Actuation and control of a turbulent channel flow ... - Semantic Scholar

PHYSICS OF FLUIDS

VOLUME 16, NUMBER 4

APRIL 2004

Actuation and control of a turbulent channel flow using Lorentz forces Kenneth S. Breuera) and Jinil Park Division of Engineering, Brown University, Providence, Rhode Island 02912

Charles Henoch Naval Undersea Warfare Center, Newport, Rhode Island 02841

共Received 22 July 2002; accepted 16 December 2003; published online 18 February 2004兲 Results concerning the design and fabrication of electromagnetic actuators, and their application to affect the wall shear stress in a fully turbulent channel flow are discussed. The actuators utilize a Lorentz force to induce fluid motion due to the interaction between a magnetic field and a current density. The actuators are comprised of spanwise-aligned rows of permanent magnets interlaced with surface-mounted electrodes, segmented to allow the Lorentz force to be propagated in the spanwise direction. Problems commonly associated with electromagnetic flow control—electrolysis, bubble formation, and electrode corrosion are substantially reduced, and in most cases eliminated by the use of a conductive polymer coating. The actuators generate velocity profiles with a penetration depth into the flow of approximately 1 mm 共set by the electrode/magnet pitch兲 and maximum velocities of approximately 4 cm/s. The actuation velocities are found to scale linearly with forcing voltage and frequency. The electrical to mechanical efficiency is found to be very low (⬇10⫺4 ), primarily due to the limitations on the magnetic field strength and the low conductivity of the working fluid 共saltwater兲. The actuators are used in a fully turbulent low Reynolds number channel flow and their effect on the turbulent skin friction is measured using a direct measurement of drag. Maximum drag reductions of approximately 10% are measured when the flow is forced using a spanwise oscillating Lorentz force. A scaling argument for the optimal amplitude of the current density is developed. The efficiency of this method for drag reduction, and its application at higher Reynolds numbers is also discussed. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1647142兴

promise in physical experiments include 共but are not limited to兲: localized heating,4 piezo-electric flaps,5 oscillatory blowing,6 synthetic jets,7 surface motion,8 and plasma discharge.9 One appealing approach for electrically conducting fluids, such as seawater, is to use the Lorentz force to induce motion in the fluid which might then lead to turbulence suppression. The Lorentz force results from the interaction between a current, I, and a magnetic field, B, and generates a body force, F L :

I. INTRODUCTION

Improved understanding of unsteady flow physics has led to a wealth of flow control applications in which fluid perturbations are deliberately introduced into a large-scale flow with the goal of affecting its global characteristics. Examples of flow control that are of interest include enhanced mixing, control of separation and reattachment, reduction of flow-induced noise, and the reduction of turbulent skin friction. The efforts toward turbulent flow control, in particular, were invigorated by numerical computations at low Reynolds number,1–3 which indicated that the turbulence and drag could be reduced by a variety of fluidic actuations. Most of these, however, were extremely difficult if not impossible to implement in any physical experiment. A pervasive challenge associated with almost all experimental flow control studies has been the development of effective, robust, and efficient actuators. Although many different actuation schemes have been proposed ‘‘on paper,’’ the successful implementation of fluidic actuation in a physical experiment is often limited by the engineering realities that must be overcome. These difficulties are present not only in the final application, but even in demonstration experiments where success is not hampered by issues of cost, weight, or efficiency. Examples of actuation methods that do show

F L⫽IBa,

where a is the characteristic length over which the current flows. For flow control, the Lorentz actuator is appealing since it has no moving parts and in principle is easy to implement. The concept of using the Lorentz force as a fluidic actuator is not new and actuators of various designs have been built and tested for a variety of flow control applications.10–12 One particularly appealing use of the Lorentz force actuators is for the control of turbulence in a lowconductivity fluid, such as saltwater. Several concepts for such control have been explored over the past few years including a series of computations13–16 in part motivated by the experiments of Nosenchuck and Brown.10 Although these approaches have not been convincingly shown to result in net drag reduction, the use of the Lorentz force actuator for turbulent drag reduction has received some renewed atten-

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Phys. Fluids, Vol. 16, No. 4, April 2004

