Control of the viscous sublayer for drag reduction - John Kim

PHYSICS OF FLUIDS

VOLUME 14, NUMBER 7

JULY 2002

Control of the viscous sublayer for drag reductiona… Changhoon Lee School of Electrical and Mechanical Engineering, Yonsei University, Seoul, Korea

John Kim Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, California 90095

共Received 12 September 2001; accepted 7 January 2002; published 5 June 2002兲 We investigate the possibility of manipulating turbulence structures in the viscous sublayer for the purpose of drag reduction using a direct numerical simulation of a turbulent channel flow. Recognizing that a great portion of production of vorticity occurs in the viscous sublayer, a body force is used to suppress spanwise velocity in the sublayer, and a significant amount of drag reduction is obtained. A more realistic body force or wall movement in the spanwise direction using instantaneous wall-shear stress in a closed-loop control is shown to reduce drag as much as 35%. Implementation of such a body force using an electromagnetic force is also discussed. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1454997兴

I. INTRODUCTION

investigate the possibility of such manipulation of the sublayer structures for the purpose of drag reduction. The paper is organized as follows. Section II describes the vorticity dynamics in the near-wall region to show that the structures in the sublayer play important roles in selfsustaining turbulent boundary layers. Section III describes body forces and a wall motion that can be used to suppress turbulence structures in the sublayer. Further discussions regarding implementation of the body forces are given in Sec. IV, followed by a short summary in Sec. V.

Streamwise vortices in turbulent boundary layers are known to be responsible for high skin-friction drag. Many attempts have been made to manipulate these vortices in order to achieve a reduction in skin friction. Opposition control,1 optimal control,2 control using neural networks,3 suboptimal control,4 and vorticity-flux control5 are just a few examples of such efforts. In these studies, substantial drag reduction has been achieved with blowing and suction at the wall as the control input. The reduction is made possible either by directly suppressing streamwise vortices in the buffer layer1 or by regulating their effects on the wall.3,4 Full-grown streamwise vortices are generally found in the buffer layer between y ⫹ ⫽10 and 50, where y ⫹ denotes distance from the wall in viscous units. This is why researchers have difficulty finding a good correlation between a flow variable measurable at the wall and streamwise vortices in the buffer layer. Relatively little attention has been paid to the viscous sublayer itself in drag-reduction studies. Here, a viscous sublayer is meant to be a region between the wall and y ⫹ ⫽10. The low- and high-speed streaky structures present in the viscous sublayer, however, play important roles in the dynamics of wall turbulence. The streaky structures 共usually consisting of spanwise inflection points in the streamwise velocity distribution兲 are unstable, and their wake-like instability leads to streamwise vortices.6,7 The mean shear in the streamwise direction, dU/dy, which has a large value in the viscous sublayer, also plays an important role in maintaining near-wall turbulence 共see Sec. II兲. We therefore hypothesize that manipulation of turbulence structures in the sublayer alone can lead to an alteration of near-wall vortical structures, which in turn leads to drag reduction. In this paper, we

II. NEAR-WALL VORTICITY DYNAMICS

The evolution equation for vorticity can be obtained by taking curl of the Navier–Stokes equations. The resulting equation is

⳵␻ i ⳵ 2␻ i ⳵␻ i ⳵ui ⫹␻ j ⫹␯ , ⫽⫺u j ⳵t ⳵x j ⳵x j ⳵x j⳵x j

where ␻ i and u i represent vorticity and velocity, respectively, with i⫽1,2,3 or interchangeably x,y,z denoting the streamwise, wall-normal, and spanwise directions, respectively. The second term on the right-hand side represents the tilting and stretching effect of vortical structures. For ␻ x and ␻ y , which for fully developed channel flow have disturbance components only, the equations can be written as

⳵␻ x ⳵u ⳵u ⳵u ⫽¯⫹ ␻ x ⫹ ␻ y ⫹ ␻ z ⫹¯ , ⳵t ⳵x ⳵y ⳵z

共2兲

⳵␻ y ⳵v ⳵v ⳵v ⫽¯⫹ ␻ x ⫹ ␻ y ⫹ ␻ z ⫹¯ , ⳵t ⳵x ⳵y ⳵z

共3兲

where u, v are streamwise and wall-normal components of velocity. The dominant term in the ␻ x -equation in the nearwall region is the sum of the second and third terms, which reduces to ⫺ ( ⳵ w/ ⳵ x)( ⳵ u/ ⳵ y). Sendstad and Moin8 have shown that the greatest contribution to ␻ j ( ⳵ u/ ⳵ x j ) is from

a兲

This paper is submitted in conjunction with the Lumley Symposium, which was held at Cornell University, 24 –25 June 2001, in honor of Professor John Lumley on the occasion of his 70th birthday.

