## Abstract

This study reexamines past studies of how drag-reducing polymer solutions modify the log-region of a developing turbulent boundary layer (TBL). The classical view was that the polymers modify the intercept constant without impacting the von Kármán coefficient, which results in the log-region being unaltered though shifted outward from the wall. However, recent work has shown this to be not accurate, especially at high drag reduction (HDR) (>40%). While the deviations to the von Kármán coefficient were conjectured to be related to polymeric properties, this had not been explored. This work examines the scatter in both log-region parameters and estimates the local polymeric properties. This shows that the scatter of the von Kármán coefficient between studies is related to the inner variable based Weissenberg number. In addition, recent polymer ocean results are included that support the implicit assumption in past studies that the maximum wall concentration should be used to define the local polymeric properties.

## 1 Introduction

The ability of a polymer solution to reduce the drag within a turbulent flow is dependent on the polymer type (e.g., molecular structure), solvent, molecular weight, and concentration. Within a turbulent boundary layer (TBL), the local mean molecular weight and polymer concentration are dependent on polymer degradation and polymer diffusion, respectively. Given a local polymer concentration and mean molecular weight, the velocity profile is modified from the Newtonian profile via polymeric stresses redistributing the momentum in the near-wall region, which results in drag reduction (DR). Consequently, examination of the near-wall velocity profile modifications relative to the Newtonian condition is primarily dependent on the drag reduction. Until recently, it was assumed that the level of drag reduction exclusively determined the modifications and, furthermore, that the log-region of the velocity profile remained unaltered [1]. However, recent computational [2] and experimental [3] work have shown that particularly at high drag reduction (HDR; >40%), the modifications are influenced by flow and polymer properties. Furthermore, Elbing et al. [3] provided high Reynolds number evidence that showed at HDR, the Reynolds number was unable to collapse available data. Since that work focused on a single flow configuration (TBL), an inability to fully collapse the data indicated that polymeric properties (e.g., relaxation time, viscosity ratio, ratio of coiled to stretched length ratio) must also be considered in examining modifications to the near-wall velocity profile in the HDR regime. This conclusion has since been expanded to even the low drag reduction (LDR; <40%) regime [4]. It has also been shown that local TBL shear forces are sufficient to break the molecular bonds of polyethylene oxide (PEO) [5], which creates uncertainty about the past evaluations of the near-wall velocity profiles in TBLs modified with drag-reducing polymer. Consequently, this paper focuses on reexamining the large body of research on injection of polymer solution into a TBL to assess the key polymer properties (e.g., relaxation time, viscosity ratio, and ratio of coiled to stretched length ratio) responsible for deviations from the classical view.

Here, we note that the primary objective of the paper is not to identify the exact dependence between the mean velocity profile deviations and polymeric properties, but rather a test of whether the recent insights [25] can provide some clarity to established data in the literature. For this analysis, only experimental TBL data modified using PEO have been considered. It was limited to experimental data to avoid potential deviations associated with how polymers were modeled in computational studies. However, direct numerical simulation studies have been leveraged in the interpretation and discussion of the results. PEO was selected as the polymer to focus our efforts on because (i) PEO has been heavily used in TBL studies, (ii) there are established relationships for critical processes including flow induced chain scission [5], and (iii) avoid differences associated with polymer molecular structure. An exhaustive search for PEO modified TBL studies prior to these recent insights that included measurements of the near-wall velocity profile along with sufficient information to estimate polymeric properties produced six sources [3,610]. These sources are listed in Table 1 along with a recent TBL study [11] that used a PEO polymer ocean (homogeneous concentration) rather than polymer injection to mitigate uncertainty about polymeric properties. The remainder of the paper is organized as follows: Sec. 2 established PEO and flow relationships, Sec. 3 overview of results being considered, Sec. 4 analysis and discussion of results, and Sec. 5 conclusions are given.

Table 1

Experimental TBL PEO studies with near-wall velocity profiles and sufficient information to estimate local polymer properties

SourcePolymerMw × 10−6 (g/mol)C* (ppm)Symbol
Fontaine et al. [6]Injection5.0480
White et al. [7]Injection3.8590
Petrie et al. [8]Injection2.0, 4.0980, 570
Hou et al. [9]Injection4.0, 5.0570, 480
Somandepalli et al. [10]Injection4.0570
Elbing et al. [3,5]Injection4.0570о
Farsiani et al. [11]Ocean0.7–4.22200–550
SourcePolymerMw × 10−6 (g/mol)C* (ppm)Symbol
Fontaine et al. [6]Injection5.0480
White et al. [7]Injection3.8590
Petrie et al. [8]Injection2.0, 4.0980, 570
Hou et al. [9]Injection4.0, 5.0570, 480
Somandepalli et al. [10]Injection4.0570
Elbing et al. [3,5]Injection4.0570о
Farsiani et al. [11]Ocean0.7–4.22200–550

The mean molecular weights are at injection or are steady-state for oceans. Symbols used in subsequent figures are also listed.

