UFDC Home  myUFDC Home  Help 



Full Text  
ON THE MODELING AND DESIGN OF ZERONET MASS FLUX ACTUATORS By QUENTIN GALLAS A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by Quentin Gallas Pour mafamille et mes amis, d'ici et de labas... (To my family and friends, from here and over there...) ACKNOWLEDGMENTS Financial support for the research project was provided by a NASALangley Research Center Grant and an AFOSR grant. First, I would like to thank my advisor, Dr. Louis N. Cattafesta. His continual guidance and support gave me the motivation and encouragement that made this work possible. I would also like to express my gratitude especially to Dr. Mark Sheplak, and to the other members of my committee (Dr. Bruce Carroll, Dr. Bhavani Sankar, and Dr. Toshikazu Nishida) for advising and guiding me with various aspects of this project. I thank the members of the Interdisciplinary Microsystems group and of the Mechanical and Aerospace Engineering department (particularly fellow student Ryan Holman) for their help with my research and their friendship. I thank everyone who contributed in a small but significant way to this work. I also thank Dr. Rajat Mittal (George Washington University) and his student Reni Raju, who greatly helped me with the computational part of this work. Finally, special thanks go to my family and friends, from the States and from France, for always encouraging me to pursue my interests and for making that pursuit possible. TABLE OF CONTENTS page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ........... ............................... ............... ............. ix LIST O F FIG U RE S .... ........................... ............ xi LIST OF SYMBOLS AND ABBREVIATIONS .................................................... xix A B S T R A C T .......................................................... ............... xxv i CHAPTER 1 IN TR OD U CTION ............................................... .. ......................... .. M o tiv atio n ..................... ... .... .. ............................................... . Overview of a ZeroNet Mass Flux Actuator............................................................3 L literature R eview ................... .. .............. .................. .......... .............. Isolated ZeroN et M ass Flux Devices ....................................... ............... 7 A application s .............................................. ....................... 8 M modeling approaches ............................................. .... ..... .......... ........ 11 ZeroNet Mass Flux Devices with the Addition of Crossflow............................15 Fluid dynam ic applications ........................................ ....... ............... 16 A eroacoustics applications ................................... ...................................... 18 M odeling approaches ............................................................................19 U resolved Technical Issues ........................................ .......................... 25 O bj ectives .................................. .......................... ....... ........... 27 Approach and Outline of Thesis ................................ .......................... 28 2 DYNAMICS OF ISOLATED ZERONET MASS FLUX ACTUATORS ..............30 Characterization and Param eter D efinitions .................................... ........................31 Lum ped Elem ent M odeling ......................................................... ............... 34 Sum m ary of Previous W ork ...................................... ........................ .......... 34 Limitations and Extensions of Existing M odel ................................................38 D im ensional A naly sis............ ... ............................................................ .. .... ... ... 44 D definition and D discussion ...................... ... ........... ............... ...44 Dimensionless Linear Transfer Function for a Generic Driver...........................46 v M modeling Issues ........................................................................................... ........51 C a v ity E ffe ct.................................................................................................. 5 1 Orifice Effect ............. .... .. ...... ..... ......... .... .. ....... ......... ............ 52 Lumped element modeling in the time domain.........................................52 L oss m echanism ..................... .. .. ..................... .... .. ........... 6 1 D rivingTransducer Effect........................................................ ............... 63 T est M atrix ........................................................................................ 69 3 E X PE R IM E N TA L SE TU P ............................................................. .....................72 E xperim mental Setup ............. .......................................................................... .... 72 C av ity P re ssu re ............................................................................... 7 5 D iaphragm D election .............................................................. .....................76 V elocity M easurem ent.............................................................. .....................79 D ataA acquisition System ......................................................... ............. 82 D ata P processing .............................................................85 Fourier Series D ecom position ............................................................................. 92 F low V isu alization ........ ....................................................................... ....... .. ..... .. 97 4 RESULTS: ORIFICE FLOW PHYSICS......................................... ............... 99 L ocal F low F ield ................................. ....... ........... ........................ 100 Velocity Profile through the Orifice: Numerical Results...............................100 Exit Velocity Profile: Experimental Results ............................................. 109 Jet Formation ............... ....... .......... ......... 116 Influence of Governing Param eters ................................................. ............... 118 Empirical N onlinear Threshold ............. ................................................... 119 Strouhal, Reynolds, and Stokes Numbers versus Pressure Loss.......................121 Nonlinear Mechanisms in a ZNMF Actuator ........... .................................128 5 RESULTS: CAVITY INVESTIGATION..... ...........................................137 C av ity P ressu re F ield .................................................................. .. .................... 137 E x p erim ental R esu lts............................................ ....................................... 13 8 N um erical Sim ulation Results................................ ................................... 141 Com putational fluid dynamics ....................................... ............... 142 Fem lab ................................... ............................... ........ 147 Com pressibility of the Cavity ........................................................ ............. 150 L E M B ased A n aly sis............................................ ....................................... 15 1 E xperim ental R esults................................................... .................. 156 Driver, Cavity, and Orifice Volum e Velocities....................................................... 162 6 REDUCEDORDER MODEL OF ISOLATED ZNMF ACTUATOR..................171 O rifice P ressu re D rop .................................................................... .................... 17 1 C control V olum e A analysis .......................................................................... 172 Validation through Numerical Results ............................... ..................175 Discussion: Orifice Flow Physics ........... ............................. ............... 181 Development of Approximate Scaling Laws ................................. ................188 E xperim ental results .............................................................. ............... 188 Nonlinear pressure loss correlation ............. ............................................ 194 R efined Lum ped Elem ent M odel..................................... ........................ ........... 198 Im plem entation ................... .. ..... ........................... ...... ............. 198 Com prison with Experim ental Data ..................................... .................202 7 ZERONET MASS FLUX ACTUATOR INTERACTING WITH AN EXTERNAL BOUNDARY LAYER ....................................................................211 On the Influence of Grazing Flow ................................ ......................... ........ 211 D im ensional A naly sis........... ........................................................... .... .... ... ....2 18 R educedO rder M odels....................................................................................... 223 Lumped Element ModelingBased SemiEmpirical Model of the External B ou n d ary L ay er ................................................................... ................ .. 2 2 4 D definition ................................................. 224 Boundary layer impedance implementation in Helmholtz resonators .......229 Boundary layer impedance implementation in ZNMF actuator...............238 Velocity Profile Scaling Laws ..................................... 241 Scaling law based on the jet exit velocity profile................ ...............244 Scaling law based on the jet exit integral parameters .............................261 V alidation and A application ............................................. ............... 270 8 CONCLUSIONS AND FUTURE W ORK.................................................... ......... 273 C o n c lu sio n s ............... ....... ... ................................................................... 2 7 3 Recommendations for Future Research...................... ......................... 276 Need in Extracting Specific Quantities .................................. ............... 276 Proper Orthogonal Decomposition.............................................. 277 Boundary Layer Impedance Characterization...........................279 M EM S Scale Im plem entation ........................................ ........ ............... 280 Design Synthesis Problem .................................................................... 282 APPENDIX A EXAMPLES OF GRAZING FLOW MODELS PAST HELMHOLTZ RESONATORS .................................. .. .. ........ .. ............283 B ON THE NATURAL FREQUENCY OF A HELMHOLTZ RESONATOR ..........291 C DERIVATION OF THE ORIFICE IMPEDANCE OF AN OSCILLATING PRESSURE DRIVEN CHANNEL FLOW .............................................................295 D NONDIMENSIONALIZATION OF A ZNMF ACTUATOR ..............................303 E NONDIMENSIONALIZATION OF A PIEZOELECTRICDRIVEN ZNMF ACTUATOR W ITHOUT CROSSFLOW ........................................... ............... 312 F NUMERICAL METHODOLOGY ............................ ... .................... 326 G EXPERIMENTAL RESULTS: POWER ANALYSIS .........................................331 L IST O F R E F E R E N C E S ........................................................................ ....................348 B IO G R A PH IC A L SK E T C H ........................................... ...........................................359 viii LIST OF TABLES Table page 21 Correspondence between synthetic jet parameter definitions.............................. 34 22 Dimensional parameters for circular and rectangular orifices............... ..............49 23 Test matrix for ZNMF actuator in quiescent medium ...........................................69 31 ZNMF device characteristic dimensions used in Test 1 .......................................75 32 LD V m easurem ent details............................................... .............................. 82 33 Repeatability in the experimental results..... ...................... ............92 41 Ratio of the diffusive to convective time scales ............. ..... .................109 51 Cavity volume effect on the device frequency response for Case 1 (Gallas et al.) from the LEM prediction. .............................................. ............................. 153 52 Cavity volume effect on the device frequency response for Case 1 (CFDVal) from the LEM prediction. .............................................. ............................. 154 53 ZNMF device characteristic dimensions used in Test 2.............. ...............156 54 Effect of the cavity volume decrease on the ZNMF actuator frequency response for C ases A B C and D .............................................. ............................... 157 71 List of configurations used for impedance tube simulations used in Choudhari et al.............. ..................... ............................................. ...... 2 16 72 Experimental operating conditions from Hersh and Walker ...............................230 73 Experimental operating conditions from Jing et al..........................................236 74 Tests cases from numerical simulations used in the development of the velocity profit les scaling law s............ ............................................................. .... .... .... .. 242 75 Coefficients of the nonlinear least square fits on the decomposed jet velocity p ro fi le ................................. ........................................................... ............... 2 5 4 76 Results from the nonlinear regression analysis for the velocity profile based sc a lin g law ..............................................................................................................2 5 9 77 Results for the parameters a, b and c from the nonlinear system .........................265 78 Integral param eters results ............................................. ............................. 266 79 Results from the nonlinear regression analysis for the integral parameters based v elo city p profile ................................................................................... 