1070-6631/2002/14(7)/2523/7/$19.00

共1兲

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© 2002 American Institute of Physics

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C. Lee and J. Kim

⫺ ( ⳵ w/ ⳵ x)( ⳵ u/ ⳵ y). The dominant term in the ␻ y equation is the third term, which can be expressed as ( ⳵ v / ⳵ z) ⫻( ⳵ u/ ⳵ y), where ␻ z is approximated by ⳵ u/ ⳵ y. The importance of this term in sustaining turbulence was investigated by Kim and Lim.9 Their result shows that suppression of this term completely annihilates near-wall turbulence structures. Neglecting the advection terms 共since they do not contribute to either production or dissipation兲 yields

冉 冉

冊 冊

⳵␻ x ⳵w ⳵u ⳵ 2␻ x ⳵ 2␻ x ⳵ 2␻ x ⯝⫺ ⫹␯ ⫹ ⫹ , ⳵t ⳵x ⳵y ⳵x2 ⳵y2 ⳵z2

共4兲

⳵␻ y ⳵v ⳵u ⳵ 2␻ y ⳵ 2␻ y ⳵ 2␻ y ⯝⫺ ⫹␯ ⫹ ⫹ . ⳵t ⳵z ⳵y ⳵x2 ⳵y2 ⳵z2

共5兲

Multiplying Eqs. 共4兲 and 共5兲 by ␻ x and ␻ y , respectively, and adding the equations together, followed by averaging over an x – z plane gives an evolution equation for partial enstrophy 共defined as the sum of squared components of the streamwise and wall-normal vorticity兲:

冉

冊

⳵ ␻ 2x ⫹ ␻ 2y ⯝ P ␻x⫹ P ␻y⫺ ⑀ ␻ ⳵t 2

共6兲

with P ␻ x ⫽⫺ ␻ x

⳵ w dU , ⳵ x dy

共7兲

P ␻ y ⫽⫺ ␻ y

⳵ v dU , ⳵ z dy

共8兲

⑀ ␻⫽ ␯

冉冉 冊 冉 冊 冉 冊 冉 冊 冉 冊 冉 冊冊

⫹

⳵␻ x ⳵x

⳵␻ y ⳵z

2

⫹

⳵␻ x ⳵y

2

⫹

⳵␻ x ⳵z

2

⫹

⳵␻ y ⳵x

2

⫹

⳵␻ y ⳵y

2

2

.

共9兲

Here, the overbar denotes averaging over an x – z plane. Note that this equation is similar to the turbulent kinetic energy equation. The role of each term on the right-hand side is very similar to that in the turbulent kinetic energy equation, i.e., the first two terms represent production of enstrophy, whereas the last term represents dissipation of enstrophy. For example, distributions of P ␻ x , P ␻ y , and ⑀ ␻ near the wall in fully developed channel flow at Re␶⫽110 are shown in Fig. 1. Note that P ␻ x and P ␻ y are positive everywhere except very close to the wall, indicating that when dU/dy is positive, ␻ x and ⫺ ⳵ w/ ⳵ x 共or ␻ y and ⫺ ⳵ v / ⳵ z) are positively correlated almost everywhere. Physically, this means that since the first term on the right-hand side of Eqs. 共4兲 and 共5兲 represents a tilting process, the tilting mechanism in the viscous sublayer extracts streamwise and wall-normal enstrophy from the mean flow. Also, as shown in Fig. 1, P ␻ y contributes twice as much to the total production as P ␻ x does. Total production is greater than dissipation in the range 6 ⬍y ⫹ ⬍17, and a large amount of production occurs in the viscous sublayer due to the large value of dU/dy near the wall. P ␻ x and P ␻ y act as sources for ␻ 2x and ␻ 2y , respectively. Both P ␻ x and P ␻ y are proportional to ␻ x ␻ y (dU/dy),

