Abstract:
One aspect of the apparatus disclosed herein is a system for modifying an aerodynamic property of an aerodynamic surface in a fluid flow. The fluid flow could comprise a free stream fluid flow, or an internal fluid flow, such as in a nozzle, diffuser, or compressor. The system of the preferred embodiment comprises a synthetic jet actuator embedded in an aerodynamic surface. In one aspect, the aerodynamic surface may be a wing. The synthetic jet actuator typically has a jet housing defining a chamber, where the chamber is in fluid communication with the fluid. This fluid communication may be accomplished via an orifice in a wall of the housing. Additionally, a portion of said housing is preferably moveable such that the volume of the chamber can be adjusted. The system also comprises a device for changing the position of the moveable portion of the housing at a predefined frequency. In this way, the synthetic jet actuator is pulse modulated in order to enhance the synthetic jet actuator&#39;s performance. In another aspect, the invention may be seen as a method of controlling a synthetic jet actuator. The method preferably comprises the steps of driving a synthetic jet actuator at a first frequency and turning the synthetic jet actuator on and off at a second frequency. The synthetic jet actuator interacting with a fluid flow to alter the fluid flow field.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims priority to copending U.S. provisional application entitled, “Enhancement of the Aerodynamic Performance of a Thick Unconventional Airfoil Using Pulse Excitation Control via Synthetic Jet Actuators,” having Ser. No. 60/142,474, filed Jul. 6, 1999, which is incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present invention generally relates to the control of fluid flow about solid surfaces and, more particularly, to fluid flow control through the use of pulse modulated actuators. 
     BACKGROUND OF THE INVENTION 
     Manipulation and control of shear flow evolution has tremendous impact for influencing system performance in diverse technological applications, including lift and drag of aerodynamic surfaces, flow reattachment to wings, and aircraft stall management. The fact that shear flows are dominated by the dynamics of a hierarchy of vortical structures, evolving as a result of inherent hydrodynamic instabilities in the flow, suggests control strategies based on manipulation of these instabilities by the introduction of disturbances at the flow boundary. It is generally recognized that suitable actuators having fast dynamic response and relatively low power consumption are the foundation of any scheme for the manipulation and control of such shear flows. 
     One particular application for shear flow manipulation devices is related to airframes or their lifting surfaces. The shear flow that is generated by an aerodynamic lifting surface presents unique challenges, and manipulation of this shear flow will greatly affect flight performance. It is commonly understood that separation of a free stream flow about a lifting surface is generally not desired. Once the flow about a lifting surface separates, the aerodynamic surface experiences a dramatic decrease in lift force produced and a corresponding dramatic rise in the drag force generated by the aerodynamic surface. 
     Because of the undesirable consequences of flow separation, active manipulation of separated flows over lifting surfaces at moderate and high angles of attack has been the focus of a number of investigations since the early eighties. The goal of such active manipulation is improving the aerodynamic performance and extending an aircraft&#39;s flight envelope by inducing complete or partial flow reattachment to the lifting surface. 
     Some efforts of designers to modify the flow about an aerodynamic surface, and lend to the reattachment of flow over a lifting surface, have centered on injection of momentum into the boundary layer of the flow. For example, the method disclosed by U.S. Pat. No. 4,802,642 to Mangiarotty involves the retardation of a flow&#39;s transition to turbulence in order to improve the aerodynamic performance of a lifting surface. The Mangiarotty apparatus propagates acoustic excitation above the Tollmien-Schlichting frequency in an attempt to disrupt Tollmien-Schlichting waves as they begin to form and thereby delay the onset of turbulence. Although the Mangiarotty method changes the drag characteristic of a lifting surface, the mean velocity field and thus apparent aerodynamic shape of the surface, remains unchanged. 
     Such devices as slots and fluid jets have also been used to inject momentum into the boundary layer in order to prevent flow separation. Although effective at delaying flow separation, none of these devices alter the apparent aerodynamic shape or mean velocity field of a given aerodynamic surface. Additionally, the locus of the flow stagnation points remain largely unchanged. 
     More recently, synthetic jet actuators have been developed for the control and manipulation of shear flows. Synthetic jet actuators are described in U. S. Pat. No. 5,758,823 to Glezer et al., issued Jun. 2, 1998, which is incorporated herein by reference. As explained in the Glezer et al. patent, a synthetic jet actuator, in its most simple form, comprises a housing defining an internal chamber. An orifice is present in a wall of the housing. The actuator further includes a mechanism in or about the housing for periodically changing the volume within the internal chamber. As the volume of the chamber is increased, ambient fluid. is drawn into the chamber. As the volume of the chamber is decreased, the ambient fluid in the chamber is ejected such that a series of fluid vortices is generated and projected in an external environment, out from the orifice of the housing. These vortices move away from the edges of the orifice under their own self-induced velocity. As the vortices travel away from the orifice, they synthesize a jet of fluid, a “synthetic jet,” through entrainment of additional ambient fluid. 
     It has been discovered that a synthetic jet actuator may be embedded in a solid body, or surface, with the jet orifice built into the body/surface. The interaction of a free stream fluid flow about the body with a synthetic jet stream will change the overall fluid flow field around the solid body. In fact, a synthetic jet actuator operated in a lifting surface will alter the apparent aerodynamic shape of the surface. This phenomenon is fully described in U. S. Pat. No. 5,957,413 to Glezer et al., issued Sep. 28, 1999, which is hereby incorporated herein by reference. 