2 6 7 A1 Experimental database for grazing flow impedance models ...............................290 Bl Calculation of Helmholtz resonator frequency..................................................293 D1 Dimensional matrix of parameter variables for the isolated actuator case ...........304 D2 Dimensional matrix of parameter variables for the general case............................308 El Dim ensional m atrix of param eter variables..........................................................314 Gl Power in the experimental time data............. ....... ...............................332 LIST OF FIGURES Figure pge 11 Schematic of typical zeronet mass flux devices interacting with a boundary layer, showing three different types of excitation mechanisms................................4 12 Orifice geom etry. ......................... ................. ............. .. .. ..5 13 H elm holtz resonators arrays....................................... ................................ 6 21 Equivalent circuit model of a piezoelectricdriven synthetic jet actuator ................35 22 Comparison between the lumped element model and experimental frequency response measured using phaselocked LDV for two prototypical synthetic jets. ...37 23 Comparison between the lumped element model () and experimental frequency response measured using phaselocked LDV (*) for four prototypical sy nth etic jets......... ............................................................................. 4 1 24 Variation in velocity profile vs. S = 1, 12, 20, and 50 for oscillatory pipe flow in a circular duct.................... ...... .... .. ... ....................... ........... 42 25 Ratio of spatial average velocity to centerline velocity vs. Stokes number for oscillatory pipe flow in a circular duct................................... ...............43 26 Schematic representation of a genericdriver ZNMF actuator ..............................47 27 Bode diagram of the second order system given by Eq. 220, for different dam ping ratio ...................................................... ................. 4 8 28 Coordinate system and sign convention definition in a ZNMF actuator ................53 29 Geometry of the piezoelectricdriven ZNMF actuator from Case 1 (CFDVal). ......55 210 Geometry of the pistondriven ZNMF actuator from Case 2 (CFDVal). ................55 211 Time signals of the jet orifice velocity, pressure across the orifice, and driver displacement during one cycle for Case 1. .................................... ............... 57 212 Time signals of the jet orifice velocity, pressure across the orifice and driver displacement during one cycle for Case 2. .................................... .................58 213 Numerical results of the time signals for A) pressure drop and B) velocity perturbation at selected locations along the resonator orifice..............................59 214 Schematic of the different flow regions inside a ZNMF actuator orifice ...............62 215 Equivalent twoport circuit representation of piezoelectric transduction. ................64 216 Speakerdriven ZNM F actuator. ........................................ ......................... 66 217 Schematic of a shakerdriven ZNMF actuator, showing the vent channel betw een the tw o sealed cavities. ........................................ ......................... 67 218 Circuit representation of a shakerdriven ZNMF actuator .............. ...............68 31 Schematic of the experimental setup for phaselocked cavity pressure, diaphragm deflection and offaxis, twocomponent LDV measurements ...............73 32 Exploded view of the modular piezoelectricdriven ZNMF actuator used in the experim mental test. ......................................................................73 33 Schematic (to scale) of the location of the two 1/8" microphones inside the Z N M F actuator cavity ....................................................................... ..................76 34 Laser displacement sensor apparatus to measure the diaphragm deflection with sign convention. .......................................................................77 35 Diaphragm mode shape comparison between linear model and experimental data at three test conditions ............................................... ............. ............... 79 36 LDV 3beam optical configuration. ........................................ ....................... 80 37 Flow chart of m easurem ent setup. ........................................... ........................83 38 Phaselocked signals acquired from the DSA card, showing the normalized trigger signal, displacement signal, pressure signals and excitation signal .............84 39 Percentage error in Error! Objects cannot be created from editing field codes. from simulated LDV data at different signal to noise ratio, using 8192 samples.....87 310 Phaselocked velocity profiles and corresponding volume flow rate acquired w ith L D V for C ase 14 ....................................................................... ..................89 311 Noise floor in the microphone measurements compared with Case 52 ..................91 312 N orm alized quantities vs. phase angle ........................................... ............... 93 313 Power spectrum of the two pressure recorded and the diaphragm displacement. ...95 314 Schem atic of the flow visualization setup...................................... .....................97 41 Numerical results of the orifice flow pattern showing axial and longitudinal velocities, azimuthal vorticity contours, and instantaneous streamlines at the tim e of m axim um expulsion. ............... .......... ......... ................................. 101 42 Velocity profile at different locations inside the orifice for Case 1......................103 43 Velocity profile at different locations inside the orifice for Case 2........................105 44 Velocity profile at different locations inside the orifice for Case 3.......................106 45 Vertical velocity contours inside the orifice during the time of maximum ex pu lsion ........................................................................ 10 7 46 Experimental vertical velocity profiles across the orifice for a ZNMF actuator in quiescent medium at different instant in time for Case 71 ............................... 110 47 Experimental vertical velocity profiles across the orifice for a ZNMF actuator in quiescent medium at different instant in time for Case 43 ..................................11 48 Experimental vertical velocity profiles across the orifice for a ZNMF actuator in quiescent medium at different instant in time for Case 69. ................................113 49 Experimental vertical velocity profiles across the orifice for a ZNMF actuator in quiescent medium at different instant in time for Case 55 ............................... 114 410 Experimental results of the ratio between the time and spatialaveraged velocity and timeaveraged centerline velocity. .......................... ....................116 411 Experimental results on the jet formation criterion. ............................................... 118 412 Averaged jet velocity vs. pressure fluctuation for different Stokes number...........120 413 Pressure fluctuation normalized by the dynamic pressure based on averaged velocity vs. St h/d ................................................................. .... ........... 122 414 Pressure fluctuation normalized by the dynamic pressure based on averaged velocity vs. Strouhal num ber. ............................................................................ 123 415 Vorticity contours during the maximum expulsion portion of the cycle from num erical sim ulations. ................................................ ................................ 124 416 Pressure fluctuation normalized by the dynamic pressure based on ingestion tim e averaged velocity vs. St h d ............................. ..... ............................ 125 417 Vorticity contours during the maximum ingestion portion of the cycle from num erical sim ulations. ................................................ ................................ 126 418 Comparison between Case 1 vertical velocity profiles at the orifice ends ...........127 419 Comparison between Case 2 vertical velocity profiles at the orifice ends. ............128 420 Comparison between Case 3 vertical velocity profiles at the orifice ends. ............128 421 Determination of the validity of the smallsignal assumption in a closed cavity. ..131 422 Loglog plot of the cavity pressure total harmonic distortion in the experimental tim e sign als. ...................................................... ................. 132 423 Loglog plot of the total harmonic distortion in the experimental time signals vs. Strouhal number as a function of Stokes number. ............................................... 134 51 Coherent power spectrum of the pressure signal for Cases 9 to 20........................138 52 Phase plot of the normalized pressures taken by microphone 1 versus m microphone 2 ............................................ ........ ................. 139 53 Pressure signals experimentally recorded by microphone 1 and microphone 2 as a function of phase in Case 59. .............................. ....... .. ............... ...... 140 54 Ratio of microphone amplitude (Pa) vs. the inverse of the Strouhal number, for different Stokes num ber. ............................................... ............................... 141 55 Pressure contours in the cavity and orifice (Case 2) from numerical simulations.. 143 56 Pressure contours in the cavity and orifice (Case 3) from numerical simulations.. 144 57 Cavity pressure probe locations in a ZNMF actuator from numerical sim ulations. .......................................... ........................... 145 58 Normalized pressure inside the cavity during one cycle at 15 different probe locations from numerical simulation results. .................................. ............... 146 2 59 Cavity pressure normalized by pVj vs. phase from numerical simulations corresponding to the experimental probing locations...................................147 510 Contours of pressure phase inside the cavity by numerically solving the 3D w ave equation using FEM LAB .......................................................... ....... ........ 148 511 Cavity pressure vs. phase by solving the 3D wave equation using FEMLAB and corresponding to the experimental probing locations...................................149 512 Loglog frequency response plot of Case 1 (Gallas et al.) as the cavity volume is decreased from the LEM prediction.................................. 153 513 Loglog frequency response plot of Case 1 (CFDVal) as the cavity volume is decreased from the LEM prediction.................................. 154 514 Experimental loglog frequency response plot of a ZNMF actuator as the cavity volume is decreased for a constant input voltage. ............................................. 158 515 Closeup view of the peak locations in the experimental actuator frequency response as the cavity volume is decreased for a constant input voltage. ............158 516 Normalized quantities vs. phase of the jet volume rate, cavity pressure and centerline driver velocity.. ............................. .... ...................................... 160 517 Experimental results of the ratio of the driver to the jet volume velocity function of dimensionless frequency as the cavity volume decreases. ............................... 164 518 Experimental jet to driver volume flow rate versus actuation to Helmholtz frequency..................................... .......................... .... ..... ......... 166 519 Current divider representation of a piezoelectricdriven ZNMF actuator. .............168 520 Frequency response of the power conservation in a ZNMF actuator from the lumped element model circuit representation for Case 1 (Gallas et al.)...............69 61 Control volume for an unsteady laminar incompressible flow in a circular orifice, from y = 1 to = 0 ............... ..................................... ............... 172 62 Numerical results for the contribution of each term in the integral momentum equation as a function of phase angle during a cycle ................ ......... ..........176 63 Definition of the approximation of the orifice entrance velocity from the orifice exit velocity ............ ... ......................................... ......................... 178 64 Momentum integral of the exit and inlet velocities normalized by Error! Objects cannot be created from editing field codes. and comparing with the actual and approxim ated entrance velocity. ........................................ ......................... 179 65 Total momentum integral equation during one cycle, showing the results using the actual and approximated entrance velocity. .................. .............................. 181 66 Numerical results of the total shear stress term versus corresponding lumped linear resistance during one cycle. ............................................... ............... 183 67 Numerical results of the unsteady term versus corresponding lumped linear reactance during one cycle. .............................................. ............................ 184 68 Numerical results of the normalized terms in the integral momentum equation as a function of phase angle during a cycle. ............................. .................187 69 Comparison between lumped elements from the orifice impedance and analytical terms from the control volume analysis. .............................................. 188 610 Experimental results of the orifice pressure drop normalized by the dynamic pressure based on averaged velocity versus St h/d for different Stokes num bers................................... ................................. ........... 191 611 Experimental results of each term contributing in the orifice pressure drop coefficient vs. St h/d ....................... .............. ................. ........ 