FIG. 1. Vorticity production and dissipation distributions near the wall. Dashed line, dissipation; solid lines, productions. All quantities are normalized by u ␶ and h.

since ⫺ ⳵ w/ ⳵ x and ⫺ ⳵ v / ⳵ z contribute to ␻ y and ␻ x , respectively. This implies that near-wall structures with the same sign of ␻ x and ␻ y can grow and survive, whereas structures with the opposite sign of ␻ x and ␻ y decay. Streamwise vortices are good examples of a turbulence structure with the same-signed ␻ x and ␻ y , since the vortices are usually slanted outward from the wall. Therefore, ␻ x and ␻ y grow together and one cannot maintain itself without the other. The streaky structures, which are usually associated with wall-normal vorticity, can easily create streamwise vorticity. Streamwise vortices in the buffer layer can produce wall-normal vorticity. This does not necessarily mean that the streaky structures directly create streamwise vortices, or vice versa. What we have shown here, however, is that in the viscous sublayer the same-signed ␻ x and ␻ y are continuously produced together, which is essential for sustaining near-wall turbulence structures. This suggests that preventing production of vorticity in the sublayer alone might suppress the self-maintaining mechanism of turbulent boundary layers. One possible mechanism by which streamwise vortices are generated is self-regeneration. Streamwise vortices typically grow in size and get stronger as the mean shear stretches them. Some investigators10–13 have shown that fullgrown streamwise vortices create opposite-signed small streamwise vortices in the region between the parent vortices and the wall through viscous interaction with the wall. Newly generated vortices, which are initially small and therefore confined inside the viscous sublayer, will continuously grow and generate another one. Some of these new vortices may grow into streamwise vortices in the buffer layer, and some disappear quickly due to viscous dissipation. Overall, the viscous sublayer can be viewed as a birthplace of vorticity. We shall show that manipulation of the viscous sublayer can result in a profound drag reduction. In addition, the presence of vortical structures in the buffer layer can be identified in the viscous sublayer, although the sublayer is very close to the wall. For example, Fig. 2 shows wall-normal vorticity distributions together with u ⬘ ,w vectors on three different x – z planes, y ⫹ ⫽5,10,15. The vorticity distribution and velocity vectors in

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Phys. Fluids, Vol. 14, No. 7, July 2002

Control of the viscous sublayer for drag reduction

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with f i ⫽⫺

FIG. 2. The wall-normal vorticity and u,w vectors at three different x – z planes: 共a兲 y ⫹ ⫽5; 共b兲 y ⫹ ⫽10; 共c兲 y ⫹ ⫽15. Dark and bright colors denote negative and positive values of vorticity, respectively.

all three are strikingly similar to each other, implying that the streamwise vortices in the buffer layer, which can be characterized by spanwise velocity and wall-normal vorticity 共‘‘streamwise’’ vortices have a wall-normal component of vorticity, as they are slightly slanted outward from the wall兲, leave their footprints in the viscous sublayer. This suggests that manipulation of turbulence structures based on information from the viscous sublayer alone could influence turbulence structures in the buffer layer. III. SUBLAYER CONTROL A. Body force control

In this section, we investigate whether suppression of near-wall disturbances in the viscous sublayer can indeed lead to suppression of the self-sustaining mechanism of turbulent boundary layers, subsequently leading to drag reduction. We focus on suppression of velocity disturbances, instead of vorticity disturbances, since manipulation of vorticity is much more difficult from an implementation point of view. We first show that a body force, which is designed to suppress velocity disturbances in the sublayer, can influence disturbances in the region outside the sublayer, resulting in a significant amount of drag reduction. The Navier–Stokes equations with the body force reads

