^{1,a)}, J. M. Castano

^{1}and D. P. Thivierge

^{1}

### Abstract

Drag reduction of high-Reynolds-number axisymmetric bodies in saltwater flow using numerous small Lorentz actuators is considered. The actuators are three-dimensional and each of them encompasses the footprint of approximately one turbulence production domain at a Reynolds number of , based on friction velocity and boundary layer thickness. The actuators seed the turbulent boundary layer locally with pulsing toroids of vorticity that straddle the periphery of the actuators. The central downward jet of the toroid counters the upward flow between naturally occurring near-wall vortex pairs. Owing to the presence of the wall, the downward central jet is deflected into wall-jets that lie underneath the toroid. The agglomerated effects of the pulsing of the power applied to the three-dimensional actuators are modeled as Stokes oscillators. An axisymmetric body containing numbers of subcentimeter-scale electromagnetic surface actuators was built. Measurements over this body show that drag reduction efficiency is higher compared to that expected in two-dimensional actuators at similar Reynolds numbers.Drag reduction depends on the parameter , where St is Stuart number and is pulsing time scale in wall-layer variables, approximately in the same manner as two-dimensional actuators do. However, the exponent is zero—not 1.0 like that in two-dimensional actuators. At , the same three-dimensional actuators would no longer match the footprints of the unit turbulence production domains and the same applied power would be a weaker perturbation on the streak vorticity. A denser clustering of three-dimensional actuators for the same input power is a likely solution for higher Reynolds numbers.

This research was sponsored by the Office of Naval Research. Professor Kenneth S. Breuer of Brown University and Professor Kwing-So Choi of the University of Nottingham are thanked for making available the Breuer *et al.* and the Pang and Choi data, respectively. Dr. Charles Henoch of the Naval Undersea Warfare Center is thanked for his help with the direct computation of applied power in the former data. Professor Choi is also thanked for his comments on the manuscript.

I. INTRODUCTION

A. Trends in the literature on two-dimensional large-scale and three-dimensional wall-layer scale actuators

II. ELECTROMAGNETIC LAYOUT OF TWO- AND THREE-DIMENSIONAL ACTUATORS

A. Two-dimensional actuators producing large-scale spanwise oscillation

B. Three-dimensional actuator producing small-scale local oscillation near the wall

III. MODELING OF THREE-DIMENSIONAL ACTUATOR PERTURBATION

A. Modeling of electromagnetic pulsing frequency

IV. EXPERIMENTS

V. RESULTS AND DISCUSSION

VI. CONCLUSIONS

### Key Topics

- Reynolds stress modeling
- 52.0
- Drag reduction
- 51.0
- Lorentz group
- 38.0
- Turbulent flows
- 37.0
- Boundary layer turbulence
- 26.0

## Figures

Schematic diagram of the electrode-magnet array in two-dimensional actuators for producing large scale cross-stream body force and secondary flow. The solid curved lines between the magnet pairs indicate the permanent magnetic field lines . The broken curved lines with the pairs of arrows between the electrode pairs indicate the pulsating applied electric current density field lines .

Schematic diagram of the electrode-magnet array in two-dimensional actuators for producing large scale cross-stream body force and secondary flow. The solid curved lines between the magnet pairs indicate the permanent magnetic field lines . The broken curved lines with the pairs of arrows between the electrode pairs indicate the pulsating applied electric current density field lines .

Schematic top view of the electrode and magnet configuration in the present three-dimensional actuator (also see Fig. 3). The widths of the electrodes and the magnets are each; the gaps between the neighboring electrodes and magnets are and , respectively. The magnets are long. The electrodes are plated onto a plastic (kapton) sheet, which is screwed to and held taut by two axial bus bars (see Fig. 3)—positive and negative—each wide, located diametrically opposite to each other. At a freestream velocity of , is 176 wall units. The inner boundary of the actuator is 528 and 1056 wall units in the spanwise and streamwise directions, respectively. The solid lines between the magnet pairs indicate the permanent magnetic field lines . The pair of broken lines between the electrode pairs indicates the pulsating applied electric current density field lines .

Schematic top view of the electrode and magnet configuration in the present three-dimensional actuator (also see Fig. 3). The widths of the electrodes and the magnets are each; the gaps between the neighboring electrodes and magnets are and , respectively. The magnets are long. The electrodes are plated onto a plastic (kapton) sheet, which is screwed to and held taut by two axial bus bars (see Fig. 3)—positive and negative—each wide, located diametrically opposite to each other. At a freestream velocity of , is 176 wall units. The inner boundary of the actuator is 528 and 1056 wall units in the spanwise and streamwise directions, respectively. The solid lines between the magnet pairs indicate the permanent magnetic field lines . The pair of broken lines between the electrode pairs indicates the pulsating applied electric current density field lines .

The experimental model. (a) A scaled drawing of one quarter length of the floating section of the axisymmetric model is shown. The floating cylinder is populated with three-dimensional actuators shown in Fig. 2. The electrodes run azimuthally with the north and the south poles lying in between. The inner cutout in (a) shows the steel rings under the magnets. The four screws per quarter length of the floating cylinder are visible on the right in (a) over the bus bar. They help align and hold down the kapton sheet of the electrodes over the magnets. The actuator dimension best matched one unit turbulence production domain at . (b) Photograph of the entire axisymmetric model and the sting support (on left).

The experimental model. (a) A scaled drawing of one quarter length of the floating section of the axisymmetric model is shown. The floating cylinder is populated with three-dimensional actuators shown in Fig. 2. The electrodes run azimuthally with the north and the south poles lying in between. The inner cutout in (a) shows the steel rings under the magnets. The four screws per quarter length of the floating cylinder are visible on the right in (a) over the bus bar. They help align and hold down the kapton sheet of the electrodes over the magnets. The actuator dimension best matched one unit turbulence production domain at . (b) Photograph of the entire axisymmetric model and the sting support (on left).