## 2 Polymeric and Flow Relationships

### 2.1 Molecular Structure.

Polyethylene oxide has been used in the majority of TBL studies [3,614] because it is a very efficient polymer solution, being able to achieve drag reduction in excess of 70% with concentrations ∼10 ppm (if mean molecular weight is sufficiently high). The PEO structural unit (monomer) is (–O–CH2–CH2–), which has a monomer molar mass (Mo) of 44.1 g/mol. The monomer structure also creates a polymer backbone consisting of carbon–carbon (C–C) and carbon–oxygen (C–O) bonds. The C–C and C–O bond strengths are nominally 4.1 nN and 4.3 nN [15], respectively. This gives the PEO backbone an average bond strength of 4.2 nN, which is used subsequently to estimate the shear rate required to break the molecular bonds within the TBL. The linear structure of PEO results in the molecular weight being directly related to the length of the polymer chain. The mean molecular weights considered in this paper range from 0.7 × 106 to 5.0 × 106 g/mol (Table 1).

While the injected polymer mean molecular weights were known, the possibility of polymer degradation via chain scission requires an ability to determine the mean molecular weight as it evolves within the TBL. The details of estimating the amount of degradation given the flow conditions are discussed subsequently. Here is a brief synopsis of how the mean molecular weight was determined from samples extracted from the TBL [5], which was used to develop the scaling law used to determine the local mean molecular weights when degradation was likely present in other studies. The samples were passed through a pipe flow apparatus to measure the deviations from the traditional friction law within the polymeric regime. The intersection of these polymeric results with the Newtonian friction law defines the onset of drag reduction, which has a corresponding onset of drag reduction shear rate $(γ *)$. For PEO, Vanapalli et al. [16] compiled data from Virk [1] and found a relationship between the shear rate at the onset of drag reduction and the mean molecular weight (Mw)
$γ *=3.35×109Mw$
(1)

This relationship has been leveraged in several studies to estimate the local mean molecular weight [3,5,11,16,17]. The uncertainty of the molecular weight estimates from Ref. [5] was ∼8%.

### 2.2 Rheological Properties.

Insights from modeling drag reducing polymer solutions suggests that there are three critical nondimensional polymer dependent parameters [2]: (i) Weissenberg number Wi, (ii) μ* the ratio of the solvent viscosity μs to the zero-shear viscosity of the polymer solution μo, and (iii) the length ratio L of the fully extended to coiled polymer molecules. While these properties are not readily available for high molecular weight PEO, they can all be estimated from established relationships.

The Weissenberg number (Wi) is a ratio of the polymer relaxation time $(λ)$ to the turbulence time scale, which is typically quantified by the inverse of the shear rate. In the TBL literature, the shear rate is typically defined as either the mean shear or the wall shear with corresponding definitions of the Weissenberg number, $Wim=λU∞/δ99$ or $Wiw=λuτ/lν$, respectively. Here, $U∞$ is the local freestream speed, $δ99$ is the 99% boundary layer thickness, $uτ (=τw/ρ)$ is the friction velocity, $τw$ is the local skin-friction on the wall, ρ is the liquid density, $lν (=ν/uτ)$ is the viscous wall unit, and ν is the kinematic viscosity. Thus, the polymer relaxation has to be estimated to determine Wi, but it is extremely difficult to accurately measure it for high molecular weight PEO. Consequently, it is typically estimated based on either the Zimm (λZ) or Kalashnikov (λK) times, which are appropriate for low and high concentrations, respectively. The Zimm time [18]
$λZ=0.4220[η]oμsRTMw$
(2)
is a good approximation for the relaxation time when PEO is dilute, where R is the ideal gas constant (8.314 J/K mol), T is the absolute temperature, and $[η]o$ is the intrinsic viscosity. The intrinsic viscosity (in cm3/g) was estimated using the Mark–Houwink relationship [19]
$[η]o=0.01248 Mw0.78$
(3)
The polymer solutions are considered dilute when the concentration is below the overlap concentration, $C*=[η]0−1$. The overlap concentrations, corresponding to the range of molecular weights examined, are included in Table 1 for reference. Above this limit, the polymer chains can no longer be considered as independent particles and the solution is semidilute. In the semidilute range, the relaxation time becomes concentration dependent. Kalashnikov time [20]
$λK=[[η]o549.5−([η]o3255)3−0.51]e−(TC/50)2C0.5$
(4)

is used to estimate the relaxation time in the semidilute range, where Tc is the temperature in ° C and C is the polymer concentration. The uncertainty of the relaxation time increases significantly when the polymer is nondilute since it becomes dependent on another challenging TBL measurement (local concentration profiles), which is why in the subsequent analysis the results are limited to dilute conditions.