⳵ui ⳵ui ⳵p ⳵ 2u i ⫽⫺u j ⫺ ⫹␯ ⫹fi ⳵t ⳵x j ⳵xi ⳵x j⳵x j

共10兲

1 ⫺y/⌬ e 共 u i ⫺u i 兲 , ⌬t

共11兲

where p denotes pressure and ⌬t is the time step used in our numerical simulation. The forcing term is designed such that any velocity fluctuations would be suppressed in a time scale of e y/⌬ ⌬t. For example, when ⌬ ⫹ ⫽3, the forcing suppresses the disturbance field in a time scale of several ⌬t in the sublayer region. The suppression time scale in the region of y ⫹ ⬎10, however, corresponds to more than 30⌬t, which is roughly 10 wall units. The flow evolves substantially over this period, thus the forcing hardly influences the flow in this region. We tested one component of the forcing at a time. We chose a fully developed channel flow for our tests, with a code similar to one used in our previous studies of turbulence control.3,4 The Fourier and Chebyshev expansions were used in the streamwise and spanwise directions, and the wall-normal direction, respectively. The Reynolds number based on the wall-shear velocity and the channel half-height was 110. The resolution was 32⫻65⫻32 in the streamwise, wall-normal, and spanwise directions. Similar but simpler spanwise suppressions were studied by Satake and Kasagi.14 Time evolution of the mean wall shear for two different ⌬ ⫹ , 3 and 5, are shown in Fig. 3. Suppression of spanwise disturbances turned out to be most effective in reducing drag for both cases.15 With ⌬ ⫹ ⫽5, all three cases reduce drag substantially, since the forcing at around y ⫹ ⫽10 was strong enough to directly influence streamwise vortices. With ⌬ ⫹ ⫽3, however, only spanwise suppression reduced drag as much as the ⌬ ⫹ ⫽5 case. Considering that with ⌬ ⫹ ⫽3 the forcing amplitude decays to 4% of the maximum value at y ⫹ ⫽10, it is remarkable that such a local spanwise forcing could eventually alter most structures of the flow such that it almost becomes laminar. We also tested the same force as Eq. 共11兲 with the force vanishing in the region, y ⫹ ⬎10 so that only the sublayer structures were influenced 共see Fig. 3兲. Almost the same amount of drag reduction was observed. Compared to the spanwise suppression, neither the streamwise nor the wall-normal suppression performed as well. This was probably because spanwise velocity is the only quantity included in both streamwise and wall-normal vorticity, which are key elements in streak instability.16 Suppression of spanwise disturbances in the sublayer prevents interaction between ␻ x and ␻ y , and thus strongly affects the streak instability. The reduction of spanwise velocity in the sublayer also interferes with self-regeneration of streamwise vortices in the sublayer, since the spanwise velocity resulting from the streamwise vortices interacting with the wall is an essential part of the self-regeneration mechanism. B. Wall-shear based control

The force introduced in the Sec. III A 关Eq. 共11兲兴, which is proportional to local velocity, is difficult to implement. We sought a more realistic control input, which would perform almost the same way. The fact that suppression of spanwise velocity in the sublayer alone could significantly reduce drag led us to seek a spanwise control based on wall-shear stress

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Phys. Fluids, Vol. 14, No. 7, July 2002

C. Lee and J. Kim

FIG. 3. Drag history for various control cases using an idealized body force 关Eq. 共11兲兴 for different penetration depths: thick solid lines, ⌬ ⫹ ⫽5; dotted lines, ⌬ ⫹ ⫽3; thin solid line, ⌬ ⫹ ⫽3 with f i ⫽0 for y ⫹ ⬎10. Note that the drag is normalized by mean drag of the no-control case.

in the spanwise direction. This is another reason that we focused on sublayer dynamics: that is, the sublayer is close enough to the wall that the sublayer information can be detected easily at the wall. Two controls based on spanwise wall-shear stress were tested: a local spanwise body force with wall-shear stress as the amplitude, and spanwise wall movement in the direction opposite to the motion near the wall. Using the Taylor series expansion, the body force in Eq. 共11兲 can be approximated by a more realistic body force f z ⫽⫺