192 612 Experimental results of the relative magnitude of each term contributing in the orifice pressure drop coefficient vs. intermediate to low St h/d ........................193 613 Experimental results for the nonlinear pressure loss coefficient for different Stokes number and orifice aspect ratio. ...................................... ............... 196 614 Nonlinear term of the pressure loss across the orifice as a function of St h/d from experim mental data. ............................................... ............................... 197 615 Implementation of the refined LEM technique to compute the jet exit velocity frequency response of an isolated ZNMF actuator. .........................................201 616 Comparison between the experimental data and the prediction of the refined and previous LEM of the impulse response of the jet exit centerline velocity. Actuator design corresponds to Case I from Gallas et al.......................................203 617 Comparison between the experimental data and the prediction of the refined and previous LEM of the impulse response of the jet exit centerline velocity. Actuator design corresponds to Case II from Gallas et al....................................205 618 Comparison between the experimental data and the prediction of the refined and previous LEM of the impulse response of the jet exit centerline velocity. Actuator design is from Gallas and is similar to Cases 41 to 50. .........................207 619 Comparison between the refined LEM prediction and experimental data of the time signals of the jet volume flow rate .......................................................209 71 Spanwise vorticity plots for three cases where the jet Reynolds number Re is in crea se d ............................. .......................................................... ............... 2 12 72 Spanwise vorticity plots for three cases where the boundary layer Reynolds num ber is increased. ....................................... ................. ..........213 73 Comparison of the jet exit velocity profile with increasing..................................214 74 Pressure contours and streamlines for mean A) inflow, and B) outflow through a resonator in the presence of grazing flow. ............ ...................... ....................218 75 LEM equivalent circuit representation of a generic ZNMF device interacting w ith a grazing boundary layer.................................................................... ...... 224 76 Schematic of an effort divider diagram for a Helmholtz resonator ......................230 77 Comparison between BL impedance model and experiments from Hersh and Walker as a function of Mach number for different SPL. ............................... 233 78 Experim mental setup used in Jing et al........................................... ............... 236 79 Comparison between model and experiments from Jing et al .............................237 710 Effect of the freestream Mach number on the frequency response of the ZNMF design from Case 1 (CFDVal) using the refined LEM. ........................................239 711 Effect of the freestream Mach number on the frequency response of the ZNMF design from Case 1 (G allas et al.) ................................................. .. ... .......... 240 712 Schematic of the two approaches used to develop the scaling laws from the jet exit velocity profile. ......................................... ..................... .......244 713 Methodology for the development of the velocity profile based scaling law.........245 714 Nonlinear least square curve fit on the decomposed jet velocity profile for Case I ...........................................................................................2 4 7 715 Nonlinear least square curve fit on the decomposed jet velocity profile for Case III ................... ......................................................................... 2 4 8 716 Nonlinear least square curve fit on the decomposed jet velocity profile for Case V ...................................................................................... . 2 4 9 717 Nonlinear least square curve fit on the decomposed jet velocity profile for Case V II ........................................................................................2 50 718 Nonlinear least square curve fit on the decomposed jet velocity profile for Case IX .................. ......................................................... ................ 2 5 1 719 Nonlinear least square curve fit on the decomposed jet velocity profile for Case X I. ........................................................................................2 52 720 Nonlinear least square curve fit on the decomposed jet velocity profile for Case X III ...................................... .................................................... 2 5 3 721 Comparison between CFD velocity profile, decomposed jet velocity profile, and modeled velocity profile, at the orifice exit, for four phase angles during a cy cle. ..............................................................................2 55 722 Test case comparison between CFD data and the scaling law based on the velocity profile at four phase angles during a cycle.................................... 260 723 Methodology for the development of the integral parameters based scaling law...262 724 Comparison between the results of the integral parameters from the scaling law and the CFD data for the test case.................................... ......................... 268 725 Example of a practical application of the ZNMF actuator reducedorder model in a numerical simulation of flow past a flat plate ............................................271 81 POD analysis applied on numerical data for ZNMF actuator with a grazing BL...278 82 Use of quarterwavelength open tube to provide an infinite impedance. ..............280 83 Representative MEMS ZNMF actuator. ....................................... ............... 281 84 Predicted output of MEMS ZNMF actuator .........................................................281 A1 Acoustic test duct and siren showing a liner panel test configuration..................285 A2 Schematic of test apparatus used in Hersh and Walker. ..........................286 A3 Apparatus for the measurement of the acoustic impedance of a perforate used by K irby and C um m ings. ........................................ .........................................288 A4 Sketch of NASA Grazing Impedance Tube..........................................................290 B l H elm holtz resonator ........................................................................ ..................29 1 C1 Rectangular slot geometry and coordinate axis definition............... .......... 295 Dl Orifice details with coordinate system .... ........... ........ .......................... 303 Fl Schematic of A) the sharpinterface method on a fixed Cartesian mesh, and B) the ZNMF actuator interacting with a grazing flow. ............................................328 F2 Typical mesh used for the computations. A) 2D simulation. B) 3D simulation...329 F3 Example of 2D and 3D numerical results of ZNMF interacting with a grazing b o u n d ary lay er ................................................................................... 3 2 9 xviii LIST OF SYMBOLS AND ABBREVIATIONS c, isentropic air speed of sound [m/s] Ca cavity acoustic compliance = V/pco2 [s2.m4/kg] CD diaphragm shortcircuit acoustic compliance =AV/Pl 0 [s2m4/kg] CD orifice discharge coefficient [1] C. skin friction coefficient = r,/O.5pV2 [1] C momentum coefficient during the time of discharge [1] C"2 successive moments of jet velocity profile [1] d orifice diameter [m] dH hydraulic diameter = 4(area)/(wetted perimeter) [m] D orifice entrance diameter (facing the cavity) [m] Dc cavity diameter (for cylindrical cavities) [m] f actuation frequency [Hz] f, driver natural frequency [Hz] fH Helmholtz frequency= (1/2;) co~ n/V = 1/ (2z (M^, +Mad )C)c [Hz] fn natural frequency of the uncontrolled flow [Hz] f, fundamental frequency [Hz] f, f2 synthetic jet lowest and highest resonant frequencies, respectively [Hz] h orifice height [m] h' effective length of the orifice = h+h0 [m] ha "end correction" of the orifice = 0.964S [m] H cavity depth (m) / boundary layer shape factor = 0/3* [1] I0 impulse per unit length [1] k wave number = c/co [m1] Kd nondimensional orifice loss coefficient [1] L0 stroke length [m] MaD diaphragm acoustic mass = A 2 [w r2 rdr [kg/4] M, orifice acoustic mass due to inertia effect [kg/m4] M o orifice acoustic mass = M + + Mad [kg/m4] MaRad orifice acoustic radiation mass [kg/m4] p' acoustic pressure [Pa] P differential pressure on the diaphragm [Pa] P, incident pressure [Pa] Pw Power [W] q' acoustic particle volume velocity [m3/s] Qc volume flow rate through the cavity = Q Qd [m3/s] Qd volume flow rate displaced by the driver = jiAV [m3/s] Q, volume flow rate through the orifice [m3/s] Q, time averaged orifice volume flow rate during the expulsion stroke [m3/s] r radial coordinate in cylindrical coordinate system [m] R radius of curvature of the surface [m] R, diaphragm acoustic resistance = 24JMaD /C [kg/m4s] R, viscous orifice acoustic resistance [kg/m4s] Roohn linear orifice acoustic resistance = R, [kg/m4s] Ron, nonlinear orifice acoustic resistance [kg/m4s] R? specific resistance [kg/m2s] Re jet Reynolds number = Vd/v [1] s Laplace variable =jco [rad/s] S Stokes number = od/v [1] St jet Strouhal number = cd/V [1] So cavity cross sectional area [m2] Sd driver cross sectional area [m2] S, orifice neck area [m2] u' acoustic particle velocity [m/s] Ub bias flow velocity through the orifice [m/s] u, wall friction velocity [m/s] U, freestream mean velocity [m/s] VCL centerline orifice velocity [m/s] V( spatial averaged jet exit velocity = Q /S, = (;r/2) V [m/s] V spatial and timeaveraged jet exit velocity during the expulsion stroke [m/s] Vac input ac voltage [V] V normalized jet velocity = vJ /U [m/s] w length of a rectangular orifice [m] w(r) transverse displacement of the diaphragm [m] W width of the cavity [m] W, centerline amplitude of the driver [m] X0 acoustic reactance = o)M [kg/m4s] X, specific reactance [kg/m2s] XI2 skewness of jet velocity profile [1] Yd vibrating driver displacement [m] yJ fluid particle displacement at the orifice [m] Z, acoustic impedance = R + jX = p'/q' [kg/m4.2] Z c acoustic cavity impedance = (jcoCac) 1= APj(Q Qj ) [kg/m4.s2] Z o acoustic impedance of the orifice = Ror,, + ko,,,i + joMo = A/Qj [kg/m4s2] ZBL acoustic impedance of the grazing boundary layer = RL + jXL [kg/m4.2] Zo, total acoustic impedance of the orifice = Zo + ZBL [kg/m4s2] Z, specific impedance = R + jXo = p'/u' [kg/m2.2] Z,p perforate specific impedance = R, + jX, = Zo/cr [kg/m2s2] a thermal diffusivity [m2/s] p nondimensional pressure gradient = (3*/rz)(dP/dx) [1] x normalized reactance [1] 3 boundary layer thickness [m] 3* boundary layer displacement thickness [m] stokes Stokes layer thickness = v/O [m] Acp normalized pressure drop = (pO Py 0.5pV) [1] AN cavity pressure [Pa] AV volume displaced by the driver [m3] 0, electroacoustic turns ratio of the piezoceramic diaphragm = Id/C, [Pa/ V] &Of phase difference between the incident sound field and the cavity sound field [deg] 7 ratio of the specific heats [1] A wavelength = co/f = 27/k [m] p dynamic viscosity = pv [kg/m's] v cinematic viscosity [m2/s] p density [kg/m3] PA area density [kg/m2] 0 boundary layer momentum thickness [m] / normalized resistance [1] a porosity of the perforate plate = Nhole x (hole area)/total area [%] a ratio of the orifice to cavity cross sectional area = S,/SC [1] z, wall shear stress [kg/m s2] xxiii V cavity volume [mm3] co radian frequency = 2;if [rad/s] Q, vorticity flux [m2/s] " damping coefficient [1] / normalized impedance = 0 + j [1] Cp normalized impedance of a perforate = O + jp [1] S compliance ratio = CD/CaC [1] 9NR mass ratio = MaD/MA [1] 91 resistance ratio = R, /RD [1] Commonly used subscripts: a acoustic domain c cavity CL centerline d driver D diaphragm ex expulsion phase of the cycle in injection phase of the cycle j jet lin linear nl nonlinear p perforate 0 specific o0 freestream xxiv Commonly used superscripts: spatial averaged spatial and time averaged fluctuating quantity Abbreviations: BL Boundary Layer CFD Computational Fluid Dynamics HWA Hot Wire Anemometry LDV Laser Doppler Velocimetry LEM Lumped Element Modeling MEMS Micro Electromechanical Systems MSV Mean Square Value PIV Particle Image Velocimetry POD Proper Orthogonal Decomposition RMS Root Mean Square ZNMF ZeroNet Mass Flux Throughout this dissertation, the term syntheticc jet actuator has the same meaning as zeronet mass flux actuator, although the former is physically more restricting to specific applications (strictly speaking, a jet must be formed). Similarly, the terms grazingflow and bias flow in the acoustic community are used interchangeably with the respective fluid dynamics terminology crossflow and mean flow, since they refer to the same phenomenon. Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON THE MODELING AND DESIGN OF ZERONET MASS FLUX ACTUATORS By Quentin Gallas May 2005 Chair: Louis Cattafesta Major Department: Mechanical and Aerospace Engineering This dissertation discusses the fundamental dynamics of zeronet mass flux (ZNMF) actuators commonly used in active flowcontrol applications. The present work addresses unresolved technical issues by providing a clear physical understanding of how these devices behave in a quiescent medium and interact with an external boundary layer by developing and validating reducedorder models. The results are expected to ultimately aid in the analysis and development of design tools for ZNMF actuators in flowcontrol applications. The case of an isolated ZNMF actuator is first documented. A dimensional analysis identifies the key governing parameters of such a device, and a rigorous investigation of the device physics is conducted based on various theoretical analyses, phaselocked measurements of orifice velocity, diaphragm displacement, and cavity pressure fluctuations, and available numerical simulations. The symmetric, sharp orifice exit velocity profile is shown to be primarily a function of the Strouhal and Reynolds numbers and orifice aspect ratio. The equivalence between Strouhal number and xxvi dimensionless stoke length is also demonstrated. A criterion is developed and validated, namely that the actuationtoHelmholtz frequency ratio is less than 0.5, for the flow in the actuator cavity to be approximately incompressible. An improved lumped element modeling technique developed from the available data is developed and is in reasonable agreement with experimental results. Next, the case in which a ZNMF actuator interacts with an external grazing boundary layer is examined. Again, dimensional analysis is used to identify the dimensionless parameters, and the interaction mechanisms are discussed in detail for different applications. An acoustic impedance model (based on the NASA "ZKTL model") of the grazing flow influence is then obtained from a critical survey of previous work and implemented in the lumped element model. Two scaling laws are then developed to model the jet velocity profile resulting from the interaction the profiles are predicted as a function of local actuator and flow condition and can serve as approximate boundary conditions for numerical simulations. Finally, extensive discussion is provided to guide future modeling efforts. xxvii CHAPTER 1 INTRODUCTION Motivation The past decade has seen numerous studies concerning an exciting type of active flow control actuator. Zeronet mass flux (ZNMF) devices, also known as synthetic jets, have emerged as versatile actuators with potential applications such as thrust vectoring of jets (Smith and Glezer 1997), heat transfer augmentation (Campbell et al. 1998; Guarino and Manno 2001), active control of separation for low Mach and Reynolds numbers (Wygnanski 1997; Smith et al. 1998; Amitay et al. 1999; Crook et al. 1999; Holman et al. 2003) or transonic Mach numbers and moderate Reynolds numbers (Seifert and Pack 1999, 2000a), and drag reduction in turbulent boundary layers (Rathnasingham and Breuer 1997; Lee and Goldstein 2001). This versatility is primarily due to several reasons. First, these devices provide unsteady forcing, which has proven to be more effective than its steady counterpart (Seifert et al. 1993). Second, since the jet is synthesized from the working fluid, complex fluid circuits are not required. Finally the actuation frequency and waveform can usually be customized for a particular flow configuration. Synthetic jets exhausting into a quiescent medium have been studied extensively both experimentally and numerically. Additionally, other studies have focused on the interaction with an external boundary layer for the diverse applications mentioned above. However, many questions remain unanswered regarding the fundamental physics that govern such complex devices. Practically, because of the presence of rich flow physics and multiple flow mechanisms, proper implementation of these actuators in realistic applications requires design tools. In turn, simple design tools benefit significantly from loworder dynamical models. However, no suitable models or design tools exist because of insufficient understanding as to how the performance of ZNMF actuator devices scales with the governing nondimensional parameters. Numerous parametric studies provide a glimpse of how the performance characteristics of ZNMF actuators and their control effectiveness depend on a number of geometrical, structural, and flow parameters (Rathnasingham and Breuer 1997; Crook and Wood 2001; He et al. 2001; Gallas et al. 2003a). However, nondimensional scaling laws are required since they form an essential component in the design and deployment of ZNMF actuators in practical flow control applications. For instance, scaling laws are expected to play an important role in the aerodynamic design of wings that, in the future, may use ZNMF devices for separation control. The current design paradigm in the aerospace industry relies heavily on steady Reynolds Averaged Navier Stokes (RANS) computations. A validated unsteady RANS (URANS) design tool is required for separation control applications at transonic Mach numbers and flight Reynolds number. However, these computations can be quite expensive and timeconsuming. Direct modeling of ZNMF devices in these computations is expected to considerably increase this expense, since the simulations must resolve the flow details in the vicinity of the actuator while also capturing the global flow characteristics. A viable alternative to minimize this cost is to simply model the effect of the ZNMF device as a time and flowdependant boundary condition in the URANS calculation. Such an approach requires that the device be characterized by a small set of nondimensional parameters, and the behavior of the actuator must be well understood over a wide range of conditions. Furthermore, successful implementation of robust closedloop control methodologies for this class of actuators calls for simple (yet effective) mathematical models, thereby emphasizing the need to develop a reducedorder model of the flow. Such loworder models will clearly aid in the analysis and development of design tools for sizing, design and deployment of these actuators. Below, an overview of the basic operating principles of a ZNMF actuator is provided. Overview of a ZeroNet Mass Flux Actuator Typically, ZNMF devices are used to inject unsteady disturbances into a shear flow, which is known to be a useful tool for active flow control. Most flow control techniques require a fluid source or sink, such as steady or pulsed suction (or blowing), vortexgenerator jets (Sondergaard et al. 2002; Eldredge and Bons 2004), etc., which introduces additional constraints in the design of the actuator and sometimes results in complicated hardware. This motivates the development of ZNMF actuators, which introduce flow perturbations with zeronet mass injection, the large coherent structures being synthesized from the surrounding working fluid (hence the name "synthetic jet"). A typical ZNMF device with different transducers is shown in Figure 11. In general, a ZNMF actuator contains three components: an oscillatory driver (examples of which are discussed below), a cavity, and an orifice or slot. The oscillating driver compresses and expands the fluid in the cavity by altering the cavity volume V at the excitation frequency f to create pressure oscillations. As the cavity volume is decreased, the fluid is compressed in the cavity and expels some fluid through the orifice. The time and spatial averaged ejection velocity during this portion of the cycle is denoted V Similarly, as the cavity volume is increased, the fluid expands in the cavity and ingests some fluid through the orifice. Common orifice geometries include simple axisymmetric hole (height h, diameter d) and rectangular slot (height h, depth d and width w), as schematically shown in Figure 12. Downstream from the orifice, a jet laminarr or turbulent, depending on the jet Reynolds number Re= Fd/v) is then synthesized from the entrained fluid and sheds vortices when the driver oscillations exceed a critical amplitude (Utturkar et al. 2003). pU,,M J dIHI Volume V U',M P.  A '0 Volume V I sin (27ift) Ft I 1r signal (f) Figure 11: Schematic of typical zeronet mass flux devices interacting with a boundary layer, showing three different types of excitation mechanisms. A) Piezoelectric diaphragm. B) Oscillating piston. C) Acoustic excitation. Even though no net mass is injected into the embedded flow during a cycle, a nonzero transfer of momentum is established with the surrounding flow. The exterior flow, if present, usually consists of a turbulent boundary layer (since most practical applications deal with such a turbulent flow) and is characterized by the freestream velocity U. and acoustic speed c., pressure gradient dP/dx, radius of curvature R, thermal diffusivity a~, and displacement 53 and momentum 0 thicknesses. Finally, the ambient fluid is characterized by its density pm and dynamic viscosity /u,. L y A B L I t h zx V d vi h I \/' Figure 12: Orifice geometry. A) Axisymmetric. B) Rectangular. Figure 11 shows three kinds of drivers commonly used to generate a synthetic jet: * An oscillating membrane (usually a piezoelectric patch mounted on one side of a metallic shim and driven by an ac voltage). * A piston mounted in the cavity (using an electromagnetic shaker, a camshaft, etc.). * A loudspeaker enclosed in the cavity (an electrodynamic voicecoil transducer). For each of them, we are interested mainly in the volume displacement generated by the driver that will eject and ingest the fluid through the orifice. Although each driver will obviously have its own characteristics, common parameters of a generic driver are its frequency of excitation f, the corresponding volume AV that it displaces, and the dynamic modal characteristics of the driver. A Porous B Acoustic Liners A /"^ Porous B __ Sface sheet V ^d Inlet Honeycomb core sheet Nacelle Fan Figure 13: Helmholtz resonators arrays. A) Schematic. B) Application in engine nacelle acoustic liners. Although noticeable differences exist, it is worthwhile to compare synthetic jets with the phenomenon of acoustic flow generation, the acoustic streaming, extensively studied by aeroacousticians in the past (e.g., Lighthill 1978). Acoustic streaming is the result of a steady flow produced by an acoustic field and is the evidence of the generation of vorticity by the sound, which occurs for example when sound impinges on solid boundaries. Quoting Howe (1998, p. 410), When a sound wave impinges on a solid surface in the absence of mean flow, the dissipated energy is usually converted directly into heat through viscous action. At very high acoustic amplitudes, however, free vorticity may still be formed at edges, and dissipation may take place, as in the presence of mean flow, by the generation of vortical kinetic energy which escapes from the interaction zone by self induction. This nonlinear mechanism can be important in small perforates or apertures. This type of flow generation could be relevant in the application of ZNMF devices where similar nonlinear flow through the orifice is expected. In particular, ZNMF devices are similar to flowinduced resonators, such as Helmholtz resonators used in acoustic liners as soundabsorber devices. As Figure 13 shows, a simple single degree offreedom (SDOF) liner consists of a perforate sheet backed with honeycomb cavities and interacting with a grazing flow. Similar liners with a second cavity (or more) are commonly used in engine nacelles to attenuate the sound noise level. More recently, Flynn et al. (1990) and Urzynicok and Fernholz (2002) used Helmholtz resonators for flow control applications. More details will be given in subsequent sections. Now that an overview of the problem has been presented along with a general description of a ZNMF device, an indepth literature survey is given to familiarize the reader with the existing developments on these subjects and to clearly set the scope of the current investigation. The objectives of this research are then formulated and the technical approach described to reach these goals. Literature Review This section presents an overview of the relevant research found in the open literature. The goal is to highlight and extract the principal features of the actuator and associated fluid dynamics, and to identify unresolved issues. First, the simpler yet practically significant case in which the synthetic jet exhausts into a quiescent medium is carefully reviewed. The case in which the synthetic jet interacts with a grazing boundary layer or crossflow is considered next. The survey reveals available experimental and numerical simulation data on the local interaction of a ZNMF device with an external boundary layer. In each subsection, the diverse applications that have employed a ZNMF actuator are first reviewed, as well as the different modeling approaches used. In the case of the presence of a grazing boundary layer, examples of applications in the field of fluid dynamics and aeroacoustics are presented where a parallel with sound absorber technology is drawn. Isolated ZeroNet Mass Flux Devices Numerous studies have addressed the fundamentals and applications of isolated ZNMF actuators. The list presented next is by no means exhaustive but reflects the major points and contributions to the understanding of such devices. Applications Mixing enhancement, heat transfer, or thrust vectoring are the major applications of isolated ZNMF devices, as opposed to active flow control applications when the actuator is interacting with an external boundary layer that will be seen in the next section. Chen et al. (1999) demonstrated the use of ZNMF actuators to enhance mixing in a gas turbine combustor. They used two streams of hot and cold gas to simulate the mixing and they measured the temperature distribution downstream of the synthetic jet to determine the effectiveness of the mixing. Their experiments showed that ZNMF devices could improve mixing in a turbine jet engine without using additional cold dilution air. Similarly, modification and control of smallscale motions and mixing processes via ZNMF actuators were investigated by Davis et al. (1999). Their experiments used an array of ZNMF devices placed around the perimeter of the primary jet. It was demonstrated that the use of these actuators made the shear layer of the primary jet spread faster with downstream distance, and the centerline velocity decreased faster in the streamwise direction, while the velocity fluctuations near the centerline were increased. In a heat transfer application, Campbell et al. (1998) explored the option of using ZNMF actuators to cool laptop computers. A small electromagnetic actuator was used to create the jet that was used to cool the processor of a laptop computer. Using optimum combination of various design parameters, the synthesized jet was able to lower the processor operating temperature rise by 22% when compared to the uncontrolled case. Not surprisingly, it was envisioned that optimization of the device design could lead to further improvement in the performance. Likewise, a thermal characterization study of laminar air jet impingement cooling of electronic