The viscosity ratio (μ*) requires the solvent viscosity (μs) and the zero-shear viscosity of the polymer solution (μo). Water was used as the solvent in all the studies considered, which allows μs to be determined directly from a measure of the water temperature (when not listed in the articles, the temperature was approximated as 20 °C). The zero-shear viscosity can be determined from rheometer data, which was available for several of the studies considered. When these measurements were not available, the relative amplitude of the viscosity $(Δ≡(μo−μ∞)/μ∞)$ was used. Noting that the high shear stress viscosity limit (μ) approaches the solvent viscosity (μs) for PEO solutions, the relative viscosity amplitude is readily related to the viscosity ratio, $μ*=1/(Δ +1)$. Kalashnikov [20] provides a relationship for computing Δ
$Δ≡μo−μ∞μ∞=[([η]o135.6)2+0.434[η]o−126]C$
(5)
The contour length (Lext) and the coiled length (Lcoil) are needed to calculate the length ratio (L) and can be estimated from mean molecular weight. The contour length is the maximum polymer chain extension and can be computed from the PEO molecular structure, $Lext=0.82lonoMw/Mo$ [21]. Here, no is the number of backbone bonds per monomer (PEO—three), lo is the mean backbone bond length (1.47 Å), and the constant (0.82) accounts for the polymer chain bond angles. The coiled length is approximated by analyzing a set of “random walks” or “conformations of a freely jointed chain.” Here, the end-to-end distance of the polymer coil is approximated as a Gaussian function dependent on the chain segment length (equal to the monomer length; 0.82lono) with the total number of segments (Mw/Mo). This analysis [21] shows that the coiled length on average can be approximated as $Lcoil=(0.82lono)(Mw/Mo)$. Consequently, the length ratio is nominally equal to (Mw/Mo)0.5.

Since high-Reynolds number TBLs can cause degradation via mechanical scission of the polymer chain [5,12], the mean molecular weight, and consequently the resulting polymer properties, can evolve within the TBL. Since recent findings [3] have shown that the polymer properties must be responsible, in part, for deviations from the classic view, the analysis requires an estimate of the local mean molecular weight. Fortunately, Elbing et al. [5] developed a scaling law for polymer degradation via chain scission for PEO within a TBL
$Mw−MwsMwo=exp {−8.6×10−5(x−xinj)γwU∞}$
(6)
Here, Mwo is the initial mean molecular weight, Mws is the steady-state mean molecular weight, x is the distance downstream from the leading edge, xinj is the downstream distance from the injection location, and $γw(=τw/ρν=uτ/lν)$ is the wall shear rate. This curve matched available data shown in Elbing et al. [5] within ±10%. The steady-state mean molecular weight can be determined from a universal scaling law [22] that determines the maximum force on a polymer chain. Scission occurs when that maximum exceeds the C–C or C–O bond strengths, and this analysis used the average bond strength of 4.2 nN. This universal scaling law [22] requires a bulk fluid-flow length scale, which should be zero at the injection location and evolve as xxinj approaches infinity to an appropriate diffusion length scale. This diffusion length scale was set at 0.64δ99, which is the half concentration length scale for a passive scalar diffusing from a line source within a TBL [23]. This produces an estimate of Mws evolving within the TBL
$MwsMo=0.49(x−xinjx)(δ99lo)Fmaxρν2(U∞δ99ν)−0.75$
(7)

where Fmax is the maximum force on the chain and it was set at the average bond strength (4.2 nN). Combining Eqs. (6) and (7) allows for an estimate of the local mean molecular weight. For the analysis, the polymer was assumed degraded once the local maximum Mw dropped below 95% of the initial Mw. Once it is considered degraded, the estimated local mean molecular weight was used to determine polymeric properties. Note that over half of the test conditions in the literature were likely impacted by degradation.

### 2.4 Polymer Diffusion.

Since most TBL studies have injected polymer solution into the developing boundary layer and many injection conditions exceed the overlap concentration, it is important to have an estimate of the local polymer concentration. For a few of the studies considered [3,6,10], concentration profiles were provided and used to determine the maximum concentration in the near-wall region. However, for velocity profiles without accompanying concentration profiles, the near-wall polymer concentration was estimated using established correlations as described here. In general, diffusion from a line source into a developing TBL can be divided into three zones; initial, intermediate, and final zones. The initial zone is characterized by a thin sheet of high concentrated solution at the wall with near-zero concentration throughout the remainder of the TBL [24]. Under ideal conditions, the maximum extent of this region is below 1 m and typically much shorter [14,25]. The established definition for the initial zone is the region where the maximum polymer concentration is greater than 0.37 (e−1) times the injection concentration [26]. The initial diffusion zone length (Lo) is estimated, assuming that $LoU∞/qinjCinj$ is a constant [27], where $qinj$ is the volumetric flux of polymer per unit span and $Cinj$ is the polymer concentration at the injection location. Past work [14] showed the constant is not universal, but it was the same order of magnitude over a wide range of operating conditions for a given initial mean molecular weight. Then, the local wall concentration (Cwall) was estimated from $Cwall/Cinj=e−(x−xinj)/Lo$. In the intermediate zone, the thin sheet from the initial zone is broken up and rapidly diffused through the TBL thickness. Vdovin and Smol'yakov [28] fit the wall concentration within the intermediate zone with the form of $Cwall/Cinj=e−α(x−xinj)/Lo+β$, which Fontaine et al. [6] found that α = 1.015 and β = −0.015. The intermediate zone extends until the concentration half-length (λC), which is defined as the wall-normal distance where the concentration drops to half Cwall, is λC = 0.64δ99 [23]. Beyond this point is the final zone where any further dilution is due to boundary layer growth. The wall-normal concentration profile in the final zone is well established as $C/Cwall=e−0.693(y/0.64δ99)2.15$ [14,23], where y is the wall-normal distance from the wall. This relationship combined with knowledge of the boundary layer growth and conservation of mass can be used to determine the local wall concentration. In general, the uncertainty of the concentration estimates decreases with increasing distance from the injection as the correlations better defined and the rate of diffusion decreases.