tion most recently with the experiments of Henoch and Stace11 and the computational studies of Berger, Lee, Kim, and Lim17 and Du, Symeonidis, and Karniadakis.14,18 The computational studies demonstrated that purely spanwise actuation, either in the form of a uniform oscillating spanwise Lorentz force or a spanwise traveling Lorentz wave, can reduce the drag in a fully turbulent channel flow by as much as 30%. The motivation for the use of an unsteady spanwise control velocity derives from the recognition that disruption of the spanwise structure of the equilibrium turbulent flow by oscillatory wall motion has profound effects on the turbulence production cycle and consequently the turbulent drag.3,19 Independent of their ability to successfully control any flow, including the turbulent boundary layer, the physical implementation of Lorentz force actuators is challenging. The generation of electrolytic bubbles and electrode corrosion have plagued previous studies, and the development of a manufacturing technique that allows for large actuator areas to be fabricated while maintaining the necessary alignment and registration between electrodes and magnets has proved difficult. The present paper describes the design, manufacture, and testing of Lorentz force actuators for use in a turbulence control experiment in which spanwise-oscillating velocities are introduced into the wall region of the flow using the Lorentz effect. The first part of the paper is focused on the actuator performance and the structure of the induced velocity fields, while the second part of the paper develops scaling arguments for turbulence control using the Lorentz force and presents results demonstrating the reduction of turbulent drag due to Lorentz force control. At this stage velocity measurements of the controlled field are not available, but these experiments are currently in progress and will be reported upon in the near future. II. LORENTZ FORCE ACTUATORS A. Design and fabrication of actuator

The actuator used in the present study is based on the actuator developed by Henoch and Stace,11 which used interlaced magnets and electrodes to generate an approximately uniform Lorentz force. In the case of Henoch and Stace, the force was aligned with the streamwise direction, while the current experiment rotates the actuator system by 90° so that the Lorentz force acts in the spanwise direction. The present system, illustrated in Fig. 1, consists of linear arrays of permanent magnets lined up in the spanwise 共z兲 direction and aligned such that the surface exposed to the flow has alternating polarity: N-S-N-S, etc. Between each of these magnet rows are a series of electrodes, also energized with alternating polarity: ⫺, ⫹, ⫺, ⫹, etc. The bulk of the electric and magnetic field lines are aligned in the streamwise direction, and the two are always crossing each other such that the Lorentz force acts in the direction spanwise to the core flow. The direction and magnitude of the Lorentz force is controlled by varying the electric field strength and polarity. The control of the spanwise structure of the Lorentz force is achieved by fabricating the actuator with multiple

Breuer, Park, and Henoch

FIG. 1. Schematic of the Lorentz force actuator showing side and top views, respectively. The Lorentz force acts in the spanwise direction, out of the page on the side view and from top to bottom on the top view. The contours of the electrodes are designed to reduce spanwise fringing fields which generate parasitic components of the Lorentz force.

phases in the spanwise direction. For the present discussion, we restrict ourselves to sinusoidal forcing voltages and, in a further restriction for simplicity’s sake, we have linked all the electrodes in a single streamwise column together so that there is no streamwise variation in the generated Lorentz force. With this subset of available forcing signals, the Lorentz force generated in the fluid is F 共 x,y,z,t 兲 ⫽ f 共 x,y 兲 sin共 ␻ t⫺␭z 兲 ,

共2兲

where ␻ is the forcing frequency and ␭ is the spanwise wavelength. The phase speed of the disturbance is given by c⫽ ␻ /␭. The ‘‘structure’’ function, f (x,y) is determined solely by the geometry of the electric and magnetic fields. Actuating all of the spanwise phases in unison 共i.e., with zero phase shift兲 generates a uniform Lorentz force, equivalent to the uniform spanwise oscillation simulated by Berger et al.17 Alternatively, operating the electrodes with the same amplitude and time signal, but with a spanwise phase shift generates a traveling wave, as simulated by Du et al.14,18 For the current experiments the actuator tiles were fabricated with sixteen independent phases in the spanwise direction, although only eight were tested independently. Other forcing options could also be explored. Each column of electrodes could be operated with both amplitude and phase differences between spanwise phases, in which case both traveling and standing waves can be generated. More complex 共nonsinusoidal兲 waveforms can also be imagined, as well as varying streamwise amplitude and phases. However, these possibilities were not considered in the present study. Since the present experiments are aimed at the control of wall-bounded turbulence, fabrication of a large area of Lorentz actuators was necessary. In order to achieve this,



FIG. 2. Composite photograph of the Lorentz force actuator showing the sub-structure 共flux plate, alignment pins and magnets兲 and the electrode layer. The flow in this picture is from left to right, and the Lorentz force is from top to bottom. The streamwise pitch of the magnets and electrodes is 3.175 mm. To generate a spanwise Lorentz force current is passed between electrodes 1 and 2, 3 and 4, etc. There are eight independent electrode pairs in the spanwise direction 共1–2,3– 4,...,15–16兲, and spanwise-modulated waveforms, such as traveling waves or standing waves with a spanwise wavelength, can be achieved by changing the phase between adjacent electrode pairs.