components in a representative geometry of the CPU compartment was reported by Guarino and Manno (2001). They used a finite controlvolume technique to solve for velocity and temperature fields (including convection, conduction and radiation effects). With jet Reynolds numbers ranging from 63 to 1500, their study confirmed the importance of the Reynolds number (rather than jet size) for effective heat transfer. Proof of the above concept was demonstrated with a numerical model of a laptop computer. In a thrust vectoring application, Smith et al. (1999) performed an experiment to study the formation and interaction of two adjacent ZNMF actuators placed beside the rectangular conduit of the primary jet. Each actuator had two modes of operation depending on direction of the synthetic jet with respect to the primary jet. It was demonstrated that the primary jet could be vectored at different angles by operating only one or both actuators in different modes. Later, Guo et al. (2000) numerically simulated these experimental results. Similarly, Smith and Glezer (2002) experimentally studied the vectoring effect between ZNMF devices near a steady jet with varying velocity, while Pack and Seifert (1999) did the same by employing periodic excitation. Others studies focused on characterizing isolated ZNMF actuators (Crook and Wood 2001; Smith and Glezer 1998). For instance, a careful experimental study by Smith and Glezer (1998) shows the formation and evolution of twodimensional synthetic jets evolving in a quiescent medium for a limited range of jet performance parameters. The synthetic jets were viewed using schlieren images via the use of a small tracer gas, and velocity fields were acquired by hot wire anemometry at different locations in space, for phaselocked and longtime averaged signals. In these experiments, along with those from Carter and Soria (2002), Bera et al. (2001) or Smith and Swift (2003a), the similarities and differences between a synthetic jet and a continuous jet have been noted and examined. Specifically, Amitay et al. (1998) and Smith et al. (1998) confirmed selfsimilar velocity profiles in the asymptotic regions via a direct comparison at the same jet Reynolds number. In terms of design characteristics, it is of practical importance to know if the ZNMF actuator synthesizes a jet via discrete vortex shedding. Utturkar et al. (2003) derived and validated a criterion for whether a jet is formed at the orifice exit of the actuator. It is governed by the square of the orifice Stokes number S2 = cod2/v and the jet Reynolds number Re= Vd/v based on the orifice diameter d and the spatiallyaveraged exit velocity V during the expulsion stroke, which holds for both axisymmetric and two dimensional orifice geometry. Their derivation is based on the criterion that the induced velocity at the orifice neck must be greater than the suction velocity for the vortices to be shed; and was verified by numerical simulations and by experiments. Their data support the jet formation criterion Re/S2 >K, where K is 0(1). In another attempt, Shuster and Smith (2004) based their criterion from PIV flow visualization for different circular orifice shape (straight, beveled or rounded) and found that it is governed by the nondimensional stroke length Lo/d and the orifice geometry, where LO is the fluid stroke length assuming a slug flow model for the jet velocity profile. Modeling approaches Few analytical models have yet characterized ZNMF actuator behavior, even for the simple case of a quiescent medium. Actually, most of the studies have been performed either via experimental efforts or numerical simulations. Several attempts have been made to reduce computational costs. For instance, Kral et al. (1997) performed twodimensional, incompressible simulations of an isolated ZNMF actuator. Interestingly, their study was performed in the absence of the actuator per se. Instead, a sinusoidal velocity profile was prescribed as a boundary condition at the jet exit in lieu of simulating the actuator, including calculations in the cavity. Both laminar and turbulent jets were studied, and although the laminar jet simulation failed to capture the breakdown of the vortex train that is commonly observed experimentally, the turbulent model showed the counterrotating vortices quickly dissipating. This suggests that the modeled boundary condition could capture some of the features of the jet, without the simulation of the flow inside the actuator cavity. In another numerical study, Rizzetta et al. (1999) used a direct numerical simulation (DNS) to solve the compressible NavierStokes equations for both 2D and 3D domain. They calculated both the interior of the actuator cavity and the external flowfield, where the cavity flow was simulated by prescribing an oscillating boundary condition at one of the cavity surfaces. However, the recorded profiles of the periodic jet exit velocity were used as the boundary condition for the exterior domain. Hence, by using this decoupling technique, they could calculate the exterior flow without simultaneously simulating the flow inside the actuator cavity. To further reduce the computational cost, the planes of symmetry were forced at the jet centerline and at the midspan location, so only a quarter of the real actuator was simulated. However, the 2D simulations were not able to capture the breakdown of the vortices as a result of the spanwise instabilities. Cavity design earned the attention of several researchers, such as Rizzeta et al. (1999) presented above; Lee and Goldstein (2002), who performed a 2D incompressible DNS study of isolated ZNMF actuators; and Utturkar et al. (2002), who did a thorough investigation of the sensitivity of the jet to cavity design using a 2D unsteady viscous incompressible solver using complex immersed moving boundaries on Cartesian grids. Utturkar et al. (2002) found that the placement of the driver inside the cavity (perpendicular or normal to the orifice exit) does not significantly affect the output characteristics. The orifice is an important component of actuator modeling. While numerous parametric studies examined various orifice geometry and flow conditions, a clear understanding of the loss mechanism is still lacking. Investigations based on orifice flows have been carried as far back as the 1950s. A comprehensive experimental study was carried out by Ingard and Ising (1967) that examined the acoustic nonlinearity of the orifice. It was observed that the relation between the pressure and the velocity transitions from linear to quadratic nature as the transmitted velocity u' crosses a threshold value nrtlc, i.e p' pcou' if u' < u rtC and else p' pu'2, where p is the density, co is the speed of sound and p' is the sound pressure level. The phase relationship between the pressure fluctuations and the velocity were also investigated. Later, Seifert and Pack (1999), in an effort to investigate the effect of oscillatory blowing on flow separation, developed a simple scaling between the pressure fluctuation inside the cavity and the velocity fluctuation. This scaling agrees with the work of Ingard and Ising (1967) and states that for low amplitude blowing u' ~ p'/pco whereas for high amplitude blowing u'~7p/p. Recently, similar to the work by Smith and Swift (2003b) who experimentally studied the losses in an oscillatory flow through a rounded slot, Gallas et al (2004) performed a conjoint numerical and experimental investigation on the orifice flow for sharp edges to understand the unsteady flow behavior and associated losses in the orifice/slot of ZNMF devices exhausting in a quiescent medium. It has been found that the flow field emanating from the orifice/slot is characterized by both linear and nonlinear losses, governed by key nondimensional parameters such as Stokes number S, Reynolds number Re, and stroke length Lo. In terms of the orifice geometry shapes, a large variety has been used, although no one has determined the most "efficient." While straight orifices are the most common, the orifice thickness to diameter ratio is widely varied. It ranges from perforate orifice plates (see discussion on Helmholtz resonators) having very small thickness with the viscous effect confined at the edges where the vortices are shed, to long and thick orifices wherein the flow could be assumed fullydeveloped (Lee and Goldstein 2002). In the case of a thick orifice, the flow can be modeled as a pressure driven oscillatory pipe or channel flow where the socalled "Richardson effect" may appear at high Stokes number of 0(10) (Gallas et al. 2003a). Furthermore, Gallas (2002) experimentally determined a limit of the fullydeveloped flow assumption through a cylindrical orifice in terms of the orifice aspect ratio h/d > 1. Otherwise, the orifice could also have round edges or a beveled shape (NASA workshop CFDValCase 2, 2004). Another design, referred to as the springboard actuator, has been proposed by Jacobson and Reynolds (1995), in which both a small and a large gap are used for the slot. In the case of the presence of an external boundary layer, Bridges and Smith (2001) and Milanovic and Zaman (2005) experimentally studied different orifice shapes such as clustered, sharp beveled, or with different angles with respect to the incoming flow. The principal changes in the flow field between the different orifices studied were mostly found in the local vicinity of the orifice actuator, and less in the far (or global) field, for the specific flow conditions used. Finally, the predominant difference between the different orifices is that of a circular orifice versus rectangular slot. Experimental studies often employ these two geometries, whereas numerical simulations preferably use the latter for computational cost considerations. In terms of analytical modeling of ZNMF actuators, few efforts have been conducted, even for the simple case of a quiescent medium. Nonetheless, Rathnasingham and Breuer (1997) developed a simple analytical/empirical model that couples the structural and fluid characteristics of the device to produce a set of coupled, firstorder, nonlinear differential equations. In their empirical model, the flow in the slot is assumed to be inviscid and incompressible and the unsteady Bernoulli equation is used to solve the oscillatory flow. Crook et al. (1999) experimentally compared Rathnasingham and Breuer's simple analytical model and found that the agreement between the predicted and measured dependence of the centerline velocity on the orifice diameter and cavity height was poor, although the trends were similar. This discrepancy is mainly due to the lack of viscous effect in the orifice model, as well as the Stokes number dependence inside the orifice that is not considered by the flow model and which could lead to a nonparabolic velocity profile. Otherwise, with the aim of achieving realtime control of synthetic jet actuated flows, Rediniotis et al. (2002) derived a loworder model of two dimensional synthetic jet flows using proper orthogonal decomposition (POD). A dynamical model of the flow was derived via Galerkin projection for specific Stokes and Reynolds number values, and they accurately modeled the flow field in the open loop response with only four modes. However, the suitability of this approach as a general analysis/design tool was not addressed. More recently in Gallas et al. (2003a), the author presented a lumped element model of a piezoelectricdriven synthetic jet actuator exhausting in a quiescent medium. Methods to estimate the parameters of the lumped element model were presented and experiments were performed to isolate different components of the model and evaluate their suitability. The model was applied to two prototypical ZNMF actuators and was found to provide good agreement with the measured performance over a wide frequency range. The results reveal that lumped element modeling (LEM) can be used to provide a reasonable estimate of the frequency response of the device as a function of the signal input, device geometry, and material and fluid properties. Additionally, based on this modeling approach, Gallas et al. (2003b) successfully optimized the performance of a baseline ZNMF actuator for specific applications. They also suggest a roadmap for the more general optimal design synthesis problem, where the end user must translate desirable actuator characteristics into quantitative design goals. ZeroNet Mass Flux Devices with the Addition of Crossflow By now letting a ZNMF actuator interact with an external boundary layer or grazing flow, a wide range of applications can be envisioned, from active control of separation in aerodynamics to sound absorber technology in aeroacoustics. Fluid dynamic applications While the responsible physical mechanism is still unclear, it has been shown that the interaction of ZNMF actuators with a crossflow can displace the local streamlines and induce an apparent (or virtual) change in the shape of the surface in which the devices are embedded and when high frequency forcing is used (Honohan et al. 2000; Honohan 2003; Mittal and Rampuggoon 2002). Changes in the flow are thereby generated on length scales that are one to two orders of magnitude larger than the characteristic scale of the jet. Furthermore, ZNMF devices have been demonstrated to help in the delay of boundary layer separation on cylinders and airfoils, hence generating lift and reducing drag or also increasing the stall margin for the latter. For cylinders, the case of laminar boundary layers has been investigated by Amitay et al. (1997), and the case of turbulent separation by Bera et al. (1998). For airfoils, research has been conducted, for example, by Seifert et al. (1993) and Greenblatt and Wygnanski (2002). However, in ZNMFbased separation control, key issues such as optimal excitation frequencies and waveforms (Seifert et al. 1996; Yehoshua and Seifert 2003), as well as pressure gradient and curvature effects still remain to be rigorously investigated (Wygnanski 1997). For instance, it has been shown by some researchers that control authority varies monotonically with V,/U, (Seifert et al. 1993, 1996, 1999; Glezer and Amitay 2002; Mittal and Rampuggoon 2002) up to a point where a further increase will