An alternative approach for assessing the diffusion (as well as drag reduction) was to use power-law curve fits between the wall concentration and a common dimensionless parameter [28] used for scaling TBL polymer drag reduction (PDR) results
$K=qinjCinjU∞(x−xinj)10−6$
(8)

where Cinj is the injection concentration in weight-parts-per-million (wppm). This parameter is not a universal scaling, but it has been heavily used for examining drag reduction and diffusion processes in a TBL [5,6,9,10,12,14,29]. Hence, when a Cwall versus K plot was available for a given experiment, the diffusion results were curve-fit within the range of interest and used to estimate the corresponding wall concentration.

### 2.5 Operation Condition.

Outer variable properties are also required for the analysis because (i) they are required for defining the Weissenberg number and (ii) there is evidence that the log-region behavior is sensitive to the Reynolds number (as well as polymer properties) [3]. The momentum thickness based Reynolds number, $Reθ=U∞θ/ν$, was estimated for all conditions discussed in this work, where θ is the momentum thickness. While most of the sources provided the momentum thickness or sufficient information of the boundary layer growth to produce an accurate estimate, some conditions (mostly from Petrie et al. [8]) required estimates assuming a velocity profile shape and the local drag reduction.

### 2.6 Log-Region Parameters.

For Newtonian fluids, the mean streamwise velocity (u) in the log-region of a TBL is well defined by the traditional law of the wall
$u+=1κln(y+)+B$
(9)
where $u+$ is the mean streamwise velocity scaled with the friction velocity, κ is the von Kármán coefficient, $y+$ is the wall-normal distance scaled with the viscous wall unit, and B is the intercept constant. The classical view [1] of how PDR modified the log-region is that below the maximum drag reduction (MDR), the von Kármán coefficient remained unaltered while the intercept constant increased in proportion to the level of drag reduction. Once MDR was achieved, then it was assumed that everything beyond the viscous sublayer takes the empirically derived relationship [30]
$u+=11.7ln(y+)−17$
(10)

that was termed the ultimate profile. While the ultimate profile has predicted the behavior of drag reduced flows with a wide varied of polymers, recently there has been doubt about the functional form being logarithmic [2,3].

With the exception of Petrie et al. [8], the von Kármán coefficient (κ) and the intercept constant (B) were determined directly from fitting the log-region region of the mean near-wall profiles to the form $u+=ln(y+)/κ+B$. The inner variable scaled velocity profiles were plotted, and the overlap region were determined via examination of an indicator function in the region of y+ > 10 and y <0.2δ99. The indicator function $(ζ=y+du+/dy+)$ is commonly used to examine the logarithmic dependence of the mean velocity profile (e.g., Refs. [2,3], and [11]). Differentiation of the log-curve fit produces
$du+dy+=1κddy+ln(y+)=1κ1y+→1κ=y+du+dy+$
(11)

Therefore, the indicator function can be interpreted as $ζ=1/κ$. For each profile in the literature, a linear fit between three adjacent points was used to determine the local $du+/dy+$, which when multiplied by the local $y+$ is the indicator function. Then, the region of nearly constant $ζ$ within the range of y+ > 10 and y <0.2δ99 was used to define 1/κ for the log-region. The logarithmic curve fit within the log-region provides κ and B.

## 3 Results

Inner variable scaled mean streamwise velocity profiles, u+ versus y+, from six different sources are shown in Fig. 1. These were selected as representative of the ranges of DRs and Reynolds numbers. The legend lists the friction Reynolds number $(Reτ=uτδ99/ν)$ for each test condition. The friction Reynolds number was selected because it conveys the range of scales present in the given flow and, consequently, the expected extent of the log-region. For example, Marusic et al. [31] examined the highest flat plate TBL studies [3235] to determine that the log-region bounds are well approximated as spanning $3Reτ0.5. If this range were applied to the profiles in Fig. 1, the log-regions range from over 1000 wall units [3] to nonexistent [7,9]. This highlights the need for caution when examining these results and that one should be looking for bulk trends rather than specific values. With that being said, this collective analysis over such a wide parameter space could not be achieved within a single experiment.

Fig. 1
Fig. 1
Close modal

The log-region for each polymer modified velocity profile examined was identified using the indicator function $(ζ=y+du+/dy+)$ as previously discussed. The 18% drag reduction condition from Fig. 1 is replotted in Fig. 2 along with the corresponding indicator function, $ζ$. Note that using the Marusic et al. [31] criteria, the log-region would extend from $340≤y+≤2000$. These results are consistent with this range since the indicator function starts to flatten out near $y+=∼300$ and then remains nearly flat. The outer edge where the indicator function appears to begin to fluctuate is likely due to image distortions at the edge of the particle image velocimetry field-of-view (Elbing et al. [3] used a unique optical arrangement that stretched the wall-normal direction with optical distortions increasing away from the image center). For reference, Fig. 2 includes lines marking the lower limit of the log-region based on Marusic et al. [31] (vertical line; $y+=340$), Newtonian profile (κ = 0.4; $ζ=2.5$), and the ultimate profile ($ζ=11.7$) [1], which is an empirical relationship for when MDR is achieved.

