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TECHNICAL PAPERS

Stokes Mechanism of Drag Reduction

[+] Author and Article Information
Promode R. Bandyopadhyay

Autonomous Systems and Technology Department, Naval Undersea Warfare Center, Newport, RI 02841bandyopadhyaypr@npt.nuwc.navy.mil

J. Appl. Mech 73(3), 483-489 (Sep 20, 2005) (7 pages) doi:10.1115/1.2125974 History: Received November 23, 2003; Revised September 20, 2005

The mechanism of drag reduction due to spanwise wall oscillation in a turbulent boundary layer is considered. Published measurements and simulation data are analyzed in light of Stokes’ second problem. A kinematic vorticity reorientation hypothesis of drag reduction is first developed. It is shown that spanwise oscillation seeds the near-wall region with oblique and skewed Stokes vorticity waves. They are attached to the wall and gradually align to the freestream direction away from it. The resulting Stokes layer has an attenuated nature compared to its laminar counterpart. The attenuation factor increases in the buffer and viscous sublayer as the wall is approached. The mean velocity profile at the condition of maximum drag reduction is similar to that due to polymer. The final mean state of maximum drag reduction due to turbulence suppression appears to be universal in nature. Finally, it is shown that the proposed kinematic drag reduction hypothesis describes the measurements significantly better than what current direct numerical simulation does.

Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Measurements of change in axial velocity due to spanwise oscillation of a turbulent boundary layer normalized by freestream velocity. Symbols: ◯=1 Hz; ◻=3 Hz, ▵=5 Hz, and ×=7 Hz of wall-oscillation frequency. Reproduced from Choi (5).

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Figure 2

Comparison of simulation and measurements of drag reduction due to spanwise wall-oscillation (4). Symbols are measurements in turbulent boundary layers: ◯=Choi (5), and ▵=Laadhari (6); ——— is DNS simulation in a channel flow due to Akhavan and co-workers (3). Reproduced from Choi (5).

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Figure 3

Schematic of drag reduction hypothesis of vorticity reorientation

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Figure 4

Drag reduction due to the proposed vorticity reorientation hypothesis

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Figure 5

The Stokes’ phase variation of total axial velocity. Symbols: ———=1 Hz; ×=3 Hz; ◻=5 Hz; ◯=7 Hz. Measurements are due to Choi (5).

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Figure 6

Total axial velocity in normal wall-layer velocity and length scales. Solid line is U+=y+. Symbols: ———=1 Hz; ×=3 Hz; ◻=5 Hz; ◯=7 Hz. Measurements are due to Choi (5).

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Figure 7

Regular law of the wall representation of the effects of increasing frequency on total axial velocity. The reference sublayer profile and the logarithmic velocity profile of unperturbed turbulent boundary layers are shown (———). Virk’s (11) ultimate polymer profile for the condition of maximum drag reduction is also shown (– – –). Symbols: ◯=1 Hz; ◻=3 Hz; ▵=5 Hz; ×=7 Hz. Measurements are due to Choi (5).

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Figure 8

Stokes’ layer scaling of change in axial velocity due to spanwise oscillation. Symbols: ◻=3 Hz; ▵=5 Hz; ×=7 Hz. Measurements are due to Choi (5).

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Figure 9

Stokes layer phase lag representation of spanwise fluid material displacement (symbols; ———=Δz∕Z) in the oscillating turbulent boundary layer compared with laminar Stokes velocity profile (--•--=w∕W). Symbols: ×=5 Hz (sample 1); ◻=5 Hz (sample 2); and ▵=2 Hz. Band in η indicates thickness of laser light sheet used for flow visualization. Data extracted from flow visualization video due to Choi (5,10).

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Figure 10

Attenuated nature of near-wall Stokes layer in an oscillating turbulent boundary layer. Symbols: ◯=1 Hz; ◻=3 Hz; ▵=5 Hz; ×=7 Hz. Measurements are due to Choi (5).

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Figure 11

Oblique “two-dimensional” Stokes waves. Freestream direction is from left to right. Frame extracted from flow visualization video due to Choi (5). Freestream velocity is 1.5m∕s; wall-oscillation frequency is 5 Hz; amplitude of spanwise oscillation is 50 mm. Figure is roughly in scale; height of laser light sheet is 1±0.5mm from wall.

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Figure 12

Extrapolation of Choi ’s (5) experimental condition (symbols). This is carried out via proposed hypothesis of drag reduction (solid line) for the determination of the condition for maximum drag reduction.

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Figure 13

Relationship between vorticity reorientation angle and Choi ’s oscillation parameter. Symbols are from extrapolation of flow visualization data shown in Fig. 9 to wall.

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Figure 14

Recovery of mean velocity profile for maximum drag reduction from an unperturbed profile by means of attenuated Stokes layer modeling. Symbols: ◯=unperturbed flow; ▵=computed maximum drag reduction. Broken and thick solid lines are Virk’s polymer maximum drag reduction profiles (11).

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Figure 15

Calculated variation of Stokes’ attenuation parameter across the boundary layer for the condition of maximum drag reduction

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Figure 16

Validation of vorticity reorientation hypothesis of drag reduction. Symbols are measurements in turbulent boundary layers: ◯=Choi (5) and ◻=Laadhari (6); - - - is DNS simulation in a channel flow due to Akhavan and co-workers (3) and ——— is presently proposed vorticity reorientation hypothesis.

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Figure 17

Comparison of the yaw angle (α line) of the resultant wall vorticity due to turbulence as per the vorticity reorientation hypothesis [Eq. 1], with hydrogen bubble measurements of the variation of angle of inclination γ (symbol) of wall streaks with respect to mean streamwise direction of flow, with fractional drag reduction (12).

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