tiles of actuators were fabricated and then the complete control surface was assembled by covering the test section with an array of tiles. Each tile measures 0.305 m square and consists of a soft iron flux plate holding rows of magnets and covered by a skin containing the electrodes and bus lines. The magnets used are neodymium–boron–iron permanent magnets 共45 M–O兲 and have a 3.175 mm square cross section. Each magnet is 50 mm long, limited only by the fact that they are very brittle, and longer lengths would have proved too difficult to handle. The magnets were placed on the flux plate, and aligned by means of non-magnetic alignment pins inserted into holes throughout the flux plate 共Fig. 2兲. This made assembly of the aligned rows of magnets straightforward 共although tedious兲. The electrode layer covers the magnet layer and consists of a commercially fabricated two-layer printed circuit board. The electrodes were patterned on the top surface, while the bus lines were fabricated on the lower surface and then covered by a dielectric layer to insulate them from the rest of the system. The electrodes were connected to the bus lines using standard plated through-holes. Care was taken in the design to minimize ohmic losses in both the bus lines and throughholes. The electrodes were then screen-printed with a conductive polymer paint which served to protect them from corrosion and to minimize the effects of electrochemistry during operation. The painted electrodes protrude approximately 100 ␮m above the substrate, sufficiently small to preserve the hydrodynamic smoothness of the surface at the conditions tested. High-flexibility lead wires were soldered to the bus lines and fed through access holes in the flux plate before the entire assembly was then glued together using epoxy. Photographs of a small section of a sample tile are shown in Fig. 2, which shows the sub-surface magnet assembly and the electrode assembly, which is in contact with the flow. The electrodes are nominally rectangular with dimensions 3.175⫻15.875 共mm兲 and 3.175 mm spacing between adjacent spanwise phases. However, in order to maximize

Actuation and control of a turbulent channel flow

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FIG. 3. Measured current in response to a step increase in voltage. Measured between two parallel electrodes with uncoated and polymer coated surfaces. The ideal current is based on the bulk resistivity of the salt water, the electrode area, and their separation.

the field lines in the streamwise direction, the electrodes were designed with rounded edges 共to reduce field concentration at sharp corners兲. In addition, a ‘‘cut-back’’ between adjacent spanwise electrodes was also introduced to increase the effective electrode separation in the spanwise direction and thus reduce the component of the electric field acting in the spanwise direction 共which results in undesirable components of the Lorentz-force acting in the wall-normal and streamwise directions, rather than the desired spanwise direction兲. These modifications can be seen in the ‘‘dog-biscuit’’ shape of the electrodes shown in Fig. 2. B. Electrode performance

Initial experiments were focused on the electrical performance of the electrodes and on the effects of the conductive polymer coating. These experiments consisted of two rectangular electrodes either gold coated or conductive polymer coated, each measuring 2 cm by 4 cm and positioned facing each other, 1 cm apart. The electrode assembly was placed in a beaker of saltwater 共␴⫽5 siemens/m兲 with approximately 2 cm of the electrode below the water surface. Figure 3 shows the step response of both the gold-coated and polymer-coated electrodes. The drive voltage, V, is changed from 0 to 3 V at t⫽0, at which time the current quickly jumps, then decays exponentially before stabilizing at a steady value of about 0.6 A. In contrast to this, the polymer-coated electrode has a more complex behavior in which the current rises very quickly, then falls slowly at first and then drops more quickly. For very short times (t ⬍0.1 s), the step response seems to indicate that the gold electrodes are slightly superior to the polymer electrodes. At longer times, (t⬇1.2 s), the polymer-coated electrodes exhibit far-inferior performance with the current falling to less than a third of the current carried by the gold electrodes. However, in the intermediate times, the polymer-coated electrodes carry more current and are thus more desirable. Although the reason for this behavior is not fully understood, it


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FIG. 4. 共a兲 The normalized current (I/V peak) measured due to sinusoidal forcing at different forcing amplitudes. The traces are all synchronized with the 2 forcing voltage so that t/T⫽0 always coincides with the start of the forcing cycle. 共b兲 The integrated power 共normalized by V peak ).

appears that the polymer-coated electrodes exhibit slower electrolysis kinetics, resulting in the slower decay of current with time. However, for longer times, the polymer electrode becomes sheathed by a dielectric layer 共perhaps microbubbles generated at the electrodes and trapped in the polymer matrix兲 which severely reduces its conductivity. We note that upon repeating the experiment, no change in electrode performance is observed, suggesting that the sheathing does not reflect any permanent changes in the electrode. The amplitude and frequency response of the polymercoated electrodes are shown in Figs. 4 and 5. Here, the currents are normalized by the driving voltage and are phasealigned with the drive voltage. These responses can be summarized by the normalized power averaged over the cycle period, T: P 2 V peak