Fig. 2
Fig. 2
Close modal

For every condition, the average of the indicator function within the identified log-region is readily used to determine the von Kármán coefficient $(κ=1/ζ)$. Then, a curve fit of the data within the log-region gives the intercept constant (B). All of the results (κ and B) from the sources listed in Table 1 are shown in Fig. 3. The markers match those listed in Table 1. The error bars convey the confidence level for each data point determined from the log-region curve fits, which accounts for both the confidence in calculations and the extent of the log-region (i.e., larger log-region the better the curve fit). Subsequent analysis will omit the error bars as they can make it difficult to identify the trends in the data. However, the error bars here combined with the indication of whether or not degradation impacted the results (open versus closed markers) conveys the challenge of identifying the impact of polymer properties on the resulting velocity field. In general, if the error bars are small, then the Reynolds number is high, and degradation has altered the mean molecular weight (i.e., closed markers). Conversely, if there was no degradation (i.e., open markers), then the uncertainty is high due, in part, to the short log-region. This is because the accuracy of the curve fits improves as the Reynolds number increases due to the log-region extent increasing with the increasing Reynolds number. However, the higher Reynolds numbers also produce higher shear rates, which stretch and break the polymer chains. Note that Elbing et al. [3] presented a similar plot with the data points grouped by Reynolds number and found no clear separation, which indicated that polymer properties must play an important role. Thus, if we have a precise measure of the log-region, then we likely have evolving mean molecular weight and concentration when studying polymer injection into a developing TBL.

Fig. 3
Fig. 3
Close modal

## 4 Analysis and Discussion

For the analysis, only data below the overlap concentration (i.e., dilute solutions) are included, which removed eight of over 200 conditions. Figure 4 shows $κ$ and B plotted versus drag reduction with the marker color corresponding to the friction Reynolds number. Elbing et al. [3] used the momentum thickness based Reynolds number $(Reθ)$, and concluded that the Reynolds number was insufficient to explain scatter at HDR. The use of friction Reynolds number was because it sets the extent of the log-region, but the scatter in Fig. 4 is even larger than that observed in Elbing et al. [3]. This reiterates the conclusions of Elbing et al. [3] that polymer properties must be responsible for at least part of the scatter observed, and conveys that the scatter is not simply the product of limitations to properly resolve the log-region. The only additional conditions considered are the polymer ocean data [11], which that data follows a similar trend to the polymer injection studies.

Fig. 4
Fig. 4
Close modal

Since Elbing et al. [3] concluded that the scatter must be due, in part, to polymer properties, each nondimensional polymer parameter (length ratio, viscosity ratio, and Weissenberg number) was examined to see if they could explain the scatter. The range of length ratio $(96≤L≤337)$ and viscosity ratio $(0.7≤μ*≤∼1)$ were significant, but they did not explain the scatter. For example, even at LDR, the largest length ratios produced results that followed the classical view (Newtonian constant) as well as the fit from Koskie and Tiederman [36]. The more likely polymer property is the Weissenberg number since it quantifies relative time scales between the flow and the polymer. As previously stated, there are two established definitions used in the literature; $Wim=λU∞/δ99$ and $Wiw=λuτ/lν$. Both Wi definitions were investigated in this study, but the scatter appeared independent of $Wim$. Consequently, Fig. 5 replots the results from Fig. 4 though with the symbol colors now corresponding to the inner variable based Weissenberg number ($Wiw$). Figure 5 shows that the higher the Weissenberg number, the longer the velocity follows the classical view (i.e., Newtonian $κ$ and an increase in B proportional to the drag reduction). In particular, the scatter in κ at HDR appears to be well correlated with the Weissenberg number. This is promising given the fact that no explanation for the scatter in κ was identified in Elbing et al. [3] (i.e., it only determined that the scatter in κ was not a Reynolds number effect). There is also some color separation for the intercept constant B, but the separation is not as pronounced as with κ. This is not surprising since past results suggested a Reynolds number dependence on B [3].

Fig. 5
Fig. 5
Close modal

The polymer ocean results [11] in Fig. 5, denoted by the star symbols, match the general trends produced by the injection conditions quite well. The primary difference between ocean and injection are that the intercept constant for several injection conditions between 55% and 70% drag reduction still follow the classical view while the ocean bends downward. While it is tempting to dismiss these few conditions as outliers, these data [3,5] were acquired at a very high Reynolds number $(Reτ≥8500)$ and are from the only study that had direct measurements of the near-wall velocity profile, concentration profile, and even local mean molecular weight for a subset of conditions. This supports the conclusion of Elbing et al. [3] that the deviation of the intercept constant from the classical view occurs at higher DR with increasing Reynolds number because those conditions are at very high Reynolds numbers $(8500≤Reτ≤14,000)$. Excluding those data points, the polymer ocean results continue a relatively smooth curve for B with medium-high Weissenberg range $(100.8 consistently above the low Weissenberg range $(10−0.8≤Wiw≤100)$. Another interesting observation can be drawn from the fact that the polymer properties for injection conditions were estimated using the maximum wall concentration. The nominal agreement between the polymer ocean results and the injection results supports an implicit assumption in most of the past PDR studies that the maximum polymer concentration controls the near-wall behavior.