likely completely disrupt the boundary layer, and where Vi can be the peak, rms or spatial averaged jet velocity during the ejection portion of the cycle. On the other hand, control authority has a highly nonmonotonic variation with F+ (Seifert and Pack 2000b; Greenblatt and Wygnanski 2003; Glezer et al. 2003. Amitay and Glezer 2002), hence the existing current debate in choosing the optimum value for F where F = f/f represents the jet actuation frequency f that is nondimensionalized by some natural frequency f, in the uncontrolled flow. In fact, it is still unclear about what definition of f, should be used, since it depends on the flow conditions. For example, f, could either be the characteristic frequency of the separation region, the vortex shedding frequency in the wake, or the natural vortex rollup frequency of the shear layer, depending on whether separation "delay" control or separation "alleviation" control is sought (Cattafesta and Mittal, private communication, 2004). As noted earlier, another key issue in ZNMF devices is the form of the excitation signal. Researchers have used single sinusoids, but lowfrequency amplitudemodulated (AM) signals (Park et al. 2001), burst mode signals (Yehoshua and Seifert 2003), and various envelopes have also been investigated (Margalit et al. 2002; Wiltse and Glezer 1993). From these studies, it seems clear that the input signal waveform should be carefully chosen function of the natural frequency of uncontrolled flow f, as discussed above. In addition, it emphasizes the fact that the dynamics of the actuator should not be ignored. Also of interest for flow control applications is the interaction of multiple ZNMF actuators (or actuator arrays) with an external boundary layer, which has been experimentally investigated by several researchers (Amitay et al. 1998; Watson et al. 2003; Amitay et al. 2000; Wood et al. 2000; Ritchie and Seitzman 2000). However, the relative phasing effect between each actuator was usually not investigated. On the other hand, Holman et al. (2003) investigated the effect of adjacent synthetic jet actuators, including their relative phasing, in an airfoil separation control application. They found that, for the single flow condition studied, separation control was independent of the relative phase, and also that for low actuation amplitudes, actuator placement on the airfoil surface could be critical in achieving desired flow control. Similarly, Orkwis and Filz (2005) numerically investigated the effect of the proximity between two adjacent ZNMF actuators in crossflow and found that favorable interactions between the two actuators could be achieved within a certain distance that separates them, but the optimal separation is different whether they are in phase or out of phase from each other. Finally, to the author's knowledge, besides a first scaling analysis performed by Rampunggoon (2001) which is based on a parameterization of the successive moments and skewness of the jet velocity profile, along with the study by McCormick (2000) that presents an electroacoustic model to describe the actuator characteristics (in a similar manner to the lumped element modeling approach used by Gallas et al. 2003b), no other loworder models have been developed for a ZNMF actuator interacting with an external boundary layer. Aeroacoustics applications For the past fifty years, people in the acoustic community have tried to predict the flow past an open cavity (Elder 1978; Meissner 1987) or a Helmholtz resonator (Howe 1981b; Nelson et al. 1981). This is a generic denomination for applications such as aircraft cavities, acoustic liners, open sunroofs, mufflers for intake and exhaust systems, or simply perforates. This research lies in the domain of acoustics of fluidstructure interactions which has generated significant attention from numerous researchers. As noted earlier, a parallel with ZNMF actuators can be draw with the study of acoustic liners, shown in Figure 13B. More specifically, the goal is usually to compute the acoustic impedance of the liner, since the notion of impedance simply relates a particle or flow velocity to the corresponding pressure. Such knowledge is required to design and implement liners in an engine nacelle. However, researchers are still facing great challenges in extracting suitable impedance models of these perforate liners, usually composed of Helmholtz resonators. In fact, because of the presence of flow over the orifice, rigorous mathematical modeling of the interaction mechanisms are very difficult to obtain, and the present state of analytical and numerical codes do not allow direct modeling of these interactions at relevant Reynolds numbers, as seen earlier in the case of ZNMF actuators. Consequently, most of the existing models of grazing flow past Helmholtz resonators are empirical or, at most, simplified mathematically models. Modeling approaches First of all, in terms of impedance models of acoustic liners, Dequand et al. (2003) and Lee and Ih (2003) provide a good review of the existing models, along with their intrinsic limitations. The main distinctions between the proposed models lay first in the orifice model, then in the characterization of the grazing flow, and finally in the addition or not of a mean bias flow through the orifice (not to be confused with grazing flow over the orifice). The cavity is often modeled as a classical resonator having a linear response (massspring system). When a bias flow is included, the prediction of its effect on the orifice impedance is usually carried out within the mechanism of soundvortex interaction. And when grazing flow is present, most of the orifice impedance models are either deduced from experimental data or rely on empiricism. With regards to orifice modeling, Ingard and Ising (1967) included effective end corrections in their impedance model that take care of the acoustic nonlinearity of the orifice (mainly dependent on the ratio of the acoustic orifice momentum to the boundary layer momentum when a grazing flow is included). Depending on the flow conditions of the application, either low frequency or high frequency assumptions are used to model the flow through the orifice. Also, standard assumption is that the orifice dimensions are much smaller than the acoustic wavelength of interest. Another important point to note is on the porosity factor of a perforate plate. Because of the direct application of such a device to engine nacelle liners, the solution for a single orifice impedance is usually derived and is then extended to multiple holes geometry. The simple relation between the specific impedance of a perforate and a single orifice, Z0p = Zo/c, holds when the orifices are not too close from each other in order to alleviate any jetting interaction effect between them. Here, the porosity factor is defined by =Nholes x (hole area)/total area, where Nhol, is the number of orifices in the perforate. Ingard (1953) states that the resonators can be treated independently of each others if the distance between the orifices is greater than half of the acoustic wavelength. Otherwise, to account for the interaction effect between multiple holes, Fok's function is usually employed (Melling 1973). The grazing flow is commonly characterized as a fullydeveloped turbulent boundary layer (or fullydeveloped turbulent pipe flow), although some investigations do not, which may lead to difficulties for comparison sake. The parameters extracted from the external boundary layer are usually the Mach number M., friction velocity u., or boundary layer thickness 3. Although most of the models are empirical or semiempirical, some are still analytical. The first models proposed were based on linear stability analysis where the shear layer (or grazing flow) is modeled using linear inviscid theory for infinite parallel flows. Later, more formal linearized models have been emphasized. For instance, Ronneberger (1972) described the orifice flow in terms of wavelike disturbances of a thin shear layer over the orifice. Howe (1981a) modeled the grazing flow interaction as a KelvinHelmholtz instability of an infinitely extended vortex sheet in incompressible flow, where the vortex strength is tuned to compensate the singularity of the potential acoustic flow at the downstream edge in order to meet the Kutta condition. Also, Elder (1978) describes the shear layer displacement as being shaped by a KelvinHelmholtz wave, while an acoustic response of the resonant system is modeled by an equivalent impedance circuit of a resonator adopted from organ pipe theory. He then treats the flow disturbances using linear shear layer instability models and the oscillation amplitude is assumed to be limited by the nonlinear orifice resistance. Nelson et al. (1981, 1983) separated the total flow field into a purely vortical flow field (associated with the shed vorticity of the grazing flow) where the vorticity of the shear layer is concentrated into point vortices traveling at a constant velocity on the straight line joining the upstream to the downstream edge, plus a potential flow (unsteady part associated with the acoustic resonance). They also provided a large experimental database in a companion paper that has been used by others (Meissner 2002; Dequand et al. 2003). Innes and Creighton (1989) used matched asymptotic expansions for small disturbances to solve the non linear differential equations, the resonator waveform containing a smooth outer part and the boundary layer a rapid change; then approximations were found in each region along with approximate values for the Fourier coefficients. Also, Jing et al. (2001) proposed a linearized potential flow model that uses the particle velocity continuity boundary condition rather than the more frequently used displacement in order to match the flowfields separated by the shear layer over the orifice. All those models however still remain linear (or nearly so) and thus carry inherent assumption limitations. The simplified mathematical models described above have been used as starting point to construct empirical models. These are based upon parameters such as the thickness h and diameter d of the orifice/perforate, plate porosity U, grazing flow velocity (mean velocity U. or friction velocity u.), Strouhal number St = cod/U (U being some characteristic velocity), or Stokes number S= od/v. The major empirical models found in the open literature are proposed by Garrison (1969), Rice (1971), Bauer (1977), Sullivan (1979), Hersh and Walker (1979), Cummings (1986), or Rao and Munjal (1986), and Kirby and Cummings (1998). They differ from each other depending on whether they include orifice nonlinear effects, orifice losses (viscous effect, compressibility), end corrections, single or clustered orifices, radiation impedance, etc. But most of all, and more interestingly, they use different functional forms for the chosen parameters that govern the physical behavior of the phenomenon, such as f(h/d,kd,St, ,U ,u,,...), as shown in Appendix A where some of these models are described in details. It should be noted that each of them are applicable for a single application over a specific parameter range (muffler, acoustic liner, etc.). Other less conventional approaches have also been attempted. For instance, Mast and Pierce (1995) used describingfunctions and the concept of a feedback mechanism. In this approach, the resonatorflow system is treated as an autonomous nonlinear system in which the limit cycles are found using describingfunction analysis. Meissner (2002) gave a simplified, though still accurate, version of this model. Similarly, following Zwikker and Kosten's (1949) theory for propagation of sound in channels, Sullivan (1979) and Parrott and Jones (1995) used transmission matrices to model parallelelement liner impedances. In another effort, Lee and Ih (2003) obtained an empirical model via nonlinear regression analysis of results coming from various parametric tests. Furthermore, acoustic education techniques have been used to determine the acoustic impedance of liners, such as a finite element method (employed by NASA, see Watson et al. 1998), that iterates on the numerical solution of the two dimensional convective wave equation to determine an impedance that reproduces the measured amplitudes and phases of the complex acoustic pressures; or a grazing flow data analysis program (employed by Boeing, see Jones et al. (2003) and references therein for details) that conducts separate computations in different regions to match the acoustic pressure and particle velocity across the interfaces that determines the modal amplitudes in each of the regions; or also a two dimensional modal propagation method based on insertion loss measurements (employed by B. F. Goodrich, see Jones et al. (2003) and references therein for details) that determines the frequencydependent acoustic impedance of the test liner. Jones et al. (2003) reviewed and compared these impedance education techniques. Finally, as noted earlier, a few studies have been performed using numerical simulations. Indeed, as can be seen in Liu and Long (1998) and Ozyoruk and Long (2000), it is computationally quite expensive, difficult to implement, and strong limitations on the geometries are required. However, a promising numerical study by Choudhari et al. (1999) gives valuable insight into the flow physics of these devices, such as the effect of acoustic nonlinearity on the surface impedance. Another important point concerns the measurement techniques used to acquire the sample data which upon most of the model are derived, from simple to more elaborate curve fitting. The two microphone technique introduced by Dean (1974) is commonly employed for in situ measurements of the local wall acoustic impedance of resonant cavity lined flow duct. This technique uses two microphones, one placed at the orifice exit of the resonator, the other flushed at the cavity bottom. Then a simple relationship for locally reactive liner between the cavity acoustic pressure and particle velocity is extracted, which is based on the continuity of particle velocities on either side of the cavity orifice (or surface resistive layer). However, the main drawbacks of this widely used