⫽

1 2 TV peak

冕

T

0

V 共 t 兲 I 共 t 兲 dt,

共3兲

The results confirm the step response results of Fig. 3. At low frequencies, the sinusoidal signal shows marked attenuation reflecting the long-time trend shown in Fig. 3. As the frequency rises, the current response becomes more sinusoidal and the delivered power rises to a plateau. Note that there is no degradation at high frequencies, and that there is almost no phase lag between the driving signal and the re-

sultant current. This confirms that the saltwater-electrode system is an almost pure resistive load with little energy storage. The amplitude response shows a slightly different characteristic. At low voltages, the electrode potential of the system is not reached and current flow is minimal. Note that this is confirmed at a driving voltage of 0.5 Vpp, which shows a nonlinear increase in current at the peak of the driving signal. For 1 Vpp and above, the signal is close to sinusoidal, and does not change with driving voltage, confirming that the system has reached a regime characterized by a linear resistive response. These results are likely to be a function of the polymer film thickness, but this was not explored in the current work. Electrochemistry at the electrodes is a necessary byproduct of this class of actuator, usually generating hydrogen and chlorine at the two electrodes. This has been a pervasive problem with the Lorentz force actuators in previous experiments resulting in both bubble generation and electrode corrosion. The gold-plated electrodes tested here were no exception and bubble formation was observed to start immediately after the voltage was applied. Although the bubble formation is problematic, more serious is the accompanying corrosion of the electrodes and it was found that after even short times, the corrosion was sufficiently serious to render the electrodes useless. In the case of the polymer-coated electrodes, how-

FIG. 5. 共a兲 Measured current flow during a complete forcing cycle between two parallel, polymer-coated electrodes. The different curves show the current measured due to sinusoidal forcing with V peak⫽1 V, and at different frequencies. The traces are synchronized to the forcing voltage so that t/T⫽0 always coincides with the start of the forcing cycle. 共b兲 The integrated power as a function of forcing frequency.




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FIG. 6. Shaded contours of spanwise velocity induced by uniform oscillatory forcing. The two frames show the spanwise velocity in the y – z plane at the maximum positive and negative limits of the forcing cycle. Contour lines are equally spaced at intervals of 0.01 m/s. White indicates maximum positive velocity of 0.05 m/s and black indicates maximum negative velocity of 0.05 m/s.

ever, no bubble formation was observed during sinusoidal operation above 0.2 Hz. Furthermore, minimal degradation of the electrode performance was observed over months of testing. Although we do not have a definitive explanation for this very encouraging result, we believe, as was commented earlier, that the polymer coat serves to slow down the kinetics of bubble formation and allow the process to remain reversible above a critical frequency so that microbubbles are formed, but then disappear without any net gas bubble generation or electrode degradation. C. Flows induced by Lorentz forces

In order to characterize the performance of the actuators in more detail, particle image velocimetry 共PIV兲 was used to measure the flows induced by the Lorentz forces. The actuator tile was placed in the center of a zero-mean-flow plexiglass tank, which measured 0.6 m by 0.6 m by 0.2 m and was filled with salt water 共␴⫽5 siemens/m兲. The actuator was divided into eight independent spanwise phases, driven by the amplified output of a Digital Signal Processing 共DSP兲 board.

electric and magnetic fields crossing in the horizontal plane and this configuration only occurs at a single point in the electrode configuration 共between two spanwise electrodes兲. The PIV data from Fig. 6 were then integrated in the z direction so that a single velocity profile, w(y), was obtained at that particular point in time. This was repeated for 16 phases during the forcing cycle, and the profiles generated are shown in Fig. 7. The velocity reaches a maximum at approximately 1 mm above the surface and dies out at a distance of about 4 mm. These length scales are solely determined by the streamwise spacing of the magnet-electrode pattern shown in Fig. 1 which sets the field penetration depth. These profiles are in excellent agreement with the velocity profiles derived by Berger et al.17 for the flow induced by a spanwise oscillating body force in a laminar flow. The maximum velocity, plotted as a function of time, indicates a sinusoidal response of the flow to the forcing voltage 共not shown here兲. These measurements were repeated over a variety of forcing voltages and frequencies ranging from 0.3 to 2 Hz

1. Uniform spanwise oscillation

The spanwise structure of the induced velocity field over a single electrode is shown in Fig. 6 for the case of uniform oscillatory forcing 共i.e., zero spanwise phase velocity, infinite wavelength兲. The PIV laser sheet aligned to illuminate the z – y plane and was positioned at the point of maximum field interaction—at the streamwise edge of the electrode. The contours of spanwise velocity are plotted in the z – y plane and are shown for two times—the positive and negative extrema of the forcing cycle. The non-uniformity in the z direction is due to the gap between adjacent electrodes in the z direction that are necessary so that the actuator can be used to propagate spanwise-traveling waves. Fortunately, the nonuniformity is minor compared to the overall fluid actuation. The wall-normal velocity, v , was also measured from the PIV data and was found to be very small, with a rms value of only 4.8% of the maximum spanwise velocity and to have little discernible spatial or temporal structure. This is to be expected, since the normal velocity is only generated by

FIG. 7. Velocity profiles due to oscillatory spanwise forcing obtained by integrating all profiles over the spanwise extent of the electrode. Sixteen profiles are plotted at equally spaced phases during the forcing cycle.