## 5 Conclusions

The primary objective for this paper was to revisit past TBL PDR studies [3,611] and use recent insights [25] to provide clarity to observed inconsistencies in the near-wall mean velocity profiles, especially at HDR. Only experimental TBL studies using PEO with sufficient information to estimate the corresponding local polymer properties were used. Most conditions injected polymer into the developing TBL, but one study [11] used a polymer ocean. A review of all relationships used to estimate critical parameters related to the polymer (molecular structure, relaxation time, overlap concentration, viscosity ratio, and length ratio), the flow (polymer diffusion, outer variables, and mean velocity profiles), and the coupling between them (i.e., flow-induced polymer degradation) was provided. Example inner variable scaled mean velocity profiles from several sources over a wide range of Reynolds numbers illustrated the classic view (i.e., von Kármán coefficient remains constant and the intercept increasing in proportion to the drag reduction until the ultimate profile is achieved) as well as deviations between various profiles. The indicator function was used to identify and measure the von Kármán coefficient for the log-region, which was shown to be consistent with the range estimate using the friction Reynolds number [31]. Then, the corresponding intercept constant was found from the best fit curve to the data within the log-region. The uncertainty for each condition incorporated confidence in the original data, any relationships applied, and the extent of the log-region.

Separating the data based on the friction Reynolds number did not show any collapse in the von Kármán coefficient (κ), which is consistent with Elbing et al. [3] that drew the same conclusion when using the momentum based Reynolds number. The length ratio, viscosity ratio, and the Weissenberg number based on the mean shear were examined and did not explain the scattered results. However, when separating the results based on the inner variable-based Weissenberg number, there is a clear separation of the data. This is significant given that no explanation for the scatter was offered in Elbing et al. [3], though by process of elimination it was concluded that polymer properties must be influencing the scatter.

The intercept constant did not find as clear of a separation in the data regardless of what parameter was used (friction Reynolds number, length ratio, viscosity ratio, or mean shear based Weissenberg number). The inner variable-based Weissenberg number showed some separation, but there were clear outliers (i.e., scatter) that remained, particularly at HDR and approaching MDR. Close to MDR, there were intercept constants with similar inner variable Weissenberg numbers ranging from the classical view prediction (increasing in proportion to the drag reduction level) to matching the ultimate profile (i.e., switching from positive to negative value). This inconsistency is not surprising since Elbing et al. [3] showed good separation of the scatter based on the momentum based Reynolds number. The data here following the classic view are at extremely high Reynolds number while the ultimate profile data is the lowest Reynolds number data.

Finally, the polymer ocean conditions [11] were compared with the polymer injection results. Here, very similar trends were observed for both the von Kármán coefficient and the intercept constant for comparable Reynolds numbers. This gives support to an implicit assumption that has been used in most previous TBL PDR studies, which is that the maximum near-wall polymer concentration sets the polymeric properties as seen by the TBL (at least in the near-wall region). This provides critical insights into the appropriate scaling of PDR modified near-wall velocity profiles.

## Acknowledgment

The author would like to thank Dr. Yasaman Farsiani and Mr. Zeeshan Saeed that have worked closely with the author on the polymer drag reduction studies, especially with establishing the procedures to produce a repeatable, well-defined polymer ocean. In addition, I want to recognize Shahrouz Mohagheghian, who assisted with the extraction of data from the literature.

## Funding Data

• National Science Foundation (NSF) (Grant No. 1604978; Funder ID: 10.13039/100000001) (Dr. Ronald Joslin, Program Manager).