     It has been discovered that the effectiveness of synthetic jet actuators are controlled and/or limited by certain parameters. Particularly, placement and strength of these jet actuators are important. Synthetic jet actuators, when used on a wing, should be placed near the point on the wing where the flow is expected to separate. Placement away from this point will reduce the effectiveness of the jet actuator. Adding to these limitations, the jet actuator must have sufficient strength to create the needed alteration of the aerodynamic shape. An undersized synthetic jet actuator is much less effective. Additionally, the ideal strength and/or placement of a synthetic jet actuator may change with flight conditions. 
     Thus there exists a need in the art to improve the performance of synthetic jet actuators. There also exists a need to optimize the performance of synthetic jet actuators that are underpowered, either due to changes in flight conditions or due to inherent limitations on the system. The apparatus and method described herein, in both the text and figures, seeks to remedy the problems in the art and provide a method and apparatus for improving the performance of actuators, and particularly, synthetic jet actuators. 
     SUMMARY OF THE INVENTION 
     Briefly described, the present invention involves the use of actuators, particularly synthetic jet actuators for modification of fluid flow about the various surfaces. The present invention is primarily concerned with pulse modulating synthetic jet actuators in order to take advantage of transient responses and thereby improve synthetic jet actuator performance. However, many different types of actuators may benefit from the pulse modulation technique described herein. 
     One aspect of the apparatus disclosed below is a system for modifying an aerodynamic property of an aerodynamic surface in a fluid flow. The fluid flow could comprise a free stream fluid flow, or an internal fluid flow, such as in a nozzle, diffuser, or compressor. The system preferably comprises a synthetic jet actuator embedded in an aerodynamic surface. In one aspect of the preferred embodiment, the aerodynamic surface may be a wing of an aircraft. The synthetic jet actuator typically has a jet housing defining a chamber, where the chamber is in fluid communication with the fluid. This fluid communication may be accomplished via an orifice in a wall of the housing. Additionally, a portion of said housing is preferably moveable such that the volume of the chamber can be adjusted. This portion of the housing may comprise a flexible diaphragm. The system also comprises a device for changing the position of the moveable portion of the housing. This may comprise a piezoelectric actuator and a power supply. The system also comprises a controller for automatically cycling the position changing device between on and off at a predefined frequency. In this way, the synthetic jet actuator is pulse modulated in order to enhance the synthetic jet actuator&#39;s performance. 
     In another aspect, the invention may been seen as a method of controlling a synthetic jet actuator. The method preferably comprises the steps of driving a synthetic jet actuator at a first frequency and turning the synthetic jet actuator on and off at a second frequency. 
     Other systems, methods, features. and advantages of the present invention will be or become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the present invention, and be protected by the accompanying claims. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The present invention can be better understood with reference to the following drawings. The drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating principles of the present invention. Moreover, like reference numerals designate corresponding parts throughout the several views. 
     FIG. 1 is a cross-sectional schematic side view of the preferred embodiment of a wing in a free stream flow with a synthetic jet actuator embedded in the wing. 
     FIG. 2A is an exploded schematic cross-sectional side view of the zero net mass flux synthetic jet actuator of FIG.  1 . 
     FIG. 2B is a schematic cross-sectional side view of the synthetic jet actuator of FIG. 2A depicting the jet as the control system causes the diaphragm to travel inward, toward the orifice. 
     FIG. 2C is a schematic cross-sectional side view of the synthetic jet actuator of FIG. 2A depicting the jet as the control system causes the diaphragm to travel outward, away from the orifice. 
     FIG. 3 is a plot of the pressure coefficient distribution for different frequencies of operating the flexible diaphragm of the synthetic jet actuator of FIG.  1 . 
     FIG. 4A is a phase-averaged plot of the incremental change in normalized circulation with respect to the unforced flow condition of the apparatus of FIG. 1, for α=17.5° and F + =0.95 ( 60   b ), and F + =10 ( 60   a ). 
     FIG. 4B is a schematic cut away side view of the apparatus of FIG. 1 after the synthetic jet actuator has been turned on. 
     FIG. 5A is a plot of the phase-averaged incremental change of the circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =10, α=17.5°, and the synthetic jet actuator is pulse modulated at a frequency f + =3.3. 
     FIG. 5B is a plot of the phase-averaged incremental change of the circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =10, α=17.5°, and the synthetic jet actuator is pulse modulated at a frequency f + =5.0. 
     FIG. 5C is a plot of the phase-averaged incremental change of the circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =10, α=17.5°, and the synthetic jet actuator is pulse modulated at a frequency f + =0.27. 
     FIG. 5D is a plot of the phase-averaged incremental change of the circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =10, α=17.5°, and the synthetic jet actuator is pulse modulated at a frequency f + =1.1. 