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FIG. 8. Maximum velocity generated by the Lorentz force actuator, plotted as a function of the forcing voltage. The velocities increase approximately linearly with driving voltage, although some saturation at high voltages is seen.

and from 0.5 to 3.0 V. The results are assembled in Figs. 8 and 9, which show the maximum velocity, W max , measured as a function of the forcing voltage and the electrical-tomechanical efficiency of the Lorentz force actuator 共plotted against the input electrical power兲. In the case of the velocities, we see that the induced velocity increases approximately linearly with the forcing voltage, but for a fixed voltage, decreases as the driving frequency increases. The actuator efficiency is defined as the ratio of the output flow power, crudely defined as P flow⬇

␳ 2 , 共 Area兲 ␦ p W max 2T

共4兲

to the input electrical power: 1 2

VI.

共5兲

FIG. 10. Velocity profiles at a fixed phase at different x locations. Forcing conditions: 6 Vpp, 1 Hz.

Here ␦ p and W max are the penetration depth and maximum spanwise velocity, measured from the PIV data. V and I are the peak voltage and current, respectively. The factors of 1/2 come from the integration over the sinusoidal forcing cycle 共the small phase shift between the voltage and current that is observed in Fig. 3 has also been neglected兲. When plotted this way, we see that the actuator is performing with relatively uniform efficiency over the entire operating space, although there is some falloff at low input power, as expected given the low-voltage behavior shown in the previous section 共Fig. 5兲. It is clear that the actuator exhibits very poor electricalto-mechanical efficiency—less than 10⫺4 . This is primarily due to two facts, namely the limited field strength available from the permanent magnets 共approximately 0.3 T effective field strength on the surface兲 and the fact that saltwater has a low conductivity 共5 siemens/m兲. As with any Lorentz force actuation, the force is generated by the current density, and since power varies as I 2 R, a high-resistivity material 共such as most fluids of practical interest兲 will require a substantial power dissipation before any practical fluid motion can be generated. The layout of the magnets and electrodes results in a streamwise dependence on both the magnetic and electric fields, and consequently, variation in the velocity induced by the Lorentz force is expected. Berger et al.17 show contours of the Lorentz force that vary in the x direction, with the maximum occurring at the edge of the electrode, and the minimum occurring at its center. Vertical profiles measured at these two locations are shown in Fig. 10 and the results confirm these numerical predictions, indicating that, although the profile shape appears self-similar, the maximum velocity at the center of the electrode is approximately 25% lower than at the electrode edge. 2. Traveling wave actuation

FIG. 9. Lorentz force actuator efficiency, measured as the ratio of the total fluid power induced to the electrical power. Data from five frequencies and five forcing amplitudes are shown.

Du and Karniadakis proposed18 that a traveling Lorentz force might be as effective for turbulent drag reduction as a stationary oscillatory Lorentz force. The actuators developed




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FIG. 11. Shaded contours of spanwise velocity induced by a traveling wave moving to the right 共left frame兲 and to the left 共right frame兲. The picture is a composite image, formed from eight snapshots taken at equally spaced intervals during the cycle. Contour lines are equally spaced at intervals of 0.01 m/s. White indicates maximum positive velocity of 0.05 m/s and black indicates maximum negative velocity of 0.05 m/s.

here are segmented in the spanwise direction and allow for arbitrary control of the eight spanwise phases. The flow induced by a traveling wave is shown in Fig. 11 which shows shaded contours of the spanwise velocity, plotted in the z – y plane. The figure is in fact a composite of eight separate measurements taken over the same electrode, each taken at a different phase during the forcing cycle. Both a rightwardmoving wave 共top frame兲 and a leftward moving wave 共lower frame兲 are shown. Notice that the scale in the y direction has been stretched. Although these measurements were taken with zero mean flow, the streamwise flow direction would be out of the page. The traveling wave consists of regions of both positive and negative spanwise velocity, and for this reason, an observer located at a fixed spanwise location experiences a compressional flow followed by a dilational flow. The local effect is thus similar to a uniform stationary oscillation as long as the wavelength over which the wave is imposed is large 共which in this case it is兲. It is reassuring to note that the left-moving and right-moving waves have almost identical but opposite structure, including the tilt to the velocity field in the direction of the wave’s motion. III. CONTROL OF FULLY TURBULENT FLOWS A. Scaling considerations

The present section describes the application of the Lorentz force actuators to the control of a fully turbulent lowReynolds number channel flow. However, before discussing the experimental setup and results, it is useful to review some fundamental scaling which will help to interpret the results. Here we develop a somewhat rudimentary scaling argument based on the underlying physical principles and a fundamental assumption that the control of a wall-bounded turbulent flow will be achieved when the Lorentz forcing is sufficiently strong to induce a spanwise velocity that is large enough to disrupt the near wall turbulent dynamics. We take as an axiom that classical turbulent wall scaling applies and thus, the only relevant scaling parameters are the friction velocity, u ␶ ⫽ 冑␶ w / ␳ , the density, ␳, and the kinematic viscosity, ␯.