method reside in the position of the microphone in front of the liner that must be in the hydrodynamicc far field" but at a distance less than the acoustic wavelength, and also in the grazing boundary layer thickness. Different experimental apparatus are given in Appendix A for clarification and illustration. As an example, five models from the literature are presented in Appendix A that are thought to be interesting, either for the quality of the experiments which upon the model fits have been based on, or for the functional form they offer in terms of the dimensionless parameters which are believed to be of certain relevance. To some extent, they are all based on experimental data. From all the models currently available, it is not obvious whether one model will perform better than another, which is mainly due to the wide range of possible applications, the limitations in the experimental data on which the semiempirical models heavily rely, and because even the mathematical models have their own limitations. However, the rich physical information carried within these semiempirical models and the corresponding data on which they are based will undoubtfully aid the development of reducedorder models in ZNMF actuator interacting with a grazing flow. Unresolved Technical Issues By surveying the literature, i.e. looking at the flow mechanism of isolated ZNMF actuators to more complex behavior when the actuator is interacting with an incoming boundary layer, along with examples of sound absorber technology, several key issues can be highlighted that still remain to be addressed. This subsection lists the principal ones. Fundamental flow physics. Clearly, there still exists a lack in the fundamental understanding of the flow mechanisms that govern the dynamics of ZNMF actuators. While the cavity design is well understood, the orifice modeling and especially the effect of the interaction with an external boundary layer requires more indepth consideration. Also, whether performing experimental studies or numerical simulations, researchers are confronted with a huge parameter space that is time consuming and requires expensive experiments or simulations. Hence the development of simple physicsbased reduced order models is primordial. 2D vs. 3D. While most of the numerical simulations are performed for two dimensional problems, threedimensionality effects clearly can be important, especially to model the flow coming out of a circular orifice as shown in Rizzeta et al. (1998) or Ravi et al. (2004) that also found distinct and non negligible threedimensional effects of the flow. Compressibility effects. Usually, the entire flow field is numerically solved using an incompressible solver. However, such an assumption, although valid outside the actuator, may be violated inside the orifice at high jet velocity and, more generally, inside the cavity due to the acoustic compliance of the cavity. Indeed, the cavity acts like a spring that stores the potential energy produced by the driver motion. Lack of highresolution experimental data. Most of the experimental studies employed either Hot Wire Anemometry (HWA), Particle Image Velocimetry (PIV) or Laser Doppler Velocimetry (LDV) to measure the flow. However, each of these techniques has shortcomings, as briefly enumerated below. In the case of HWA, since the flow is highly unsteady and by definition oscillatory, its deployment must be carefully envisaged, especially considering the derectification procedure used to obtain the reversal flow. Since it is an intrusive technique that may perturb the flow, other issues are that it is a single point measurement (hence the need to traverse the whole flow field), problems arise with measurements near zero velocity (transition from free to forced convection), and the accuracy may be affected by the calibration (sensitivity), the local temperature, or some conductive heat loss. With regards to PIV, although the main advantage resides in the fact that it is a nonintrusive flow visualization technique that captures instantaneous snapshots of the flow field, the micro/meso scale of ZNMF devices requires very high resolution in the vicinity of the actuator orifice in order to obtain reasonable accuracy in the data. This is difficult to achieve using a standard digital PIV system. Finally, a large number of samples are required in order to get proper accuracy in the data from LDV measurement, and excellent spatial resolution is difficult to achieve due to the finite length of the probe volume. Also, since LDV is a single point measurement, a traversing probe is required in order to map the entire flow field. Lack of accurate loworder models. Clearly, the few reducedorder models that are present so far are not sufficient to be able to capture the essential dynamics of the flow generated by a ZNMF actuator. Better models must be constructed to account for the slot geometry and the impact of the crossflow on the jet velocity profile. The five models of grazing flow past Helmholtz resonators summarized in Appendix A reveal the disparity in the impedance expressions as well as in the range of applications (see Table Ai). Clearly, the task of extracting a validated semiempirical model is far from trivial. But leveraging past experience is critical to yielding accurate loworder models for implementation of a ZNMF actuator. Objectives The literature survey presented above has permitted the identification of key technical issues that remain to be resolved in order to fully implement ZNMF actuators into realistic applications. Currently, it is difficult for a prospective user to successfully choose and use the appropriate actuator that will satisfy specific requirements. Even though many designs have been used in the literature, no studies have systematically studied the optimal design of these devices. For instance, how large should the cavity be? What type of driver is most appropriate to a specific application? Possibilities include a low cost, low power piezoelectricdiaphragm, an electromagnetic or mechanical piston that will provide large flow rate but may require significant power, or a voicecoil speaker typically used in audio applications? What orifice geometry should be chosen? Options include sharp versus rounded edges, large versus short thickness, an axisymmetric versus a rectangular slot? Clearly, no validated tools are currently available for end users to address these questions. Generally, a trial and error method using expensive experimental studies and/or time consuming numerical simulations have been employed. The present work seeks to address these issues by providing a clear physical understanding of how these devices behave and interact with and without an external flow, and by developing and validating reducedorder dynamical models and scaling laws. Successful completion of these objectives will ultimately aid in the analysis and development of design tools for sizing, design and deployment of ZNMF actuators in flow control applications. Approach and Outline of Thesis To reach the stated objectives, the following technical approach has been employed. First, the identification of outstanding key issues and the formulation of the problem have been addressed in this chapter by surveying the literature concerning the modeling in diverse applications of ZNMF actuators and acoustic liner technology. The relevant information about the key device parameters and flow conditions (like the driver configuration, cavity, orifice shape, or the external boundary layer parameters) are thus extracted. Before investigating how a ZNMF device interacts with an external boundary layer, the case of an isolated ZNMF actuator must be fully understood and documented. This is the subject of Chapter 2. An isolated ZNMF device is first characterized and the relevant parameters are defined. Then, the previous work done by the author in Gallas et al. (2003a) is summarized. Their work discusses a lumped element model of a piezoelectricdriven ZNMF actuator. One goal of the present work is to extend their model to more general devices and to remove, as far as possible, some restricting limitations, especially on the orifice loss coefficient. Consequently, a thorough nondimensional analysis is first carried out to extract the physics behind such a device. 29 Also, some relevant modeling issues are discussed and reviewed, for instance on the orifice geometry effects and the driving transducer dynamics. Then, to study in great details the dynamics of isolated ZNMF actuators, an extensive experimental investigation is proposed where various test actuator configurations are examined over a wide range of operating conditions. The experimental setup is described in Chapter 3. CHAPTER 2 DYNAMICS OF ISOLATED ZERONET MASS FLUX ACTUATORS Several key issues were highlighted in the introduction chapter that will be addressed in this thesis. This Chapter is first devoted to familiarize the reader with the dynamics of ZNMF actuators, their behavior and inherent challenges in developing tools to accurately model them. One goal, before addressing the general case of the interaction with an external boundary layer, is to understand the nonlinear dynamics of an isolated ZNMF actuator. This chapter is therefore entirely dedicated to the analysis of isolated ZNMF actuators issuing into a quiescent medium, as outlined below. The device is first characterized and the relevant parameters defined in order to clearly define the scope of the present investigation. The previous work performed by the author in Gallas et al. (2003a) is next summarized. Their work discusses a lumped element model of a piezoelectricdriven ZNMF actuator that relates the output volume flow rate to the input voltage in terms of a transfer function. Their model is extended to more general devices and solutions to remove some restricting limitations are explored. Based on this knowledge, a thorough dimensional analysis is then carried out to extract the physics behind an isolated ZNMF actuator. A dimensionless linear transfer function is also derived for a generic driver configuration, which is thought to be relevant as a design tool. It is shown that a compact expression can be obtained regardless of the orifice geometry and regardless of the driver configuration. Finally, relevant modeling issues pointed out in the first chapter are discussed and reviewed. Some issues are then addressed, more particularly on the modeling of the orifice flow where a temporal analysis of the existing lumped element model is proposed along with a physicallybased discussion on the orifice loss mechanism. Issues on the dynamics of the driving transducer are discussed as well. Finally, a test matrix constructed to study the ZNMF actuator dynamics is presented. Characterization and Parameter Definitions Figure 11 shows a typical ZNMF actuator, where the geometric parameters are shown. First of all, it is worthwhile to define some precise quantities of interest that have been used in the published literature and try to unify them into a generalized form. For instance, people have used the impulse stroke length, some spatially or time averaged exit velocities, or Reynolds numbers based either on the circulation of vortex rings or on an averaged jet velocity to characterize the oscillating orifice jet flow. Here, an attempt to unify them is made. The inherent nature of the jet is both a function of time oscillatoryy motion) and of space (velocity distribution across the orifice exit area). It is also valuable to distinguish the ejection from the ingestion portion of a cycle. Many researchers (Smith and Glezer 1998, Glezer and Amitay 2002) characterize a synthetic jet based on a simple "slug velocity profile" model that includes a dimensionless stroke length Lo/d and a Reynolds number Re, = VCLd/v based on the velocity scale (average orifice velocity) such that VCL =L= f vcL (t)dt, (21) where vL (t) is the centerline velocity, T = 1/f is the period, thereby T/2 representing half the period or the time of discharge for a sinusoidal signal, and L0 is the distance that a "slug" of fluid travels away from the orifice during the ejection portion of the cycle or period. In addition, Smith and Glezer (1998) have employed a Reynolds number based on the impulse per unit length (i.e., the momentum associated with the ejection per unit width), Reo = Io/ /d, where the impulse per unit width is defined as Io = pdL T2 (t)dt. (22) Or similarly, following the physics of vortex ring formation (Glezer 1988), a Reynolds number, Rer = Fo/v, is used based on the initial circulation associated with the vortex generation process, with F0 defined by 1 IT/2 1T 0 C (t)dt= 2 CL. (23) Alternatively (Utturkar et al. 2003), a spatial and timeaveraged exit velocity during the expulsion stroke is used to define the Reynolds number Re = Vd/v, where the time averaged exit velocity V, is defined as 2 1 T/2 2 T/2 V T fo v(t, x)dtdS = o T (t)dt, (24) T f S, TJo where v(t) is the spatial averaged velocity, S, is the exit area of the orifice neck, and x is the crossstream coordinate (see Figure 12 for coordinates definition). For general purposes, instead of limiting ourselves to a simple uniform "slug" profile, the latter definition is considered throughout this dissertation. Notice that for a "slug" profile, it can be shown that the average orifice velocity scale defined above in Eq. 21 and Eq. 24 is related by VL =2V. Similarly, L, id = VL/(fd) is closely related to the inverse of the Strouhal number St since L0 VCL 2V V 1 L0 CL (25) d fd cod/2i cod St and since 1 J Vd v Re (26) St cod v d o2 S2 the following relationship always holds 1 Re Lod (27) St S2 co r where r is the time of discharge (= T 2 for a sinusoidal signal) and S = od 2/v is the Stokes number. The use of the Stokes number to characterize a synthetic jet and the relationship to the Strouhal number were previously mentioned in Utturkar et al. (2003) and Rathnasingham and Breuer (1997). The corresponding relations between the different definitions are summarized in Table 21. Correspondingly, the volume flow rate coming out of the orifice during the ejection part of the cycle can be defined as QJ = J v(t,x)dtdS, = VjS,,. (28) And clearly, since we are dealing with a