     FIG. 6 is a plot of the phase-averaged incremental change in normalized circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =10, α=17.5°, and the synthetic jet actuator is pulse modulated at a frequency f + =3.3. FIG. 6 is also a plot of the phase-averaged incremental change in normalized circulation with respect to the unforced flow condition of the apparatus of FIG. 1, where the synthetic jet actuator diaphragm is oscillated at F + =3.3, α=17.5°. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     It will be obvious to those skilled in the art that many modifications and variations may be made to the preferred embodiments of the present invention as described hereafter without substantially departing from the spirit and scope of the present invention. All such modifications and variations are intended to be included herein within the scope of the present invention, as is set forth in the appended claims. 
     For example, the pulse modulation techniques described below are described with relation to synthetic jet actuators. However, the pulse modulation techniques, which are very beneficial to the synthetic jet actuators, are also applicable to many other types of actuators. Such other actuators are intended to be included within the scope of the present invention. 
     A. Synthetic Jet Actuator Embedded In A Wing 
     FIG. 1 depicts the preferred embodiment of a system  10  for modification of fluid flow  12  about an aerodynamic surface  11 . In the preferred embodiment  10 , a synthetic jet actuator  13  is employed to modify the lift and drag characteristics of the aerodynamic surface  11  by preventing stall and/or inducing flow reattachment. The aerodynamic surface  11  typically comprises a wing, or aerodynamic fin. Of course, the aerodynamic surface is not so-limited and may comprise any solid body or solid surface that is placed in a free stream flow. By way of example, the present system  10  could be used along the surface of an aircraft fuselage, or along the surface of an automobile body panel. The present system  10  is also adaptable for altering flow characteristics of internal flows, such as in a nozzle, diffuser, or compressor. 
     In the preferred embodiment  10 , the aerodynamic surface  11  is depicted as a cross-sectional side view of a three-dimensional wing structure in a free stream flow  12 . Since the depiction of the aerodynamic surface  11  is two-dimensional, the aerodynamic surface will be referred to as both a wing and an airfoil interchangeably herein. The wing  11  generally comprises an upper surface, or skin  14  and a lower surface, or skin  16 . The upper and lower surfaces  14 ,  16  meet at the wing&#39;s trailing edge  17  and at the wing&#39;s leading edge  18 . 
     The wing  11  depicted in FIG. 1 comprises an airfoil having a slightly positive camber. The chord line  19  of the airfoil  11  is depicted in FIG. 1 as a dashed line. Of course, the airfoil  11  may be symmetric or of any other camber. The actual natural camber of the aerodynamic surface  11  is not important to the present invention. 
     As depicted in FIG. 1, the wing  11  is disposed at some angle of attack  21 . The particular angle of attack  21  is not important to the present invention, although the angle of attack may affect several parameters of the present invention. However, as depicted in FIG. 1, the angle of attack  21  is approximately 20 degrees. Twenty degrees is not necessarily a preferred angle, as no particular angle of attack is really preferred over any other. 
     B. Construction Of The Synthetic Jet Actuator 
     In the preferred embodiment  10 , a synthetic jet actuator  13  is embedded in the wing  11 . FIG. 2A is an exploded side view of the synthetic jet actuator  13  of the system  10  such that the components of the synthetic jet actuator  13  may be better described. The synthetic jet actuator  13  of FIG. 2A comprises a housing  22  defining and enclosing an internal chamber  23 . The housing  22  and chamber  23  can take virtually any geometric configuration, but for purposes of discussion and understanding, the housing  22  is shown in cross-section in FIG. 2A to have a rigid side wall  24 , a rigid front wall  26 , and a rear diaphragm  27  that is flexible to an extent to permit movement of the diaphragm  27  inwardly and outwardly relative to the chamber  23 . The front wall  26  has an orifice  28  of any geometric shape. The orifice  28  diametrically opposes the rear diaphragm  27  and connects the internal chamber  23  to an external environment having ambient fluid  29 . The walls  24 ,  26  are typically welded or glued together. The diaphragm  27  is preferably glued to the side walls  24 . Of course, the method of adhesion is not critical to the present invention and welding/glue is only the preferred method and not the only method of attachment. 
     Recall that the synthetic jet actuator  13  is embedded in a wing  11 , in the preferred embodiment  10 . As a result, the ambient fluid  29  will typically comprise the free stream fluid flow  12  (see FIG.  1 ). Also because the synthetic jet actuator  13  is embedded in the wing  11 , the front wall  26  of the synthetic jet actuator  13  is preferably the same as the upper skin  14  of the wing  11  (see FIG.  1 ). Thus, in the preferred embodiment, the upper skin  14  of the wing  11  has an orifice  28  therein. In this way, the side wall  24  and diaphragm  27  structure of the synthetic jet actuator  13  can be built directly onto the inside surface of the wing  11 . The side walls  24  can be attached to the inner surface of the wing  11  in any suitable manner, but welding is preferred. 
     In order to operate the synthetic jet actuator  13 , the diaphragm  27  is preferably constructed so as to oscillate in time-harmonic motion. While this motion may be effected in a variety of manners, preferably, a piezoelectric actuator  31  is attached to the diaphragm  27 . Piezoelectric elements  31  may be attached in any suitable manner, but most commonly they are attached by an adhesive glue. The piezoelectric element  31  should be supplied with power in order to begin vibration. In the preferred embodiment, a control system  32  is connected to the piezoelectric actuator  31  in order to cause the piezoelectric element  31  to vibrate. The control system  32  is preferably connected via standard wiring. Note that the control system  32  is depicted schematically in the drawings and the line  35  connecting the control system  32  the piezoelectric actuator  31  only represents a connection between the two components. The actual location of the control system  32  and the connecting line  35  will be different from as depicted and the figure is depicted schematically for clarity. 