We consider a single electrode/magnet-pair with characteristic pitch, a. The volume of fluid affected by this actuator is approximately a 2 w 共w is the cross-stream width of the actuator兲. Note that here, and throughout this analysis, we ignore constant factors of order one. The electrical resistance, ⍀, of this fluid scales approximately with the distance between electrode pairs and inversely proportional to its cross-sectional area and the fluid conductivity, ␴: ⍀⫽

1 . w␴

共6兲

The Lorentz force, F L , generated by a current, I, passed through this volume of fluid was defined earlier 关Eq. 共1兲兴, and in order for the control to be effective, this force must be strong enough to induce the fluid to move with a spanwise velocity of approximately u ␶ , oscillating back and forth with a period T. Thus the required force on the fluid, F F , is F F⬇

共 mass兲共velocity兲 ␳ a 2 wu ␶ ⫽ . 共period) T

共7兲

Equating the Lorentz force generated, F L , to the fluid force required for control, F F , we can solve for the current: I⫽

␳ awu ␶ , BT

共8兲

and a corresponding current density of J⫽

␳u␶ I ⫽ . aw BT

共9兲

The ratio of the Lorentz inertia, JBa, to the fluid inertia, ␳ u ␶2 , is the definition of the Stuart number, and thus we obtain an expression for the optimal Stuart number at which we expect Lorentz force control to be effective: St⫽

JBa

␳ u ␶2

⫽

a . u ␶T

共10兲

Note that this definition differs slightly from that used in the numerical simulations which uses the channel half height, h, as the length scale. The definition here is more closely related to the underlying physics of the problem, and is some-


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FIG. 12. Schematic of flow facility, showing the accumulation and collection tanks, development section, hanging test plate, and return flow path.

what more appealing since the length scale used is related to the local generation of the Lorentz force, not a somewhat arbitrary global scale. If we define a and T in terms of turbulent wall units: a⫽a ⫹ ␯ /u ␶ ; T⫽T ⫹ ␯ /u ␶2 , then the definition of the optimal Stuart number is simply St⫽

a⫹ T⫹

.

共11兲

This result is similar to that derived by Berger et al.,17 but derived from physical arguments rather than the governing equations of motion as was used in their case. Although this predicts the functional relationship between parameters, the range over which one can freely vary the parameters is clearly not infinite. We know, from both basic physical intuition as well as previous numerical simulations on wallbounded turbulence control,1,14,17 that the region where control is most effective is the near-wall region extending approximately 40l * , away from the wall (l * ⫽ ␯ /u ␶ ). We also expect the optimal Stuart number, as defined, to be of order one—representing a balance between the induced control velocity and the natural inertia of the near wall region. Thus we can predict that the optimal forcing period will be approximately a ⫹ , or T ⫹ ⬇40. Given the simplicity of this scaling argument, this prediction is in surprisingly good agreement with the value of T ⫹ ⬇100 found in the DNS studies of spanwise oscillation flow control3,14,17 and, as will be shown in the next section, the present results. Lastly, we can extend this scaling argument slightly further in service of the current experiment, in which the electrode spacing, a, has a fixed size, but is nevertheless used in flows with different Reynolds numbers. Thus, its ‘‘apparent’’ size, a ⫹ , increases as the Reynolds number of the flow increases. It is trivial to re-write the definition of the Stuart number 共11兲 as St⫽

a R* , h T⫹

共12兲

where h is the channel half-height and R * is the turbulent Reynolds number: R * ⫽u ␶ h/ ␯ . Thus we see that, for a fixed value of a, we predict that the optimal value of the period will also rise linearly with Reynolds number, and that for a

fixed non-dimensional period, the optimal Stuart number will also scale with R * . These predictions will be confirmed 共at least for two values of R * ) by the experimental data presented below. Control efficiency. From this analysis, we can also predict the control efficiency. The power 共per unit area兲 required to operate the control system is P control⫽

I 2 ⍀ ␳ 2 au ␶2 ⫽ . wa B 2 T 2 ␴

共13兲

The power saved will be some fraction, K, of the total turbulent drag: P saved⫽K ␳ u 2␶ U,

共14兲

where U is the free-stream velocity. Thus the total system efficiency is given by