## References

1.
Virk
,
P. S.
,
1975
, “
Drag Reduction Fundamentals
,”
AIChE J.
,
21
(
4
), pp.
625
656
.10.1002/aic.690210402
2.
White
,
C. M.
,
Dubief
,
Y.
, and
Klewicki
,
J.
,
2012
, “
Re-Examining the Logarithmic Dependence of the Mean Velocity Distribution in Polymer Drag Reduced Wall-Bounded Flow
,”
Phys. Fluids
,
24
(
2
), p.
021701
.10.1063/1.3681862
3.
Elbing
,
B. R.
,
Perlin
,
M.
,
Dowling
,
D. R.
, and
Ceccio
,
S. L.
,
2013
, “
Modification of the Mean Near-Wall Velocity Profile of a High-Reynolds Number Turbulent Boundary Layer With the Injection of Drag-Reducing Polymer Solutions
,”
Phys. Fluids
,
25
(
8
), p.
085103
.10.1063/1.4817073
4.
Elsnab
,
J. R.
,
Monty
,
J. P.
,
White
,
C. M.
,
Koochesfahani
,
M. M.
, and
Klewicki
,
J. C.
,
2019
, “
High-Fidelity Measurements in Channel Flow With Polymer Wall Injection
,”
J. Fluid Mech.
,
859
, pp.
851
886
.10.1017/jfm.2018.873
5.
Elbing
,
B. R.
,
Solomon
,
M. J.
,
Perlin
,
M.
,
Dowling
,
D. R.
, and
Ceccio
,
S. L.
,
2011
, “
Flow-Induced Degradation of Drag-Reducing Polymer Solutions Within a High-Reynolds-Number Turbulent Boundary Layer
,”
J. Fluid Mech.
,
670
, pp.
337
364
.10.1017/S0022112010005331
6.
Fontaine
,
A.
,
Petrie
,
H.
, and
Brungart
,
T.
,
1992
, “
Velocity Profile Statistics in a Turbulent Boundary Layer With Slot-Injected Polymer
,”
J. Fluid Mech.
,
238
, pp.
435
466
.10.1017/S0022112092001770
7.
White
,
C. M.
,
Somandepalli
,
V. S. R.
, and
Mungal
,
M. G.
,
2004
, “
The Turbulence Structure of Drag-Reduced Boundary Layer Flow
,”
Exp. Fluids
,
36
(
1
), pp.
62
69
.10.1007/s00348-003-0630-0
8.
Petrie
,
H.
,
Fontaine
,
A.
,
Money
,
M.
, and
Deutsch
,
S.
,
2005
, “
Experimental Study of Slot-Injected Polymer Drag Reduction
,”
Proceedings of the Second International Symposium on Seawater Drag Reduction
, Busan, South Korea, May 23–26, pp.
605
620
.
9.
Hou
,
Y. X.
,
Somandepalli
,
V. S. R.
, and
Mungal
,
M. G.
,
2008
, “
Streamwise Development of Turbulent Boundary-Layer Drag Reduction With Polymer Injection
,”
J. Fluid Mech.
,
597
, pp.
31
66
.10.1017/S0022112007009718
10.
Somandepalli
,
V. S. R.
,
Hou
,
Y. X.
, and
Mungal
,
M. G.
,
2010
, “
Concentration Flux Measurements in a Polymer Drag-Reduced Turbulent Boundary Layer
,”
J. Fluid Mech.
,
644
, pp.
281
319
.10.1017/S0022112009992382
11.
Farsiani
,
Y.
,
Saeed
,
Z.
,
Jayaraman
,
B.
, and
Elbing
,
B. R.
,
2020
, “
Modification of Turbulent Boundary Layer Coherent Structures With Drag Reducing Polymer Solution
,”
Phys. Fluids
,
32
(
1
), p.
015107
.10.1063/1.5127293
12.
Winkel
,
E. S.
,
Oweis
,
G. F.
,
Vanapalli
,
S. A.
,
Dowling
,
D. R.
,
Perlin
,
M.
,
Solomon
,
M. J.
, and
Ceccio
,
S. L.
,
2009
, “
High-Reynolds-Number Turbulent Boundary Layer Friction Drag Reduction From Wall-Injected Polymer Solutions
,”
J. Fluid Mech.
,
621
, pp.
259
288
.10.1017/S0022112008004874
13.
Elbing
,
B. R.
,
Winkel
,
E. S.
,
Ceccio
,
S. L.
,
Perlin
,
M.
, and
Dowling
,
D. R.
,
2010
, “
High-Reynolds-Number Turbulent-Boundary-Layer Wall-Pressure Fluctuations With Dilute Polymer Solutions
,”
Phys. Fluids
,
22
(
8
), p.
085104
.10.1063/1.3478982
14.
Elbing
,
B. R.
,
Dowling
,
D. R.
,
Perlin
,
M.
, and
Ceccio
,
S. L.
,
2010
, “
Diffusion of Drag-Reducing Polymer Solutions Within a Rough-Walled Turbulent Boundary Layer
,”
Phys. Fluids
,
22
(
4
), p.
045102
.10.1063/1.3371957
15.
Grandbois
,
M.
,
Beyer
,
M.
,
Rief
,
M.
,
Clausen-Schaumann
,
H.
, and
Gaub
,
H. E.
,
1999
, “
How Strong is a Covalent Bond?
,”
Science
,
283
(
5408
), pp.
1727
1730
.10.1126/science.283.5408.1727
16.
Vanapalli
,
S. A.
,
Islam
,
M. T.
, and
Solomon
,
M. J.
,
2005
, “
Scission-Induced Bounds on Maximum Polymer Drag Reduction in Turbulent Flow
,”
Phys. Fluids
,
17
(
9
), p.
095108
.10.1063/1.2042489
17.
Farsiani
,
Y.
,
Saeed
,
Z.
, and
Elbing
,
B. R.
,
2020
, “