     The control system  32  of the preferred embodiment comprises an amplifier  33  and a computer  34 . In the preferred embodiment  10  of a pulse modulated synthetic jet actuator  13 , as will be fully described below, the computer  34  generates a modulated sine wave and sends this waveform to the amplifier  33 . The modulated sine wave is preferably generated by first constructing a sine wave of high frequency and then constructing a TTL signal (basically a square wave of values zero and five) at a lower frequency. The product of these is the modulated sine wave for the amplifier  33 . 
     Upon receiving the modulated sine wave from the computer  34 , the amplifier  33  generates a power output to the piezoelectric actuator  31  via connecting line  35 . This power supplied by the amplifier  33  causes the piezoelectric actuator  31  to vibrate in a manner corresponding to the modulated sine wave, and thereby move the diaphragm  27  in time-harmonic motion. 
     The control system  32  may be housed in the wing  11 ; however, it is preferred that the control system  32  be housed in the fuselage of the aircraft (not depicted) and connected to the synthetic jet actuator  13  via wiring installed in the interior of the wing  11 . 
     Of course, the generation of the electrical power to the piezoelectric actuator  31  can be accomplished and controlled by any suitable device, for example but not limited to, a signal generator may serve the same purpose. Of course, it is preferred that the control system  32 , in whatever embodiment is selected, has the capability of causing the diaphragm  27  to oscillate at a range of predefined frequencies and have the capability of selectively turning on and off the power to the piezoelectric actuator  31  at a range of predefined frequencies. 
     The method of causing the diaphragm  27  to modulate is not limited in the system  10  described herein. As an example of an alternative method of causing diaphragm motion, the diaphragm  27  may be equipped with a metal layer that acts as a first electrode, and a metal second electrode may be disposed adjacent to, but spaced from the metal layer so that the diaphragm  27  can be moved via an electrical bias imposed between the electrode and the metal layer. Generally, a positive charge is imparted to one electrode and a negative charge to the other electrode. The attractive force between the two electrodes can be harnessed and used to oscillate the diaphragm  27 . This method of actuation may be referred to as electrostatic actuation. 
     As another alternative to the use of a diaphragm  27  as a wall of the housing  22 , one wall of the housing  22  could comprise a piston structure. Thus, the volume of the chamber  23  could be altered via control of the piston. In this configuration, the control system  32  would supply power to the piston and cause the piston to move into and out of the chamber  23  in periodic fashion. 
     C. Operation Of The Synthetic Jet Actuator 
     The operation of the synthetic jet actuator  21  will now be described with reference to FIG.  2 B and FIG.  2 C. FIG. 2B depicts the synthetic jet actuator  13  as the diaphragm  27  is controlled to flex into the chamber  23 . The chamber  23  has its volume decreased and fluid is ejected through the orifice  28 . As the fluid exits the chamber  23  through the orifice  28 , the flow separates at sharp orifice edges  36  and creates vortex sheets  37 , which roll into vortices  38  and begin to move away from the orifice edges  36  in the direction indicated by arrow  39 . 
     FIG. 2C depicts the synthetic jet actuator  13  as the diaphragm  27  is controlled to move outward with respect to the chamber  23 . The chamber  23  has its volume increased and ambient fluid  29  rushes into the chamber  23  as depicted by the set of arrows  41 . The diaphragm  27  is controlled by the control system  32  (FIG. 2A) so that when the diaphragm  27  moves away from the chamber  23 , the vortices  38  are already removed from the orifice edges  36  and thus are not affected by the ambient fluid  41  being drawn into the chamber  23 . Meanwhile, an ambient fluid jet  42  is synthesized by the vortices  38 . The vortices  38  create the ambient fluid jet  42  by entraining ambient fluid  29 , due to their rotation, from large distances away from the orifice  28 . 
     Referring back to FIG. 1, in the preferred embodiment  10 , the orifice  28  of the synthetic jet actuator  13  is positioned in order to be flush with the upper skin  14  of the wing  11 . The location of the orifice  28  along the airfoil skin  14  can be determined based on the particular effect on the flow  12  desired. As indicated above, the airfoil  11  is typically placed in a free stream fluid flow  12 . The flow of the fluid  12  about the airfoil  11  is depicted by the set of streamlines  43 . 
     When the synthetic jet actuator  13  is operational, the synthetic jet actuator  13  forms a fluid flow as depicted by arrow  42  in FIG.  2 B. Because the synthetic jet actuator  13  does not inject any new fluid into the free stream flow  12 , a closed recirculating flow region is formed adjacent to the airfoil skin  14 . The recirculating flow region is closed and modifies the apparent aerodynamic shape of the airfoil. Of course, the modification of the aerodynamic shape of the airfoil results in alteration of the flow about the airfoil, alteration of the streamlines  43 , and alteration of the aerodynamic characteristics of the airfoil. In fact, as will be explained in more detail below, if the flow about the wing  11  is detached, simply turning on the synthetic jet actuator  13  may cause the flow  12  to reattach to the wing surface  14 . In short, actuation of the synthetic jet actuator  13  in the airfoil  11  leads to flow reattachment and the establishment of a higher (positive) lift force on the airfoil  11 , which is accompanied by a change in the vorticity flux and a net increase in circulation. 