␩⫽

冉冊冉冊冉冊

KB 2 ␴ P saved KB 2 ␴ T 2 U ⫽ ⫽ P control ␳ a ␳␯

T ⫹2 a

⫹

U 1 . u ␶ u 2␶ 共15兲

Since U/u ␶ is more or less constant 共varying only by a factor of 2 over a very wide range of Reynolds numbers兲, this result indicates that not only is the efficiency poor, but that at higher speeds, it only gets worse. The main culprit is the fact that while the actuation requirements increase 共the necessary actuation velocity increases with u ␶ ), the magnet strength, B, is fixed so that all of the extra actuation authority must come from the current density. The poor conductivity of saltwater makes this current very expensive to use. B. Drag reduction measurements

The effect of the spanwise oscillatory Lorentz force on the turbulent drag was measured in a low speed salt-water channel, shown in Fig. 12. The flow is driven by the gravity head developed in the upstream tank, and moderated by a back-pressure tank downstream of the test section. The test section measures 4.5 cm from wall to wall (h⫽2.25 cm), 91 cm (40h) in the spanwise direction, and is 3.6 m long (1600h) in the streamwise direction, ensuring fully developed turbulent flow. The active surface comprises one wall




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FIG. 13. Percent drag reduction vs normalized Stuart number, St/R * , at a fixed value of T ⫹ ⫽100. Data from two different Reynolds numbers are shown.

FIG. 14. Percent drag reduction vs the non-dimensional forcing period scaled by the Stuart and Reynolds numbers: T ⫹ St/R * . The amplitude is fixed at a Stuart number of 1.8 for Re*⫽289, and 1.1 for Re*⫽418.

of the last 150 cm of the test section, and is completely covered by tiles of Lorentz force actuators, as described above. This actuated surface is suspended by thin wires and allowed to move under the force of the fluid shear stress. In this manner a direct measurement of drag can be made over the actuated surface. The surface is large enough and heavy enough so that turbulent fluctuations in drag are averaged out and the entire plate simply deflects under the fluid shear. The motion of the plate is measured in both the streamwise and spanwise directions by three optical displacement transducers. Typical displacements due to the mean shear force 关 O(0.5)N at R * ⫽300] were of approximately 300 ␮m. The system was calibrated using two methods. First, with no flow, known forces were applied to the plate and the corresponding displacement measured. Following this, the plate displacement was measured due to the mean turbulent drag 共with no control兲 at different centerline water speeds, the results compared very well with standard correlations for turbulent skin friction. The drag system was measured to have a sensitivity 共to measure local changes in drag兲 better than 0.01N 共2% of the mean drag force兲. The effects of Lorentz control on the turbulent shear stress were tested at two Reynolds numbers: R * ⫽289 and 418 共centerline velocities of 0.234 and 0.280 m/s兲. At these conditions, the friction velocity, u ␶ was 0.0134 and 0.0194 m/s, respectively, and the nondimensional magnet electrode pitch was 40 and 58 l * . Profiles of the mean and rms velocities in the streamwise and wall-normal directions were taken using PIV and confirm that the baseline flow is representative of a fully developed low-Reynolds number turbulent channel flow.20 Figure 13 shows the percentage change in the turbulent drag measured using the full-channel drag balance as a function of the forcing amplitude. The forcing period was fixed at T ⫹ ⫽100. The amplitude is represented as a non-dimensional Stuart number, defined by 共10兲. To correct for the fact that T ⫹ was fixed, the Stuart number is normalized by the Reynolds number, R * , as suggested by the scaling analysis developed above 共11兲. It is clear that significant drag reduction is achieved over a variety of forcing amplitudes, reaching a

maximum of approximately 10% at a value of St/R * of approximately 0.005 before beginning to decrease again. However, the sparcity of measurements at this peak makes the exact determination of the peak impossible at this time. The voltages required to generate these Stuart numbers are in the 3 V range 共3 V at Re*⫽289 corresponds to a value of St⫽1.8 and St/R * ⫽0.006), meaning that the spanwise velocities induced by the Lorentz force actuators are about 4 cm/s or approximately 2u ␶ . This value is slightly smaller than the optimal amplitude found by Berger et al. for ‘‘idealized’’ forcing, but very similar to the optimal amplitude for the ‘‘realistic’’ forcing case. The flow induced by the actuator is, of course, modified by the base flow fluctuations and the profiles are not likely to remain as clean as was measured in the quiescent flow tank. It is reassuring to see that the results at two different Reynolds numbers match reasonably well, particularly at low values of the forcing amplitude. However, there are not enough data, especially at the higher forcing amplitudes for the R * ⫽418 case, to definitively capture the location of the maximum in the drag reduction. The 10% reduction in drag is the reduction for the entire active surface, and one should keep in mind that this includes the edge of the channel 共where corner flows likely contaminate the flow兲 and the development region where the flow must adjust from its canonical state to the fully controlled state. Thus, the local drag reduction is likely much higher than 10%. Figure 14 shows the drag reduction as a function of the forcing period at a forcing amplitude fixed at each Reynolds number 共St⫽1.8 at R * ⫽289 and 1.1 at R * ⫽418). Directed again by the scaling arguments above, the drag reduction is plotted versus T ⫹ St/R * to correct for both the Stuart number and Reynolds number effects. Again, the collapse of the data is quite compelling, although there is considerable scatter at the mid values of T ⫹ St/R * . As with the variation in amplitude, a wide range of drag reduction is observed, reaching a maximum of approximately 10%. It is also interesting to note that at both high and low values of the forcing period, an increase in drag is observed although the data are somewhat sparse to make any definitive remarks regarding this.