⳵w 1 ye ⫺y/⌬ ⌬t ⳵y

冏

,

共12兲

wall

which is a good approximation to the idealized body force near the wall. This force was tested in the same channel flow and the result is shown in Fig. 4. Although this force did not reduce drag by the same amount as did the idealized force, it reduced drag by as much as 35%. Realizing that the spanwise velocity can be approximated by the Taylor series expansion only up to y ⫹ ⫽3 or 5, it is remarkable that this control reduces drag significantly. This suggests that suppres-

sion of spanwise disturbances very close to the wall alone is sufficient to alter near-wall turbulence structures 共including those in the buffer layer兲 and to achieve a significant drag reduction. Contours of streamwise vorticity in a y – z cross section for a control using the above-mentioned force 关Eq. 共12兲兴 is shown in Fig. 5. The upper wall is a controlled wall with the lower wall intact. It can be seen that streamwise vorticity near the upper wall was suppressed almost completely compared to the uncontrolled wall. Near the lower wall, one can see several small streamwise vortices in the sublayer region. Distribution of the rms fluctuations of velocity and vorticity near the upper wall are shown in Fig. 6. All quantities are normalized by u ␶ of the uncontrolled flow and h, the channel half-height. It is shown that not only the spanwise velocity component but also the streamwise and wall-normal components were substantially suppressed. Streamwise and wall-normal vorticity, streamwise in particular, were also significantly reduced, suggesting that the growth of ␻ x and ␻ y , through interaction between them, was greatly prohibited by

FIG. 4. Drag history for cases using the wall-shear based body force or wall movement compared with the idealized body force case. Thick solid line denotes the idealized force case with ⌬ ⫹ ⫽3, thin solid line, the wall-shear based body force 关Eq. 共12兲兴 with ⌬ ⫹ ⫽3, and dotted line, the wall-shear based wall movement control 关Eq. 共13兲兴. The drag is normalized by mean drag of the no-control case.

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Phys. Fluids, Vol. 14, No. 7, July 2002

Control of the viscous sublayer for drag reduction

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FIG. 5. Contours of streamwise vorticity for the wallshear based body force control 关Eq. 共12兲兴 in a z – y plane. Contours of negative values are dashed. Only the upper wall is controlled.

suppression of spanwise velocity in the sublayer. Distribution of vorticity production and dissipation are shown in Fig. 7. Compared with the no-control case 共Fig. 1兲, the wall-information based spanwise control effectively suppresses production in the sublayer region. Note the difference in the ordinate scale in Figs. 1 and 7. This result supports the notion that suppression of turbulence structures in the sublayer can result in suppression of the self-sustaining mechanism in the entire near-wall region.

FIG. 6. Turbulence intensities near the controlled wall for the wall-shear based body force control case 关Eq. 共12兲兴 compared with the no-control case: 共a兲 the rms velocity; 共b兲 rms vorticity. Dashed lines denote no-control case and thick solid lines, control case. All quantities are normalized by u ␶ of the no-control case and h.

Next, we tested spanwise wall movement in the direction opposite to the near-wall spanwise velocity. The wall boundary condition for spanwise velocity was given by

冉 冏

w n⫹1 兩 wall⫽⫺ w n

wall⫹ ␦

冏 冊

⳵wn ⳵y

,

共13兲

wall

where the superscript n denotes time steps. Note w 兩 wall is a function of x and z as it depends on ⳵ w/ ⳵ y 兩 wall(x,z). It was found that ␦ should be carefully chosen and the optimum value is about 3 wall-units distance. Also, application of the above-mentioned boundary condition for every time step led to a numerical instability, since a sudden change in the wall boundary condition generated a very thin boundary layer structure in velocity distribution in one time step, and the wall shear did not have proper information on the structures near the wall. To overcome this problem, the feedback was kept the same for several time steps. The optimal time period of sensing can be estimated as follows: Time required for viscous diffusion in distance l in the wall-normal direction, T, should be approximately l 2 / ␯ . When translated into the 2 wall units, T ⫹ ⯝l ⫹ . If l ⫹ is chosen to be 2, then T ⫹ is about 4. Mean-shear stress behavior is shown together with other cases in Fig. 4. The spanwise wall motion did not perform well compared to body forces, but still reduced drag by 20%. The reason for this partial success can be understood by

FIG. 7. Vorticity production and dissipation distributions near the controlled wall for the wall-shear based body force control case 关Eq. 共12兲兴: dashed line, dissipation; solid lines, productions. All quantities are normalized by u ␶ of the no-control case and h.