Drag Reduction Performance of Mechanically Degraded Dilute Polyethylene Oxide Solutions
,”
ASME J. Fluids Eng.
,
142
(
9
), p.
091201
.10.1115/1.4047118
18.
Zimm
,
B. H.
,
1956
, “
Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss
,”
J. Chem. Phys.
,
24
(
2
), pp.
269
278
.10.1063/1.1742462
19.
Bailey
,
F. E.
, Jr.
, and
Callard
,
R. W.
,
1959
, “
Some Properties of Poly(Ethylene Oxide) in Aqueous Solution
,”
J. Appl. Polym. Sci.
,
1
(
1
), pp.
56
62
.10.1002/app.1959.070010110
20.
Kalashnikov
,
V. N.
,
1998
, “
Dynamical Similarity and Dimensionless Relations for Turbulent Drag Reduction by Polymer Additives
,”
J. Non-Newtonian Fluid Mech.
,
75
(
2–3
), pp.
209
230
.10.1016/S0377-0257(97)00093-1
21.
Larson
,
R. G.
,
1999
,
The Structure and Rheology of Complex Fluids
,
Oxford University Press
,
New York
.
22.
Vanapalli
,
S. A.
,
Ceccio
,
S. L.
, and
Solomon
,
M. J.
,
2006
, “
Universal Scaling for Polymer Chain Scission in Turbulence
,”
,
103
(
45
), pp.
16660
16665
.10.1073/pnas.0607933103
23.
Poreh
,
M.
, and
Cermak
,
J. E.
,
1964
, “
Study of Diffusion From a Line Source in a Turbulent Boundary Layer
,”
Int. J. Heat Mass Transfer
,
7
(
10
), pp.
1083
1095
.10.1016/0017-9310(64)90032-8
24.
Collins
,
D. J.
, and
Gorton
,
C. W.
,
1976
, “
An Experimental Study of Diffusion From a Line Source in a Turbulent Boundary Layer
,”
AIChE J.
,
22
(
3
), pp.
610
612
.10.1002/aic.690220333
25.
Giles
,
W. B.
,
1968
, “
Similarity Laws of Friction-Reduced-Flows
,”
J. Hydronautics
,
2
(
1
), pp.
34
40
.10.2514/3.62771
26.
Gebel
,
C.
,
Reitzer
,
H.
, and
Bues
,
M.
,
1978
, “
Diffusion of Macromolecular Solutions in the Boundary Layer
,”
Rheol. Acta
,
17
(
2
), pp.
172
175
.10.1007/BF01517708
27.
Vdovin
,
A. V.
, and
Smol'yakov
,
A. V.
,
1982
, “
Turbulent Diffusion of Polymers in a Boundary Layer
,”
J. Appl. Mech. Tech. Phys.
,
22
(
4
), pp.
526
531
.10.1007/BF00906264
28.
Vdovin
,
A. V.
, and
Smol'yakov
,
A. V.
,
1978
, “
Diffusion of Polymer Solutions in a Turbulent Boundary Layer
,”
J. Appl. Mech. Tech. Phys.
,
19
(
2
), pp.
196
201
.10.1007/BF00850033
29.
White
,
C. M.
, and
Mungal
,
M. G.
,
2008
, “
Mechanics and Prediction of Turbulent Drag Reduction With Polymer Additives
,”
Annu. Rev. Fluid Mech.
,
40
(
1
), pp.
235
256
.10.1146/annurev.fluid.40.111406.102156
30.
Virk
,
P. S.
,
Mickley
,
H. S.
, and
Smith
,
K. A.
,
1970
, “
The Ultimate Asymptote and Mean Flow Structure in Toms' Phenomenon
,”
ASME J. Appl. Mech.
,
37
(
2
), pp.
488
493
.10.1115/1.3408532
31.
Marusic
,
I.
,
Monty
,
J. P.
,
Hultmark
,
M.
, and
Smits
,
A. J.
,
2013
, “
On the Logarithmic Region in Wall Turbulence
,”
J. Fluid Mech.
,
716
, p.
R3
.10.1017/jfm.2012.511
32.
Hultmark
,
M.
,
Vallikivi
,
M.
,
Bailey
,
S. C. C.
, and
Smits
,
A. J.
,
2012
, “
Turbulent Pipe Flow at Extreme Reynolds Numbers
,”
Phys. Rev. Lett.
,
108
(
9
), p.
094501
.10.1103/PhysRevLett.108.094501
33.
Hutchins
,
N.
,
Chauhan
,
K.
,
Marusic
,
I.
,
Monty
,
J. P.
, and
Klewicki
,
J.
,
2012
, “
Towards Reconciling the Large-Scale Structure of Turbulent Boundary Layers in the Atmosphere and Laboratory
,”
Boundary-Layer Meteorol.
,
145
(
2
), pp.
273
306
.10.1007/s10546-012-9735-4
34.
Kulandaivelu
,
V.
,
2012
, “
Evolution of Zero Pressure Gradient Turbulent Boundary Layers From Different Initial Conditions
,” Ph.D. thesis,
University of Melbourne
, Melbourne, Australia.
35.
Winkel
,
E. S.
,
Cutbirth
,
J. M.
,
Ceccio
,
S. L.
,
Perlin
,
M.
, and
Dowling
,
D. R.
,
2012
, “
Turbulence Profiles From a Smooth Flat-Plate Turbulent Boundary Layer at High Reynolds Number
,”
Exp. Therm. Fluid Sci.
,
40
, pp.
140
149
.10.1016/j.expthermflusci.2012.02.009
36.
Koskie
,
J. E.
, and
Tiederman
,
W. G.
,
1991
, “
Polymer Drag Reduction of a Zero-Pressure-Gradient Boundary Layer
,”
Phys. Fluids
,
3
(
10
), pp.
2471
2473
.10.1063/1.858187