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IV. CONCLUDING REMARKS

A Lorentz force actuator has been designed, fabricated, and tested, and has been demonstrated to be able to induce unidirectional velocities in the near-wall region of a wallbounded flow. The actuator has been tested over a range of low-frequency conditions and indicates good performance and reasonably uniform velocity profiles, in good agreement with computational results.17 In the current configuration, the actuator was designed to generate purely unidirectional 共spanwise兲 motion so that a particular class of turbulence control experiments could be tested. However, modifications to the magnet and electrode layout can be explored to generate a wide variety of control velocity patterns, many of which have been tested for turbulence control.10 However, the constraint that the magnets and electrodes must lie on the surface makes complex actuator design very challenging as fringing fields and undesirable field interactions can make it hard to generate the desired control velocities while still suppressing spurious components. From a purely experimental standpoint, the fabrication technology demonstrated includes a number of important and novel features including the alignment of the magnets and electrodes, the protection of the electrodes from corrosion and the epoxy encapsulation of the system from the corrosive fluid environment. These represent innovations that collectively have yielded a compact and robust actuation system that has not, to our knowledge, been demonstrated before. The actuators have the distinct advantage that they are bodyconformal and contain no moving parts. The overall cost of the actuation surface is not excessive, particularly if the assembly cost is low, as would be the case for a massmanufactured system. The control authority of the system is quite sufficient for such applications as the control of wall bounded turbulence or separation control, both at low to moderate Reynolds numbers. The velocities, at reasonable actuation voltages 共3 V for example兲 are about 4 cm/s which, although not outstanding, are not insignificant. Turbulence control using the actuators has been clearly demonstrated at the low Reynolds numbers comparable to the numerical results of Refs. 14, 17, and 18. Although the values of the drag reduction predicted by the DNS have not been reproduced, the trends are very similar, including the scaling suggested by Berger et al.17 and re-derived here from scaling arguments. The experiments are expected to have lower performance than the DNS due to edge effects and the development distance, neither of which are present in the numerical simulations 共thanks to periodic boundary conditions兲. In addition, the realities of the physical actuator geometry cannot approach that of the idealized simulation, contributing to the lower than optimal performance. The success of the Lorentz force control should not be unexpected—the and previous related numerical simulations14,17 19 experiments have demonstrated the fundamental principle of oscillatory control of turbulence, and so one would be more surprised if the experiments had not yielded any controlled flow. Nevertheless, the success of the present experiment does demonstrate a system for effective and robust flow control in salt water using the Lorentz force that avoids the


difficulties that have marked previous attempts. More experiments, including velocity measurements are currently under way to reveal more of the nature of the controlled flow and hopefully to suggest new directions for achieving even greater gains. Although drag reduction has been demonstrated to work reliably and as predicted by theory, one cannot ignore the fact that the electrical-to-mechanical efficiency is poor. As mentioned earlier, this is primarily a consequence of the low conductivity of the working fluid 共saltwater兲, which is not an easily changed parameter. Furthermore, a relatively simple scaling argument suggests that the system efficiency can only get worse as the Reynolds number increases even if the actuators could be manufactured small enough. The basic problem is that the actuation velocity required increases as the free-stream velocity increases and, since the magnetic field strength does not scale, the only way to generate the higher velocities is through the current flux. The poor conductivity of water makes this an expensive and inefficient approach. For this reason, the actuator is unlikely to find much application if it is required to be widely deployed over a surface, or to be operated for extensive periods of time. Nevertheless, its ease of operation and reliability may make it attractive for some applications where power conversion efficiency is not the most important metric of performance. An interesting final observation is that the fluid control efficiency 共the drag reduced compared to the fluid power introduced to the flow in the form of spanwise velocities兲 is excellent and is of order one. Thus, if a more efficient actuator capable of generating the necessary spanwise velocity could be devised, the control scheme would be energetically very attractive. ACKNOWLEDGMENTS

This work was supported by DARPA/TTO under the supervision of Richard Wlezien, David Walker, Patrick Purtell, and Ronald Joslin. The authors would like to acknowledge Kevin Wu, Maureen McCamley, Jim Paduano, George Karniadakis, Vasileios Symeonidis, and Jin Xu for much assistance and many valuable discussions. Francis Choi and Evan Chan helped with the measurements of the electrode performance and finally, the general technical support of Chet Klepadlo and Dan Thivierge has been invaluable. 1

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