Schematic of the axisymmetric experimental model showing the location of the floating electromagnetic actuator section and the boundary layer trip. The stainless steel tube sting support passes through the entire model, which is hollow, through the tail end on right [see Fig. 3(b)].

Schematic of the axisymmetric experimental model showing the location of the floating electromagnetic actuator section and the boundary layer trip. The stainless steel tube sting support passes through the entire model, which is hollow, through the tail end on right [see Fig. 3(b)].

Laser Doppler measurements (symbols) of the mean velocity profile at the downstream end of the floating section in wall layer scales when the actuators have been turned off. The solid line is a smooth flat-wall log law .

Laser Doppler measurements (symbols) of the mean velocity profile at the downstream end of the floating section in wall layer scales when the actuators have been turned off. The solid line is a smooth flat-wall log law .

Laser Doppler measurements (symbols) of the streamwise turbulence profile at the downstream end of the cylindrical floating section when actuators have been turned off compared with flat-plate zero-pressure-gradient measurements due to Klebanoff (Ref. 39) (solid line; Reynolds number of ) and Purtell *et al.* (Ref. 37) (broken lines, which show the upper and lower bounds of data in the range of Reynolds numbers from 485 to 5100).

Laser Doppler measurements (symbols) of the streamwise turbulence profile at the downstream end of the cylindrical floating section when actuators have been turned off compared with flat-plate zero-pressure-gradient measurements due to Klebanoff (Ref. 39) (solid line; Reynolds number of ) and Purtell *et al.* (Ref. 37) (broken lines, which show the upper and lower bounds of data in the range of Reynolds numbers from 485 to 5100).

Laser Doppler measurements (symbols) of the surface-normal turbulence profile at the downstream end of the floating section when actuators have been turned off compared with the measurements due to Klebanoff (Ref. 39) (solid line, ).

Laser Doppler measurements (symbols) of the surface-normal turbulence profile at the downstream end of the floating section when actuators have been turned off compared with the measurements due to Klebanoff (Ref. 39) (solid line, ).

Reynolds number dependence of the efficiency of drag reduction in two-dimensional and three-dimensional actuators. Filled diamonds and squares are from channel flow measurements in Ref. 16 at and 418 (defined with half channel height and ), respectively. Cut X symbols are DNS data in turbulent boundary layers at from Ref. 18. Triangles and x symbols are the DNS data (Ref. 18) scaled to Reynolds numbers of 289 and 418, respectively. The solid and broken lines are least-square linear fits. Open circles are present data on three-dimensional actuators in turbulent boundary layers over axisymmetric bodies at .

Reynolds number dependence of the efficiency of drag reduction in two-dimensional and three-dimensional actuators. Filled diamonds and squares are from channel flow measurements in Ref. 16 at and 418 (defined with half channel height and ), respectively. Cut X symbols are DNS data in turbulent boundary layers at from Ref. 18. Triangles and x symbols are the DNS data (Ref. 18) scaled to Reynolds numbers of 289 and 418, respectively. The solid and broken lines are least-square linear fits. Open circles are present data on three-dimensional actuators in turbulent boundary layers over axisymmetric bodies at .

Comparison of drag reduction between two- and three-dimensional actuators in turbulent boundary layers. The value of is 1.0 in two-dimensional actuators and is taken to be 0 in the present three-dimensional actuators. Filled diamonds: present work on axisymmetric bodies with numerous three-dimensional actuators at . Open circles are measurements due to Pang and Choi (Ref. 17) at . Plus symbols are DNSs due to Berger *et al.* (Ref. 18) at . Solid line is a trend line due to Pang and Choi (Ref. 17) drawn through theirs and Berger *et al.*’s (Ref. 18) two-dimensional actuator data.

Comparison of drag reduction between two- and three-dimensional actuators in turbulent boundary layers. The value of is 1.0 in two-dimensional actuators and is taken to be 0 in the present three-dimensional actuators. Filled diamonds: present work on axisymmetric bodies with numerous three-dimensional actuators at . Open circles are measurements due to Pang and Choi (Ref. 17) at . Plus symbols are DNSs due to Berger *et al.* (Ref. 18) at . Solid line is a trend line due to Pang and Choi (Ref. 17) drawn through theirs and Berger *et al.*’s (Ref. 18) two-dimensional actuator data.

Laser Doppler measurements of the surface-normal turbulence near wall when the electrodes are powered (filled symbol) or not (open symbol). The solid line is a least square third order polynomial fit to the reference (open symbol) data. The pulsing frequency is bipolar, the freestream speed is , the conductivity of the water is , the electrode voltage and currents are and , respectively.

Laser Doppler measurements of the surface-normal turbulence near wall when the electrodes are powered (filled symbol) or not (open symbol). The solid line is a least square third order polynomial fit to the reference (open symbol) data. The pulsing frequency is bipolar, the freestream speed is , the conductivity of the water is , the electrode voltage and currents are and , respectively.

## Tables

Lorentz actuators: Summary of experiments reported.

Lorentz actuators: Summary of experiments reported.

Lorentz actuators: Summary of simulations reported.

Lorentz actuators: Summary of simulations reported.

Estimates of the frequency of Lorentz pulsing in the present experiments based on bursting frequency and on small-scale local Stokes flow modeling.

Estimates of the frequency of Lorentz pulsing in the present experiments based on bursting frequency and on small-scale local Stokes flow modeling.

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