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FIG. 8. A schematic of an electromagnetic tile that produces a near-wall spanwise body force. Solid and dashed arrows denote magnetic and electric fields, respectively.

realizing that relying solely on viscous diffusion 共the case with wall movement as the control input兲 for modification of near-wall structures is not as effective as the body force, since interactions between vorticity components are occurring at a much higher rate, on the order of 兩 dU/dy 兩 . IV. DISCUSSION

The spanwise movement of the wall is difficult to implement, and no one has ever built such a control device, although there have been some attempts to make such wall actuators.17 Creation of a local spanwise body force, however, may be possible by using an electromagnetic force. Recent applications of arrays of electrodes and magnets to generate the Lorentz body force18,19 illustrate a possibility of using the Lorentz force for active turbulence control in conducting fluids. Creation of a local force in one direction 共spanwise for the present actuation兲, however, is not easy, since the Lorentz force created by an array of electrodes and magnets is three dimensional in general. In order to create a body force primarily in the spanwise direction, an electromagnetic tile could be arranged so that the influence of other components is minimized. An example of such a tile arrangement is shown in Fig. 8. The wall-normal and streamwise components are small, since the magnetic field and electric fields are concentrated in the region where the force is mainly spanwise. The direction of the force can be changed by flipping the signs of the electrodes. A surface covered by an array of the tile shown in Fig. 8 can create a distribution of spanwise force over the surface. The size of each tile that creates the force localized to the sublayer region should be about 30⫻30 in wall units in the streamwise and spanwise directions. The spanwise force decays roughly exponentially along the wall-normal direction with the maximum value at the wall, different from Eq. 共12兲 which has the maximum at y ⫽⌬. This should not cause a serious problem, since the force at or near the wall cannot influence the near-wall flow, due to the no-slip condition at the wall. We tested an exponentially decaying body force instead of Eq. 共12兲 with a similar amplitude, and it was found that in order to achieve the same amount of drag reduction, we would need an underrelaxation of the wall-shear stress information with the value at the previous time step. With the use of an electromagnetic force for flow control, efficiency has been a concern, especially in the applica-

tion of low-conducting fluids such as seawater. The maximum rms force of Eq. 共12兲 is max f rms ⫽

⌬ ⳵w e⌬t ⳵ y

冏

.

共14兲

rms

With ⌬t ⫹ ⫽0.4,⌬ ⫹ ⫽3 and ( ⳵ w ⫹ / ⳵ y ⫹ ) 兩 rms⫽0.4, the normalized rms force, defined as the rms Stuart number Strms max 2 (⬅ f rms /u␶ /␦), is equal to 110, which is greater than openloop controls tested before.18 However, when a larger time scale is used instead, the efficiency can be improved since performance of the control does not deteriorate as quickly. For example, when the force amplitude was reduced to 1/10th of the current value, almost the same amount of drag reduction was obtained, implying that the required Stuart number can be reduced to 11. The only difference was in how quickly drag was reduced. Even with 1/20th amplitude, 20% reduction was observed. Furthermore, once the drag is fully reduced, the required Stuart number becomes much smaller since the wall-shear stress is almost completely suppressed. Figure 6共b兲 clearly shows that streamwise vorticity at the wall, which is the rms spanwise wall-shear stress, is suppressed to 1/40th of the level of the uncontrolled flow, implying that the required rms Stuart number reduces to 0.275. This order of the Stuart number with the corresponding amount of drag reduction would lead to a net gain. V. SUMMARY

We investigated vorticity dynamics in the viscous sublayer and found that a significant portion of vorticity production occurs in the sublayer. In spite of strong dissipation adjacent to the wall, streamwise and wall-normal vorticity is continuously produced in the sublayer, which is essential in sustaining near-wall turbulence. By suppressing spanwise disturbances in the sublayer, we can suppress near-wall turbulence structures, which led to a substantial drag reduction. A more realistic body force that uses the spanwise wall-shear information was shown to reduce drag by as much as 35%. The present results confirm the notion that manipulation of the sublayer alone can alter the entire near-wall structures in turbulent boundary layers. We plan to implement the Lorentz force using the aforementioned electromagnetic tiles to investigate whether one can achieve a net drag reduction. Finally, it ought to be mentioned that the present results are wholly based on low-Reynolds number simulations, and that whether the same control schemes would be as effective for

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Phys. Fluids, Vol. 14, No. 7, July 2002

high-Reynolds number boundary layers remains to be investigated. We plan to investigate the effect of the Reynolds number as well as the effect of surface roughness in our future study. ACKNOWLEDGMENTS

