^{1,a),b)}, Marc Perlin

^{2}, David R. Dowling

^{3}and Steven L. Ceccio

^{2,3}

### Abstract

The current study explores the influence of polymer drag reduction on the near-wall velocity distribution in a turbulent boundary layer (TBL) and its dependence on Reynolds number. Recent moderate Reynolds number direct numerical simulation and experimental studies presented in White et al. [Phys. Fluids24, 021701 (Year: 2012)]10.1063/1.3681862 have challenged the classical representation of the logarithmic dependence of the velocity profile for drag-reduced flows, especially at drag reduction levels above 40%. In the present study, high Reynolds number data from a drag reduced TBL is presented and compared to the observations of White et al. [Phys. Fluids24, 021701 (Year: 2012)]10.1063/1.3681862. Data presented here were acquired in the TBL flow on a 12.9-m-long flat plate at speeds to 20.3 m s−1, achieving momentum thickness based Reynolds number to 1.5 × 105, which is an order of magnitude greater than that available in the literature. Polyethylene oxide solutions with an average molecular weight of 3.9 × 106 g mol−1 were injected into the flow at various concentrations and volumetric fluxes to achieve a particular level of drag reduction. The resulting mean near-wall velocity profiles show distinctly different behavior depending on whether they fall in the low drag reduction (LDR) or the high drag reduction (HDR) regimes, which are nominally divided at 40% drag reduction. In the LDR regime, the classical view that the logarithmic slope remains constant at the Newtonian value and the intercept constant increases with increasing drag reduction appears to be valid. However, in the HDR regime the behavior is no longer universal. The intercept constant continues to increase linearly in proportion to the drag reduction level until a Reynolds-number-dependent threshold is achieved, at which point the intercept constant rapidly decreases to that predicted by the ultimate profile. The rapid decrease in the intercept constant is due to the corresponding increase in the profile slope in the HDR regime. There was significant scatter in the observed slope in the HDR regime, but the scatter did not appear to be Reynolds number dependent. Finally, the ultimate profiles for flows at maximum drag reduction were examined and did not exhibit a logarithmic functional relationship, which is the classical empirical relationship suggested by Virk [J. Am. Inst. Chem. Eng.21, 625–656 (Year: 1975)]10.1002/aic.690210402.

The authors would like to thank the technical staff of the U.S. Navy's William B. Morgan Large Cavitation Channel for their help conducting the experiment, co-workers at the University of Michigan (Dr. Ghanem Oweis, Dr. Eric Winkel, Dr. Shiyao Bian, Dr. Keary Lay, Ms. Laëtitia Decoster, Ms. Ciara Stella, and Mr. Kent Pruss) that assisted in preparation and execution of the experiment and Mr. Duncan Brown of the Johns Hopkins University Applied Physics Laboratory for valuable discussions on PDR. This research was sponsored by DARPA under Contract Nos. HR0011-04-1-001 and HR0011-06-1-0057 (Dr. Thomas Beutner, Program Manager) and by ONR under Contract No. N00014-06-1-0244 (Dr. L. Patrick Purtell, Program Manager). The content of this document does not necessarily reflect the position or the policy of the United States Government, and no official endorsement should be inferred.

I. INTRODUCTION

II. EXPERIMENTAL METHODS

A. Test facility and model

B. Instrumentation

C. Polymer preparation and delivery to the TBL

D. Test matrix

III. NO-INJECTION VELOCITY DISTRIBUTION

IV. DRAG REDUCED FLOWS

A. Modification to the log-region of the TBL

V. ULTIMATE PROFILE

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Drag reduction
- 59.0
- Reynolds stress modeling
- 46.0
- Polymers
- 29.0
- Channel flows
- 12.0
- Polymer flows
- 11.0

## Figures

Schematic of the test model with gravity oriented upward (i.e., showing the working surface). The locations of the skin-friction balances, PIV systems, and the injector are shown.

Schematic of the test model with gravity oriented upward (i.e., showing the working surface). The locations of the skin-friction balances, PIV systems, and the injector are shown.

Non-injection velocity data from the current experiment scaled with inner variables. Included are results from (+) X = 1.96 m, (×) X = 5.94 m, and (*) X = 10.68 m. Also included are curves for the viscous sublayer (U + = Y +), traditional log-law region (U + = ln(Y +)/κ + B) and the ultimate profile (U + = 11.7 ln(Y +) – 17.0).

Non-injection velocity data from the current experiment scaled with inner variables. Included are results from (+) X = 1.96 m, (×) X = 5.94 m, and (*) X = 10.68 m. Also included are curves for the viscous sublayer (U + = Y +), traditional log-law region (U + = ln(Y +)/κ + B) and the ultimate profile (U + = 11.7 ln(Y +) – 17.0).

Examples of scaled velocity profiles with varying levels of drag reduction. The injection concentration was 4000 wppm for all the conditions shown. (%DR = 18.2, X = 1.96 m, U ∞ = 19.9 m s−1, q = 2 qs, Re θ = 4.6 × 104; %DR = 35.2, X = 5.94 m, U ∞ = 20.0 m/s, q = 10 qs, Re θ = 1.2 × 105; %DR = 53.5, X = 1.96 m, U ∞ = 19.9 m/s, q = 10 qs, Re θ = 4.5 × 104; %DR = 64.8, X = 5.94 m, U ∞ = 6.7 m/s, q = 10 qs, Re θ = 4.6 × 104.) The maximum drag reduction asymptote or ultimate profile (heavy dotted line) and the law of the wall (solid line) are shown also.