     Separation control through the use of synthetic jet actuators  13 , as outlined above, results in a substantial increase in the stall margin of an airfoil  11  at high angles of attack  21  with a significant improvement in the lift and a corresponding reduction in pressure drag. However, the effectiveness of a synthetic jet actuator  13  is affected by both the positioning and the strength of the jet actuator  13  for a given angle of attack  21 . For example, as the distance between the synthetic jet actuator  13  and the separation line is decreased, the power required to effect reattachment is reduced. Similarly, as the distance from the separation line is increased, the power required is also increased. 
     D. Synthetic Jet Actuator Performance Enhancement: Optimizing The Frequency Of Synthetic Jet Actuator Operation 
     The frequency at which the diaphragm  27  is driven by the control system  32  also has a significant impact on the effectiveness of the synthetic jet actuator. Previous work on airfoil separation control has primarily emphasized actuation frequencies that were typically of the same order as the natural shedding frequency of the separated flow over the airfoil (i.e., F + =O(1)) regardless of the choice of actuators. As will be explained in detail below, the desired frequency of operation for the diaphragm  27  is approximately at least an order of magnitude larger than the natural shedding frequency (i.e., F + =O( 10 )). 
     It has also been discovered that the synthetic jet actuator  13  of FIG. 1 works most efficiently and effectively if the synthetic jet actuator  13  is tuned such that the desired frequency of operation, in this case F + =O( 10 ), is also the resonant frequency of the synthetic jet actuator  13 . The preferred method sizing a synthetic jet actuator  13  begins with determining the natural shedding frequency of the airfoil  11  to be used. In a general sense, the natural shedding frequency can be determined from the chord of the airfoil  11  and the free stream velocity of the flow  12 . The natural shedding frequency is understood to exhibit the following relationship: 
       f   s   ∝{fraction (U∞/c)}   
     Where f s  is the natural shedding frequency, U ∞  is the free stream fluid velocity, and c is the chord length of the airfoil  11 . 
     Once the shedding frequency is determined, the actuator designer may then set the frequency at which the actuator diaphragm  27  will be driven. As stated above, the preferred frequency of actuation as at least ten times the value of the natural shedding frequency (i.e., F + =O( 10 )). So, the preferred size of the synthetic jet actuator  13  is set such that the resonant frequency of the synthetic jet actuator  13  is on the order of F + =O( 10 ). 
     The resonant frequency of the synthetic jet actuator  13  may be determined from the material properties of the piezoelectric element  31  and the diaphragm  27  in combination with the volume of the chamber  23 . The chamber  23  has a specific resonant frequency that can be determined mathematically or experimentally. The resonant frequency of the entire actuator  13  is a function of the volume size, the material of the piezoelectric element/diaphragm, and the clamping method of the piezoelectric element  27  to the housing  24 . As the material properties are typically set, it is easiest to tune the actuator by adjusting the size of the chamber  23 . 
     As stated, the preferred frequency of operation of the synthetic jet actuator is at an order of magnitude larger than the natural shedding frequency of the airfoil. It will now be explained why this frequency of operation is preferred. 
     FIG. 3 is a plot of the pressure coefficient distributions around the airfoil  11  of the preferred embodiment  10  when the synthetic jet actuator  13  is driven at approximately the natural shedding frequency (F + =O( 1 )), ten times the natural shedding frequency (F + =O( 10 )), and in the absence of synthetic jet actuator  13  operation. For purposes of this disclosure, a synthetic jet actuator  13  is “driven” with a sinusoidal waveform with a certain frequency by causing the diaphragm  27  to oscillate at that frequency. 
     The pressure coefficient is set forth along the vertical, y-axis while the normalized distance along the length of the airfoil  11  is set forth along the horizontal, x-axis. The distance is expressed as a percent of chord traversed. For F + =O( 10 ), the pressure distribution exhibits a large suction peak  51  near the leading edge on the suction side of the airfoil  11  followed by a rapid recovery  52  of the pressure towards the trailing edge  17 . When low frequency forcing (F + =O( 1 )) is used, the pressure distribution exhibits a smaller and sharper suction peak  53  near the leading edge  18 , and as a result, the contribution of this peak to the lift is smaller than for the high frequency forcing. Downstream of the low frequency suction peak  53  the pressure difference between the suction and pressure sides is larger than for high frequency forcing resulting in a larger contribution to lift, but also a larger contribution to pressure drag. 
     In short, operation of the synthetic jet actuator  13  at approximately F + =O( 10 ) yields better performance from a lift-to-pressure drag ratio perspective. For this reason alone, it may be preferred to operate the synthetic jet actuator  13  of the preferred embodiment  10  at a frequency approximately ten times as high as the natural shedding frequency of the airfoil  11 . FIG. 4A demonstrates an additional reason why high frequency forcing may be preferable. 