A portion of this work was carried out while C.L. was visiting UCLA. This work has been supported by KOSEF Grant No. 304-03-2 and ONR Grant No. N00014-95-1-0352. Computer time has been provided by the NSF’s Partnership for Advanced Computational Infrastructure 共PACI兲 Program, and the KORDIC Supercomputing Center in Korea. 1

H. Choi, P. Moin, and J. Kim, ‘‘Active turbulence control for drag reduction in wall-bounded flows,’’ J. Fluid Mech. 262, 75 共1994兲. 2 T. Bewley and P. Moin, ‘‘Optimal control of turbulent channel flows,’’ Active Control of Vibration and Noise, ASME DE 75, 221 共1995兲. 3 C. Lee, J. Kim, D. Babcock, and R. Goodman, ‘‘Application of neural networks to turbulence control for drag reduction,’’ Phys. Fluids 9, 1740 共1997兲. 4 C. Lee, J. Kim, and H. Choi, ‘‘Suboptimal control of turbulent channel flows for drag reduction,’’ J. Fluid Mech. 358, 245 共1998兲. 5 P. Koumoutsakos, ‘‘Vorticity flux control for a turbulent channel flow,’’ Phys. Fluids 11, 248 共1999兲. 6 F. Waleffe and J. Kim, ‘‘How streamwise rolls and streaks self-sustain in a shear flow,’’ in Self-Sustaining Mechanisms of Wall Turbulence, edited by R. L. Panton 共Computational Mechanics, Southampton, UK, 1997兲, p. 309. 7 W. Schoppa and F. Hussain, ‘‘Genesis and dynamics of coherent structures in near-wall turbulence: a new look,’’ in Ref. 6, p. 385. 8 O. Sendstad and P. Moin, ‘‘The near wall mechanics of three-dimensional

Control of the viscous sublayer for drag reduction

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turbulent boundary layers,’’ Report No. TF-57, Department of Mechanical Engineering, Stanford University, Stanford, CA, 1992. 9 J. Kim and J. Lim, ‘‘A linear process in wall-bounded turbulent shear flows,’’ Phys. Fluids 12, 1885 共2000兲. 10 J. Kim, P. Moin, and R. D. Moser, ‘‘Turbulence statistics in fully developed channel flow at low Reynolds number,’’ J. Fluid Mech. 177, 133 共1987兲. 11 J. W. Brooke and T. J. Hanratty, ‘‘Origin of turbulence-producing eddies in a channel flow,’’ Phys. Fluids A 5, 1011 共1993兲. 12 P. S. Bernard and J. M. Wallace, ‘‘Vortex kinematics, dynamics, and turbulent momentum transport in wall bounded flows,’’ in Ref. 6, p. 65. 13 T. J. Hanratty and D. V. Papavassiliou, ‘‘The role of wall vortices in producing turbulence,’’ in Ref. 6, p. 83. 14 S. Satake and N. Kasagi, ‘‘Turbulence control with a wall-adjacent thin layer of spanwise damping force,’’ in the Tenth Symposium on Turbulent Shear Flows, The Pennsylvania State University, 14 –16 August 1995. 15 The mechanism by which riblet suppresses the near-wall turbulence is similar in a sense that spanwise motion associated with the streamwise vortices is suppressed, hence drag is reduced. See H. Choi, P. Moin, and J. Kim, ‘‘Direct numerical simulation of turbulent flow over riblet,’’ J. Fluid Mech. 255, 503 共1993兲. 16 J. M. Hamilton, J. Kim, and F. Waleffe, ‘‘Regeneration mechanisms of near-wall turbulence structures,’’ J. Fluid Mech. 287, 317 共1995兲. 17 F. Sherman, S. Tung, C.-J. Kim, C.-M. Ho, and J. Woo, ‘‘Flow control by using high-aspect-ratio, in-plane microactuators,’’ Sens. Actuators A 73, 169 共1999兲. 18 C. Henoch and J. Stace, ‘‘Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force,’’ Phys. Fluids 7, 1371 共1995兲. 19 T. Berger, J. Kim, C. Lee, and J. Lim, ‘‘Electromagnetic force control of turbulent boundary layers for drag reduction,’’ Phys. Fluids 12, 631 共2000兲.

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