure 24.

Profiles of the $W^{+}$ layer width distributions as estimated from the present measurements. The solid line is for $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines, $DR>0$ ; dash–dot line, DNS at $\unicode[STIX]{x1D6FF}^{+}=2004$ .

When interpreted as a force, the turbulent inertia term in (1.3) behaves like a momentum source for $y^{+}<y_{m}^{+}$ , and like a momentum sink for $y^{+}>y_{m}^{+}$ (Klewicki et al. Reference Klewicki, Fife, Wei and McMurtry2007). (Recall that $y_{m}^{+}$ is the position where the Reynolds stress attains its maximum, or, equivalently, where $TI$ crosses zero.) Owing to

(3.14) $$\begin{eqnarray}\displaystyle \int _{0}^{\unicode[STIX]{x1D6FF}}\frac{\text{d}\overline{u^{\prime }v^{\prime }}^{+}}{\text{d}y^{+}}\,\text{d}y^{+}=0, & & \displaystyle\end{eqnarray}$$

the net momentum source and sink contributions sum to zero at any given $\unicode[STIX]{x1D6FF}^{+}$ . This integral balance occurs owing to a concentration of smaller, but more intense, source-like motions interior to $y_{m}^{+}$ , and larger scale, but less intense, sink-like motions beyond $y_{m}^{+}$ . In this regard, Newtonian flow analysis shows that when the $TI$ profiles are weighted (premultiplied) by $W^{+}$ and plotted on a logarithmic scale, the source and sink contributions to (3.14) appear equally distributed about $y_{m}^{+}$ (for example, Klewicki Reference Klewicki2013b ). It is thus useful to investigate this balancing behaviour for drag-reducing flows. The estimate for this profile is given in figure 25(a). Owing to the uncertainty in the present $W^{+}$ estimates, it is also useful to compare the weighted total stress gradient profile of figure 25(a) with that employing the pure $y$ -scaling surrogate, i.e. the weighted profile, $y^{+}\text{d}RPS^{+}/\text{d}y^{+}$ versus $y^{+}$ , as was previously employed by Klewicki et al. (Reference Klewicki, Ebner and Wu2011). This profile is shown in figure 25(b). While both of these weighted profiles fail to resolve the peak momentum source near the wall, it is clear that they both consistently indicate that the source contributions to the momentum distribution move farther from the wall in viscous units for $DR>0$ flow. In the source region, these profiles also show that the $DR>0$ profiles consistently reside above the Newtonian flow profiles. This indicates that the increase in the gradient of the combined polymer and Reynolds stress more than compensates for the observed reduction in the polymer influenced $W^{+}$ profiles of figure 24. Unfortunately, the lack of data in the near-wall region does not allow for a precise quantification of these source contributions. Similarly, the $W^{+}\text{d}RPS^{+}/\text{d}y^{+}$ profiles show large uncertainties for large $y^{+}$ ; notice that the profile of figure 25(a) is truncated to only show the inner part of layer $IV$ .

Figure 25.

Profiles of the weighted $W^{+}$ profiles: (a) $W^{+}\text{d}RPS^{+}/\text{d}y^{+}$ versus $y^{+}$ , (b) $y^{+}\text{d}RPS^{+}/\text{d}y^{+}$ versus $y^{+}$ . The solid line denotes the $DR=0$ profile at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines denote $DR>0$ cases.

3.10 Rescaling the mean vorticity profile

Further insights are also gained by re-examining the mean vorticity profile within the context of the normalizations predicted by the MMB theory. Specifically, comparison with the $DR=0$ channel flow allows for an assessment of the effects of polymer on the establishment and scaling behaviours of the mean vorticity distribution. Here we reiterate that the present channel flow has only one mean vorticity component. This component is parallel to the wall and perpendicular to the flow direction, $\unicode[STIX]{x1D6FA}_{z}=-\text{d}U^{+}/\text{d}y^{+}$ , and thus its wall value is directly related to the mean wall shear stress.

Figure 26.

Absolute values of the mean vorticity profiles scaled according to the MMB theory by using the inner-normalized peak location, $y_{m}^{+}$ , of the $RPS^{+}$ as the characteristic scale. The solid line is for $DR=0$ at $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ ; dashed lines, $DR>0$ ; dash–dot line, DNS at $\unicode[STIX]{x1D6FF}^{+}=2004$ .