     FIG. 4A also addresses the differences between low and high frequency actuation of the synthetic jet actuator  13  of the preferred embodiment  10 . FIG. 4A shows the temporal variations in the aerodynamic loads via phase averaged circulation for F + =O( 1 ) and F + =O( 10 ) (using curves  60   b  and  60   a , respectively). Change in the normalized circulation with respect to the unforced flow condition is represented along the vertical. y-axis. Normalized circulation may be reflected by the equation: {circumflex over (Γ)}=Γ/(U () ·c). A normalized time parameter is depicted along the horizontal, x-axis. where T is the period time of the actuation frequency. The synthetic jet actuator  13  is turned on at the up arrow  56  and turned off at the down arrow  57 . 
     Note that at time t/T&lt;75, before the first arrow  56 , the change in the normalized circulation with respect to the unforced flow condition is approximately zero. Obviously, with no change in the circulation about the airfoil  11 , there is no change in the lift. Upon turning on the synthetic jet actuator  13 , the flow  12  reattaches and the lift force increases, as reflected by the negative change in circulation values generated from approximately t/T=200 to t/T=450. However, when flow reattachment begins, the change in the normalized circulation exhibits a transient condition for both control frequencies. Note that there is a sharp positive peak  59  followed by a sharp negative peak  61  and a smaller positive peak  62 . Then, for the high frequency forcing, the circulation ultimately reaches a steady level. In contrast, low frequency forcing results in oscillations of Δ{circumflex over (Γ)} at the forcing frequency with peak-to-peak fluctuations of up to 45% of the mean level for the attached flow. It can also be noted that when the synthetic jet actuator  13  is turned off  57 , both plots show another transient response. The high frequency plot shows a slight negative peak  63   a  and the low frequency forcing plot shows a large negative peak  63   b.    
     FIG. 4A demonstrates an important distinction between operation of the synthetic jet actuator  13  at a frequency of the same order as the natural shedding frequency, F + =O( 1 ), and at the preferred frequency, F + =O( 10 ). When the synthetic jet actuator  13  is operated at F + =O( 10 ), the flow  12  about the airfoil  11  will re-attach to the airfoil  11  and several coherent vortical structures at the operating frequency of the synthetic jet actuator  13  will form along the upper skin  14  of the airfoil  11 . It is noted however, that these vortices quickly lose their identity and vanish well before they reach the trailing edge  17  of the airfoil  11 . 
     When the actuation frequency is F + =O( 1 ), the reattachment of the flow  12  is characterized by the formation of large vortical structures that scale with the chord  19  of the airfoil  11  and persist well beyond the trailing edge  17  of the airfoil  11 . It appears that because the formation frequency of these vortices couples with the natural shedding frequency of the airfoil  11 , these vortices are actually enhanced with downstream distance. It is the formation and shedding of these vortical structures that leads to time-periodic variation in lift, as is shown graphically in FIG.  4 A. 
     In order to minimize the variations in circulation (and consequently the lift force) exhibited in FIG. 4A, the synthetic jet actuator  13  of the preferred embodiment  10  should preferably be operated at a frequency at least an order of magnitude larger than the natural shedding frequency of the airfoil  11 . The fluctuations of circulation caused by the low frequency forcing are not generally desirable. It is preferable that the circulation generated by the synthetic jet actuator  13  attain and maintain an approximately constant level during actuator  13  operation. Thus, once again, operation of the synthetic jet actuator  13  of the preferred embodiment  10  at higher frequencies is preferred. 
     E. Further Performance Enhancement: Pulse Modulation Of Synthetic Jet Actuators 
     i. Basis For Pulse Modulation 
     FIG. 4A also demonstrates an important feature of the flow dynamics associated with controlled separation. The discussion will now focus primarily on the case where the diaphragm  27  of the synthetic jet actuator  13  is driven at a frequency much greater than the natural shedding frequency of the airfoil  11 . In FIG. 4A, the “spikes” at approximately t/T=100 ( 59 ) and the slight negative peaks at approximately t/T=500 ( 63   a ,  63   b ) are important features of any actuation control methodology that can be exploited for further improvements in synthetic jet actuator  13  performance. 
     It is known that when the flow  12  about the airfoil  11  is separated from the upper surface  14  of the airfoil  11 , the vorticity distribution in the wake is comprised of a train of vortical structures of alternating sign (clockwise vorticity is taken to be negative) having a nominal passage frequency equivalent to the shedding frequency of the aerodynamic surface  11 . Nevertheless, the total vorticity flux across the wake during one period of the unforced shedding frequency is approximately zero. Turning on the synthetic jet actuator.  13  leads to flow reattachment and the establishment of a higher (positive) lift force on the airfoil  11 , which must be accompanied by a change in the vorticity flux and a net increase in circulation associated with positive (counter-clockwise) vorticity. 
     However as demonstrated clearly in FIG. 4A, following the activation  56  of the synthetic jet actuator  13 , a strong clockwise vortex, indicating a reduction in lift  59 , is generated by the actuator  13  and shed along the upper surface  14  of the airfoil  11 . This strong negative vortex is followed closely by a stronger counter-clockwise (positive) vortex, indicating the re-establishment of lift and shown graphically in FIG. 4A as the peak  61 . These two large vortices  59 ,  60  are followed by a series of smaller vortices of alternating signs and diminishing strength. It appears that the reduced wake of the attached airfoil I ultimately reaches a state of symmetric vorticity distribution. 