Examples of scaled velocity profiles with varying levels of drag reduction. The injection concentration was 4000 wppm for all the conditions shown. (%DR = 18.2, X = 1.96 m, U ∞ = 19.9 m s−1, q = 2 qs, Re θ = 4.6 × 104; %DR = 35.2, X = 5.94 m, U ∞ = 20.0 m/s, q = 10 qs, Re θ = 1.2 × 105; %DR = 53.5, X = 1.96 m, U ∞ = 19.9 m/s, q = 10 qs, Re θ = 4.5 × 104; %DR = 64.8, X = 5.94 m, U ∞ = 6.7 m/s, q = 10 qs, Re θ = 4.6 × 104.) The maximum drag reduction asymptote or ultimate profile (heavy dotted line) and the law of the wall (solid line) are shown also.

The intercept B plotted versus the percent drag reduction. The Reynolds number based on the momentum thickness (Re θ ) ranged from 9.7 × 102 to 1.5 × 105. Results shown are from the (✩) current study, (○) Koskie and Tiederman, 24 (◊) Fontaine et al., 25 (▵) White et al., 26 (□) Petrie et al., 27 (⊲) Hou et al., 28 and (⊳) Somandepalli et al. 29 The symbol color corresponds to the Reynolds number range as shown in the legend. Dashed lines indicate the Newtonian value (B = 5.0) and the ultimate profile (B = −17). The solid line (B = 5 + (0.2)(%DR)) was a linear fit to data presented in Petrie et al. 27

The intercept B plotted versus the percent drag reduction. The Reynolds number based on the momentum thickness (Re θ ) ranged from 9.7 × 102 to 1.5 × 105. Results shown are from the (✩) current study, (○) Koskie and Tiederman, 24 (◊) Fontaine et al., 25 (▵) White et al., 26 (□) Petrie et al., 27 (⊲) Hou et al., 28 and (⊳) Somandepalli et al. 29 The symbol color corresponds to the Reynolds number range as shown in the legend. Dashed lines indicate the Newtonian value (B = 5.0) and the ultimate profile (B = −17). The solid line (B = 5 + (0.2)(%DR)) was a linear fit to data presented in Petrie et al. 27

The von Kármán constant as a function of percent drag reduction from several studies (symbols and colors are the same as in Figure 4 ). The dashed lines correspond to the values corresponding to Newtonian (κ = 0.41) and the ultimate profile (κ = 1 / 11.7 ≈ 0.0855). The solid line is a linear fit to data presented in Koskie and Tiederman. 24

The von Kármán constant as a function of percent drag reduction from several studies (symbols and colors are the same as in Figure 4 ). The dashed lines correspond to the values corresponding to Newtonian (κ = 0.41) and the ultimate profile (κ = 1 / 11.7 ≈ 0.0855). The solid line is a linear fit to data presented in Koskie and Tiederman. 24

Indicator function (ζ) versus inner-variable-scaled wall-normal distance at a free-stream speed of (upper) 6.7 and (lower) 20.3 m/s. All the data shown were acquired at the first measurement station (X = 1.96 m) except for the condition at %DR = 64.8, which was acquired at X = 5.94 m. The Reynolds numbers for %DR = 42.3, 53.0, 64.8, 18.2, 44.0, and 53.5 conditions are Re θ = 1.7 × 104, 1.8×104, 4.6 × 104, 4.6 × 104, 4.6 × 104, and 4.5 × 104, respectively. At each speed a no-injection condition is provided for comparison with the drag reduced results.

Indicator function (ζ) versus inner-variable-scaled wall-normal distance at a free-stream speed of (upper) 6.7 and (lower) 20.3 m/s. All the data shown were acquired at the first measurement station (X = 1.96 m) except for the condition at %DR = 64.8, which was acquired at X = 5.94 m. The Reynolds numbers for %DR = 42.3, 53.0, 64.8, 18.2, 44.0, and 53.5 conditions are Re θ = 1.7 × 104, 1.8×104, 4.6 × 104, 4.6 × 104, 4.6 × 104, and 4.5 × 104, respectively. At each speed a no-injection condition is provided for comparison with the drag reduced results.

## Tables

Sources of experimental measurements of the mean velocity distribution in PDR TBL flows. Included are the type of polymer used, maximum level of drag reduction and the range of momentum-thickness based Reynolds number.

Sources of experimental measurements of the mean velocity distribution in PDR TBL flows. Included are the type of polymer used, maximum level of drag reduction and the range of momentum-thickness based Reynolds number.

Summary of conditions where the velocity distributions were measured. The velocities shown are the average free-stream velocity over the length of the test model. Local free-stream velocities varied due to boundary layer growth on the model and tunnel walls (6.7 m s−1 = 6.62, 6.65, 6.77 m s−1; 20.3 m s−1 = 19.9, 20.0, 20.3 m s−1 for X = 1.96, 5.94, 10.68 m, respectively).

Summary of conditions where the velocity distributions were measured. The velocities shown are the average free-stream velocity over the length of the test model. Local free-stream velocities varied due to boundary layer growth on the model and tunnel walls (6.7 m s−1 = 6.62, 6.65, 6.77 m s−1; 20.3 m s−1 = 19.9, 20.0, 20.3 m s−1 for X = 1.96, 5.94, 10.68 m, respectively).

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