MMB-based analysis of the canonical smooth-wall Newtonian channel flow reveals that the limiting values of the mean vorticity profile can be made the same at any given $\unicode[STIX]{x1D6FF}^{+}$ by multiplying $|\unicode[STIX]{x1D6FA}_{z}^{+}|$ by $\unicode[STIX]{x1D716}^{-1}=\sqrt{(\unicode[STIX]{x1D6FF}^{+})}$ and multiplying $y^{+}$ by $\unicode[STIX]{x1D716}$ (Klewicki Reference Klewicki2013a ). This scaling is associated with the similarity solution for the mean velocity profile that is formally admitted by the mean momentum equation and that asymptotically underlies the classical logarithmic law (Klewicki & Oberlack Reference Klewicki and Oberlack2015). It exists on the inertial domain that was used in the present estimation of $\unicode[STIX]{x1D705}$ in figure 13. An important feature of this solution is that its $y$ -coordinate stretching effectively aligns the onset of the inertial domain for all $\unicode[STIX]{x1D6FF}^{+}$ . As discussed above relative to the layer velocity increments, at any finite $\unicode[STIX]{x1D6FF}^{+}$ slightly more than half of the mean vorticity is contained in the region $\unicode[STIX]{x1D716}y^{+}\lesssim 2.6$ , with the remainder being spread over a layer $IV$ domain that approaches $\unicode[STIX]{x1D6FF}$ in width as $\unicode[STIX]{x1D6FF}^{+}\rightarrow \infty$ (Klewicki Reference Klewicki2013a ). A key finding of the canonical flow analysis is that from the MMB one can analytically surmise that the inner-normalized widths of layers $II$ and $III$ , as well as $y_{m}^{+}$ , all scale like $\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ . Relative to non-canonical flows, Mehdi et al. (Reference Mehdi, Klewicki and White2013) were the first to hypothesize that when the four-layer structure depicted in figure 1 is qualitatively retained, the same scaling framework will remain operative. Here we note that figure 22 clearly reveals that this is the present case. Physically, this indicates that the flow structure retains the same characteristic balance exchange of mean forces across layer $III$ (leading to an inertially dominated layer $IV$ ), as well as the associated change from a momentum source to momentum sink-like behaviour of the non-viscous stress gradients across layer $III$ . Under such cases, however, the characteristic scale (analogous to $\unicode[STIX]{x1D716}$ ) must generally be determined empirically. In this regard, $y_{m}^{+}$ provides a convenient and physically equivalent surrogate scale, and rough-wall experiments provide clear evidence that $y_{m}^{+}$ effectively recovers the invariant scaling for $|\unicode[STIX]{x1D6FA}_{z}|$ described above (Mehdi et al. Reference Mehdi, Klewicki and White2013; Ebner et al. Reference Ebner, Mehdi and Klewicki2016; Squire et al. Reference Squire, Morrill-Winter, Hutchins, Schultz, Klewicki and Marusic2016).

The scaling recipe just described is applied to the present polymer data in figure 26. In this figure the absolute value of the inner-normalized mean vorticity is weighted by $y_{m}^{+}$ , the inner-normalized position of the peak in the $RPS$ , and similarly $y^{+}$ is divided by $y_{m}^{+}$ . As is apparent, interior to $y^{+}/y_{m}^{+}\simeq 3$ , the polymer-influenced profiles clearly diverge from those for Newtonian flow. Specifically, with increasing $DR$ the profiles in this region consistently shift upwards, and then merge with or cross over the Newtonian profile closer to the wall. A similar trend occurs for decreasing $\unicode[STIX]{x1D6FF}^{+}$ in Newtonian flow. This is, therefore, interpreted to provide further support that polymer drag reduction has an effect that, in a number of respects, is similar to that of reducing the Reynolds number (White et al. Reference White, Dubief and Klewicki2018). Beyond $y^{+}/y_{m}^{+}\simeq 3$ , the $DR=0$ and all of the $DR>0$ profiles in figure 26 are nearly indistinguishable. Note, however, that the Newtonian and lower $DR$ profiles tend towards coalescence along a straight line ( $-1$ power law) progressively closer to the wall with decreasing $DR$ . The $y^{+}/y_{m}^{+}\simeq 3$ position approximately denotes the outer edge of layer $III$ for the highest $DR$ case. Thus, per the MMB theory, this corresponds to the onset of the inertial sublayer for this flow. Overall, the results of figure 26 are taken to provide compelling support for the present MMB-based theoretical description and the concomitant scaling prescriptions for the mean vorticity profile. They also corroborate the present means for identifying the inertial layer bounds used to estimate the value of $\unicode[STIX]{x1D705}$ . Lastly, it is relevant to note that for the flows indicated in figure 26 the position $y^{+}/y_{m}^{+}\simeq 3$ is also quite close to $y^{+}=4\sqrt{\unicode[STIX]{x1D6FF}^{+}}$ , and thus these results are also consistent with the recent estimate of White et al. (Reference White, Dubief and Klewicki2018) for the inertial layer onset for $DR>0$ .