     FIG. 4B schematically depicts the vortices shed by the airfoil  11  at start of the synthetic jet actuator  13 . The vortices are numbered to correspond to the peaks in FIG. 4A, as described above. FIG. 4B demonstrates schematically the shedding of the “trapped” vorticity of the separated flow  59 , which is closely followed by the “starting vortex”  61 . These two vortices are followed by a train of vortices of alternating signs and diminishing strength  62   a ,  62   b  to re-adjust the flow  12  around the airfoil  11 . 
     When the synthetic jet actuator  13  of the preferred embodiment  10  is turned off  57 , the flow  12  separates again and the airfoil  11  loses its lift. This reduction in lift is accompanied by a decrease in circulation and the shedding of negative (clockwise) vorticity, as shown in FIG. 4A as the change in circulation tends to zero after the down arrow  57 . However, immediately following the termination  57  of the control, a counter-clockwise vortex, indicating a momentary increase in lift is emitted along the upper surface  14  of the airfoil  11  before the separated vorticity field is established. This large counter-clockwise vortex is graphically depicted as the positive peak in change in circulation  63   a ,  63   b  in FIG.  4 A. 
     The increase in the circulation at the end of synthetic jet actuator  13  operation is one key to optimizing the performance of the synthetic jet actuator  13 , even when the streamwise placement of the actuator  13  along the surface  14  of the airfoil  11 , or strength of the actuator  13 , is sub-optimal. For example, it has been discovered that a relatively small reduction of the strength of the synthetic jet actuator  13  results in only a partially reattached flow  12  and a substantial degradation of the lift generated by the synthetic jet actuator  13 . The pressure distribution, when the synthetic jet actuator  13  is relatively weak, exhibits a much smaller suction peak near the leading edge  18  followed by a separation bubble that extends throughout most of the upper surface  14  of the airfoil  11 . 
     It is noteworthy, however, that while the asymptotic levels of the circulation decrease substantially with the actuator momentum, the transients  63   a ,  63   b  associated with each control input do not appreciably change. Thus, it has been discovered that a synthetic jet actuator  13  operated at frequencies substantially higher than the natural shedding frequency of the wing  11  are amenable to optimization through pulse modulation and consequently, through exploitation of these transients, that are present even for weak synthetic jet actuators, in order maximize an increase in airfoil lift due to flow reattachment. 
     Basically, when the synthetic jet actuator  13  is turned off  57 , the flow begins to separate from the surface  14  of the airfoil  11 . However, flow separation is not instantaneous and, in the transient phase, enough vorticity is accumulated to actually increase the airfoil lift (see  63   a ,  63   b ). Thus, if the synthetic jet actuator  13  is pulse modulated at the proper frequency, the synthetic jet actuator  13  can be turned back on before the flow detaches, thereby taking advantage of the transient “spike” in lift  63   a ,  64   b , trap the additional vorticity over the airfoil  11  and cause the flow to stay attached such that the lift does not drop to the levels common when flow is detached. 
     This phenomenon of an increase in lift following the time when actuation is ceased would be true for other types of actuators in addition to synthetic jet actuators. So, while synthetic jet actuators will greatly benefit from such pulse modulation, other actuators may also be modulated in order to take advantage of this transient affect. 
     ii. Implementation of Pulse Modulation 
     In order to improve the performance of the synthetic jet actuator  13  of the preferred embodiment  10  at reduced levels of actuator momentum, the actuator diaphragm  27  resonance waveform (nominally driven at F + =O( 10 )) is preferably pulse modulated. Further, it is preferred that the period t′ and duty cycle û of the modulating pulse train are independently controlled. The duty cycle û is a measure of how long the synthetic jet actuator  13  is operational. 
     To effectuate pulse modulation of synthetic jet actuators  13 , the computer  34  of the control system  32  is preferably programmed to produce a modulated sine wave and send this waveform to the amplifier  33 . The modulated sine wave is preferably generated by first constructing a sine wave of high frequency (F + =O( 10 )) and then constructing a TTL signal at the frequency at which the actuator  13  will be pulsed (this frequency is represented by f +  herein). The two waves are multiplied by each other to generate the modulated sine wave for the amplifier  33 . Of course, other waveforms may be used to modulate the synthetic jet actuator  13 . For example, the computer  34  may be programmed to use a “saw-tooth” waveform or a triangle waveform. 
     Upon receiving the modulated sine wave from the computer  34 , the amplifier  33  generates a power output to the piezoelectric actuator  3   1 . This power supplied by the amplifier  33  causes the piezoelectric actuator  31  to vibrate in a manner corresponding to the modulated sine wave and thereby move the diaphragm  27  in time-harmonic motion, and to pulse according to the frequency f + . 
     The duty cycle of the pulse frequency f +  is a function of the angle of attack  21  and chord of a given airfoil  11 . The frequency f +  at which the synthetic jet actuator is to be modulated is a function of the airfoil angle of attack  21 , the free stream fluid velocity, and the chord of the airfoil  11 . One having skill in the art can easily determine the optimum modulation frequency f +  as set forth below and in FIGS. 5A-5D. By way of example, for an angle of attack of 17.5 degrees, the preferred duty cycle, û=0.25. For this example, the period (and corresponding frequency) is preferably restricted to approximately t′=0.303 and f + 32 1/t′=3.3. The selection of these values is demonstrated below. 
     FIG. 5A is a plot of change in the normalized circulation with respect to the unforced flow condition versus normalized time for the preferred embodiment  10 , where the synthetic jet actuator  13  is modulated at a frequency of f + =3.3 (for an angle of attack of 17.5 degrees). The pulse modulated plot is depicted as the line without symbols  74 . The synthetic jet actuator  13  of the preferred embodiment  10  is turned on at time where the up arrow  76  is positioned, approximately t/T=75. Note that prior to turning the synthetic jet actuator  13  on (i.e. beginning the time-harmonic oscillation of the diaphragm 27 at F + =O(10)), the change in the normalized circulation is oscillating about a value of zero. 
     When the synthetic jet actuator  13 , under pulse modulation, is turned on, a much attenuated transient response is exhibited. Particularly, note that the plot representing unmodulated synthetic jet actuator  13  operation exhibits a much greater peak  77  than the corresponding plot of the transient response  78  of a pulse modulated synthetic jet actuator  13 . Clearly, the fluctuation of lift is much less during this transient phase for pulse modulated synthetic jet actuation. One primary benefit of this feature is that there is much less strain on the aircraft structure during this period. 
     Note also that the pulse modulated synthetic jet actuator  13  induces a much greater negative change in circulation (and therefore positive lift) on the airfoil  11  in the steady state. In fact, pulse modulation of the synthetic jet actuator  13  yields an increase of ˜45% in the lift coefficient (when it reaches steady state) compared to continuous high-frequency excitation  75 . 
     The frequency f + =3.3 is the preferred frequency because, for the specific preferred embodiment described herein, frequencies higher or lower than this value are generally not as effective at increasing lift. Through experimentation, one with skill in the art will determine if this frequency is appropriate for his/her particular application. The inventors have determined that, for most embodiments, the preferred modulation frequency, f + , is of the same order of magnitude as the natural shedding frequency, or less. For example, FIG. 5B depicts the situation where the modulating frequency is increased to f + =5 for the preferred conditions set forth above. The time between successive pulses of the modulating wave train is too short to capture the unsteady vortical structures. The effectiveness of the modulation is minimal and the circulation returns to the same levels obtained with a continuous synthetic jet actuator operation. 
     On the other hand, if the modulation frequency is set to a value of f + =0.27 (FIG.  5 C), corresponding to the “natural” passage frequency of the vortices during the initial (transient) stages of the reattachment/separation processes, the results are also less than ideal. The resulting quasi-steady circulation exhibits oscillations that are similar in magnitude and duration to the transient stages of the reattachment with shedding of similar vortical structures. The phase of each pulse of the modulating wave train is timed so that it re-triggers reattachment before the flow separates again. Note that the circulation exhibits low-frequency variations (having a period of the order of 60T). 
     When f +  is increased to 1.1 the elapsed time between pulses within the modulating wave train is decreased (FIG. 5D) and the large oscillations in the circulation are substantially attenuated. This suggests that the modulating pulses are timed to prevent continuous shedding of large vortical structures and the corresponding variations in circulation. The recovery of an asymptotic circulation of approximately Δ{circumflex over (Γ)}=0.45 also suggests that the forcing allows the accumulation and maintenance of (clockwise) vorticity on the suction side  14  of the airfoil  11  even though the reattachment is unsteady and the circulation oscillates with peak-to-peak variations of 42% of its asymptotic mean level. 
     Thus, it can be seen that pulse modulation of the synthetic jet actuator  13  of the preferred embodiment  10  greatly improves the performance of the synthetic jet actuator  13 . This may be especially advantageous if the synthetic jet actuator  13  is not at the optimum location on the airfoil  11 , or is sub-optimally powered. 
     Interestingly, an increase in lift, similar to that shown in the depiction of the change in the normalized circulation in FIG. 5A, may not be obtained by simply oscillating the diaphragm  27  of the synthetic jet actuator  13  of the preferred embodiment  10  with a sine wave having a frequency F + =f + =3 .3. Although this would bypass the need for pulse modulation, it simply does not work. 
     FIG. 6 depicts a plot of phase-averaged change in normalized circulation with respect to the unforced flow condition for both continuously forced synthetic jet actuator operation  79  and for pulse modulated synthetic jet actuator operation  80 . Note that the continuously forced synthetic jet actuator operation plot  79  represents the situation where the actuator diaphragm is oscillated at a frequency of the same order of magnitude of the natural forcing frequency, F + =3.3. The pulse modulated synthetic jet actuator operation plot  80  represents the situation where the actuator diaphragm  27  is oscillated at frequency of magnitude F + =O( 10 ), and the actuator itself is then pulse modulated at a frequency of f + =3.3. In FIG. 6, the same jet momentum coefficient (c μ ) is used for both the continuously forced jet actuator and the pulse modulated jet actuator. FIG. 6 also shows the input waveforms that lead to identical jet momentum coefficient (c μ ) over one period of the actuation. 
     FIG. 6 clearly shows that the pulse modulated synthetic jet actuator  13  yields a much greater increase in normalized circulation than the continuously driven synthetic jet actuator. It is also important to note that the transient response  81  of the pulse modulated synthetic jet actuator  13  is much attenuated.