3.11 Diagnostic plot

As described in § 3.9, the MMB can be transformed into a single parameter-free form that is valid across a continuous hierarchy of scaling layers. The width distribution of these layers, $W^{+}(y^{+})$ , is an increasing function of wall-normal distance that in Newtonian flow spans $4\lesssim W^{+}\lesssim \unicode[STIX]{x1D6FF}^{+}/3$ over $7\lesssim y^{+}\lesssim \unicode[STIX]{x1D6FF}^{+}/2$ . For the part of the layer hierarchy on the inertial domain ( $2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}\lesssim y^{+}\lesssim \unicode[STIX]{x1D6FF}^{+}/2$ ) the self-similar structure of the MMB is characterized by $W^{+}\propto y^{+}$ ; a condition that becomes exact as $\unicode[STIX]{x1D6FF}^{+}\rightarrow \infty$ . A logarithmic mean velocity profile stems from this self-similar structure. Zhou & Klewicki (Reference Zhou and Klewicki2015) investigated the correspondences between the self-similar inertial domain just described, and that found for other measures of self-similar structure in the canonical boundary layer. Notably, to within the uncertainty of their data they found an exact correspondence between $2.6\sqrt{\unicode[STIX]{x1D6FF}^{+}}\lesssim y^{+}\lesssim \unicode[STIX]{x1D6FF}^{+}/2$ and the domain over which the so-called diagnostic plot of Alfredsson, Orlu & Segalini (Reference Alfredsson, Orlu and Segalini2012) exhibits its (approximately) linear decrease.

Figure 27.

Versions of the diagnostic plot. Streamwise r.m.s. velocity normalized with $U(y)$ versus $U(y)$ normalized by (a) $U_{cl}$ and (b) $U$ velocity at the end of layer $III$ , $U_{III}$ . See figure 26 and caption for line definitions. Symbols in (a) denote position of the onset of layer $IV$ for each profile, and the vertical line denotes the value of $U/U_{cl}$ at the end of layer $III$ in the DNS $DR=0$ flow at $\unicode[STIX]{x1D6FF}^{+}=2004$ .

Originally developed as a means to identify data anomalies in the near-wall profile of $\sqrt{\overline{u^{\prime 2}}}$ , the diagnostic plot represents the r.m.s. $u^{\prime }$ normalized by the corresponding $U$ value versus $U/U_{cl}$ (Alfredsson et al. Reference Alfredsson, Orlu and Segalini2012). In this regard, $U/U_{cl}$ acts as a flow structure based surrogate for the normalized distance from the wall. This representation of the diagnostic plot is given in figure 27(a). Referencing the Newtonian fluid flow cases ( $\unicode[STIX]{x1D6FF}^{+}\approxeq 1800$ experiment, $\unicode[STIX]{x1D6FF}^{+}=2004$ DNS), it is apparent that these profiles complete their distinct change in slope near $U/U_{cl}=0.68$ at the present $\unicode[STIX]{x1D6FF}^{+}$ . As observed by Zhou & Klewicki (Reference Zhou and Klewicki2015) in boundary layers over $1450\leqslant \unicode[STIX]{x1D6FF}^{+}\leqslant 19\,600$ this knee in the curve slowly moves to lower $U/U_{cl}$ with increasing $\unicode[STIX]{x1D6FF}^{+}$ , and the onset of the Reynolds-number-invariant (approximately linear) region closely corresponds to the start of the inertial domain found from consideration of the MMB. For the DNS, this position is denoted by the solid vertical line in figure 27(a), and by the black diamond symbol for the $\unicode[STIX]{x1D6FF}^{+}\simeq 1800$ experiment. Similarly, the coloured symbols denote the onset of layer $IV$ for each of the $DR>0$ cases. For increasing $DR$ it is apparent that these marked positions successively coincide with where the $DR>0$ profiles nominally merge with the $DR=0$ profiles. In contrast to the present drag-reduced flow, the roughness study by Castro, Segalini & Alfredsson (Reference Castro, Segalini and Alfredsson2013) indicates that roughness shifts the profiles above the Newtonian case, whereas the shift is below for $DR>0$ ( $U/U_{cl}<0.5$ ), as shown in figure 27(a). Similarly, 27(b) uses the normalization $U/U_{III}$ (see table 2) to illustrate that all of the curves do indeed begin their shallower slope near $U/U_{III}=1$ . These profiles also clearly reveal that, relative to the invariant $DR=0$ profile slope, the $DR>0$ profiles exhibit a continuous variation with increasing $DR$ . Overall, the results of figure 27 provide clear evidence that, with increasing $DR$ , the inertial sublayer continuously moves outwards in $y^{+}$ , its width diminishes, and the invariant $DR=0$ relationship between $\sqrt{\overline{u^{\prime 2}}}$ and $U$ on the inertial sublayer continuously changes. Note, however, that this analysis relies on the empirical observation that the scaling domain for the diagnostic plot coincides with the bounds of the inertial domain as determined from the MMB theory. Thus it is a correlative finding, as opposed to an analytical result.