zeronet mass flux actuator, the following relationship always holds j,total = j,ex + Q, = 0, (29) where the suffices 'ex' and 'in' stand for 'expulsion' and 'ingestion', respectively. Table 21: Correspondence between synthetic jet parameter definitions L, 1 1 Re d O) St S2 Re, Rero Re As seen from the above definitions, once a velocity or time scale has been chosen, a length scale must be similarly selected for the orifice or slot. Figure 13 show two typical orifice geometries encountered in a ZNMF actuator, and give the geometric parameters and coordinates definition. Notice that the orifice is straight in both cases. No beveled, rounded or other shapes are taken into account, although other geometries have been investigated (Bridges and Smith 2001; Smith and Swift 2003b; Milanovic & Zaman 2005; Shuster and Smith 2004). Throughout this dissertation, the primary length scale used is the diameter or depth of the orifice d. The spanwise orifice width w is used as needed for discussions related to a rectangular slot, and the height h is a third characteristic dimension. Clearly, if d is chosen as the characteristic length scale, then w/d and h/d are key nondimensional parameters. Lumped Element Modeling Summary of Previous Work A lumped element model of a piezoelectricdriven synthetic jet actuator exhausting in a quiescent medium has been recently developed and compared with experiments by Gallas et al. (2003a). In lumped element modeling (LEM), the individual components of a synthetic jet are modeled as elements of an equivalent electrical circuit using conjugate power variables (i.e., power = generalized flow x generalized effort). The frequency response function of the circuit is derived to obtain an expression for Q /IVC the volume flow rate per applied voltage. LEM provides a compact analytical model and valuable physical insight into the dependence of the device behavior on geometric and material properties. Methods to estimate the parameters of the lumped element model were presented and experiments were performed to isolate different components of the model and evaluate their suitability. The model was applied to two prototypical synthetic jets and found to provide very good agreement with the measured performance. The results reveal the advantages and shortcomings of the model in its present form. With slight modifications, the model is applicable to any type of ZNMF device. Orifice Cavity(v) Vac Piezoceramic Composite Diaphragm electroacoustic coupling ... . . .j a.D RaD R a 'RM M .aD RaD MaD "RaN MaN i d c' ,' aO > / '* / l MaRad * a\c ,b electrical acoustic/fluidic domain domain Figure 21: Equivalent circuit model of a piezoelectricdriven synthetic jet actuator. The equivalent circuit model is shown in Figure 21. The structure of the equivalent circuit is explained as follows. An ac voltage V, is applied across the piezoceramic to create an effective acoustic pressure that drives the diaphragm into oscillatory motion. This represents a conversion from the electrical to the acoustic domain and is accounted for via a transformer with a turns ratio An ideal transformer (i.e., power conserving) converts energy from the electrical to acoustic domain and converts an electrical impedance to an acoustic impedance. The motion of the diaphragm can either compress the fluid in the cavity (modeled, at low frequencies, by an acoustic compliance Ca) or can eject/ingest fluid through the orifice. Physically, this is represented as a volume velocity divider, Qd = Q + Q The goal of the actuator design is to maximize the magnitude of the volume flow rate through the orifice per applied voltage Q, /Va, given by (Gallas et al. 2003a) Q, (s) dos Q (s) d (210) Vac(S) a4S4 + +a2s +1as+1' where d0 is an effective piezoelectric constant obtained from composite plate theory (Prasad et al. 2002), s = jco, and al,a2,...,a are functions of the material properties and dimensions of the piezoelectric diaphragm, the volume of the cavity V, orifice height h, orifice diameter d, fluid kinematic viscosity v, and sound speed co, and are given by ai = CD (RkOn + RaR +R CaC (kROn + Rao), a2 = CD (MRad +MA, +M D) +CC (MORad +MN) +CacCaDRaD (Ron +RaN), (211) a3 = C D MaD ( RDon + R, )+(MaORad +MN ) RD ], and a4 = CaCCDMa (MoRad +MA). In Eq. 211, CD, Ra and MaD are respectively the acoustic compliance, resistance and mass of the diaphragm. Cc is the acoustic compliance of the cavity. R., MA, and MaRad are respectively the acoustic resistance, mass and radiation mass of the actuator orifice, while R represents the nonlinear resistance term associated with the orifice flow discharge and is a function of the volume flow rate Q,. 35 70 30  60 > 25  50  E E 20   40  0 00 1000 100 2000 200 3000 0 00 1000 100 Frequency (Hz) Frequency (Hz) Figure 22: Comparison between the lumped element model and experimental frequency response measured using phaselocked LDV for two prototypical synthetic O0 500 1000 1500 2000 2500 3000 500 Frequency (Hz) Frequency (Hz) Figure 22: Comparison between the lumped element model and experimental frequency response measured using phaselocked LDV for two prototypical synthetic jets (Gallas et al. 2003a). The lumped parameters in the circuit in Figure 21 represent generalized energy storage elements (i.e., capacitors and inductors) and dissipative elements (i.e., resistors). Model parameter estimation techniques, assumptions, and limitations are discussed in Gallas et al. (2003a). The capability of the technique to describe the measured frequency response of two prototypical synthetic jets is shown in Figure 22. The case in the left half of the figure reveals the 4thorder nature of the frequency response. The two resonance peaks are related to the diaphragm natural frequency f, and the Helmholtz frequency f,, thereby demonstrating the potential significance of compressibility effects. The case in the right half of the figure reveals how the model can be "tuned" to produce a device with a single resonance frequency with large output velocities. The important point is that the model gives a reasonable estimate of the output of interest (typically within +20%) with minimal effort. The power of LEM is its simplicity and its usefulness as a design tool. LEM can be used to provide a reasonable estimate of the frequency response of the device as a function of the signal input, device geometry, and material and fluid properties. Limitations and Extensions of Existing Model The study performed in Gallas et al. (2003a) was restricted to axisymmetric orifice geometry and the oscillating pressure driven flow inside the pipe was assumed to be laminar and fullydeveloped. Also, a piezoelectricdiaphragm was chosen to drive the actuator. A straightforward extension of their model is that of a rectangular slot model. Appendix C provides a derivation of the solution of oscillating pressure driven flow in a 2D channel, assuming the flow is laminar, incompressible and fullydeveloped. The low frequency approximation then yields the lumped element parameters. Hence, for a 2D channel orifice the acoustic resistance and mass are found to be, respectively, 3 auh 3ph Rv = u and MaN = (212) 2w(d 2) 5w(d 2) Similarly, also of interest is the acoustic radiation impedance for a rectangular slot. The acoustic radiation mass MOaad is modeled for kd <1 as a rectangular piston in an infinite baffle by assuming that the rectangular slot is mounted in a plate that is much larger in extent than the slot size (Meissner 1987), XaRad = coMaRad kd k (213) wd ; In (2d w) 2; (1 kV /6) where Xlad corresponds to the acoustic radiation reactance. Another extension of their work can be made with regards to the driver employed. As shown in the next section, a convenient expression of the actuator response can be made in terms of the nondimensional transfer function Q /Qd the ratio of the jet to driver volume flow rate. Hence, by decoupling the driver dynamics from the rest of the actuator one can easily implement any type of driver, under the condition that its dynamics are properly modeled. In the LEM representation, the driving transducer is represented in terms of a circuit analogy; it thus requires that the transducer components must be fully known, whether the driver transducer is a piezoelectricdiaphragm, a moving piston (electromagnetic or mechanical), or an electromagnetic voicecoil speaker. A more detailed discussion on this issue is provided towards the end of this chapter. The most restricting limitations of the lumped element model in its current state, as presented above, are found in the orifice modeling. First, the model cannot handle orifice geometries other than a straight pipe (or 2D channel, as seen above), i.e. no rounded edges or beveled shapes can be considered. However, by analogy with minor losses in fluid piping systems, this should only affect the nonlinear resistance term Raon associated with the discharge from the orifice, and not RooR, = Ra that represents the viscous losses due to the assumed fullydeveloped pipe flow. The nonlinear resistance term Ron, is approximated by modeling the orifice as a generalized Bernoulli flow meter (White 1979; McCormick 2000), 0.5KdpQ oR n= (214) n where Q, is the amplitude of the jet volume flow rate, and Kd is a dimensionless loss coefficient that is assumed, in this existing model, to be unity. In practice, Kd is a function of orifice geometry, Reynolds number, and frequency. Hence, a detailed analysis on the loss coefficient for various orifice shapes should yield a more accurate expression in terms of modeling the associated nonlinear resistance. This is actually one of the goals of this dissertation and this is systematically investigated in subsequent chapters. A second restricting assumption found in the orifice model of Gallas et al. (2003a) comes from the required fullydeveloped hypothesis of the flow inside the orifice. Clearly this limits the orifice design to a sufficiently large aspect ratio h/d or low stroke length compare to the orifice height h. The lumped parameters of the orifice impedance are based on the steady solution for a fullydeveloped oscillating pipe/channel flow (see Appendix C). In addition, the author experimentally found (Gallas 2002) that reasonable agreement was achieved between the lumped element model and the measured dynamic response of an isolated ZNMF actuator when the orifice aspect ratio h/d approximately exceeded unity. Figure 23 below reproduces this fact for four different aspect ratios, where the orifices considered were axisymmetric, and the model prediction of the centerline velocity was compared to phaselocked LDV measurements versus frequency. Note that the diaphragm damping coefficient CD was empirically adjusted to match the peak magnitude at the frequency governed by the diaphragm natural frequency. Clearly, a careful study of the entrance effect in straight pipe/channel flow should greatly enhance the completeness and validity of the orifice model in its current form, such a model being able to be applied to all sorts of straight orifices, from long neck to short perforates. Again, additional insight into this issue is discussed at the end of the chapter. 70 70 = 0.015 h/d 3/1 =3 = 0.013 h/d =5/1= 5 60  60 50 L 50  40   40  20 f   20 30 0 1 0 L 10  10  S500 1000 1500 0 500 1000 1500 Frequency (Hz) Frequency (Hz) = 0.005 hd =1/3 =0.33 0.005 hd5/31.66 30  30   ** 205    <          205   L   15   \    15          5 5 '  0 500 Frequency (Hz) 1000 1500 0 500 Frequency (Hz) 1000 Frequency (Hz) Figure 23: Comparison between the lumped element model () and experimental frequency response measured using phaselocked LDV (*) for four prototypical synthetic jets, having different orifice aspect ratio h d (Gallas 2002). Finally, another constraint in the current model is about the low frequency approximation. By definition LEM is fundamentally limited to low frequencies since it is the main hypothesis employed. The characteristic length scales of the governing physical phenomena must be much larger than the largest geometric dimension. For example, for the lumped approximation to be valid in an acoustic system, the acoustic wavelength (A = 1 k) must be significantly larger than the device itself (kd < 1). This assumption permits decoupling of the temporal from the spatial variations, and the governing partial differential equations for the distributed system can be "lumped" into a set of coupled ordinary differential equations. 0.8 S: \ 0.6 S=1  S S=12\ S0.5  S=20   \\\ 0.4 s =50  0.3 1 0.2  0.1 0 0.2 0.4 0.6 0.8 1 x/(d/2) Figure 24: Variation in velocity profile vs. S = 1, 12, 20, and 50 for oscillatory pipe flow in a circular duct. However, it is well known that the flow inside a long pipe/channel is frequency dependent, as shown in Figure 24 and Figure 25. From Figure 24, it can be seen that, as the Stokes number S goes to zero, the velocity profile asymptotes to Poiseuille flow, while as S increases, the thickness of the Stokes layer decreases below d/2, leading to an inviscid core surrounded by a viscous annular region where a phase lag is also present between the pressure drop across the orifice and the velocity profile. Figure 25 shows that the ratio of the spatial average velocity i, (t) to the centerline velocity vC (t), which is 0.5 for Poiseuille flow, is strongly dependant on the Stokes number. Although it has been shown (Gallas et al. 2003a) that the acoustic reactance is approximately constant with frequency, the acoustic resistance, which does asymptote at low frequencies to the steady value given by the lumped element model, gradually increases with frequency. Therefore, this frequencydependence estimate should not be disregarded, and care must be taken in the frequency range at which ZNMF actuators are running to apply LEM. For instance, the frequency dependence given by Figure 25 can be easily implemented in the present model to provide estimates for the acoustic impedance of the orifice, as discussed in Gallas et al. (2003a). 0.95  0.9 0.85  0.8 / I > 0.75  0.7  0.65  0.6  0.55 f ' 0.5 1 10 100 S=(I d2 1/2 SI cod I)1 Figure 25: Ratio of spatial average velocity to centerline velocity vs. Stokes number for oscillatory pipe flow in a circular duct. To summarize this section, the model given in Gallas et al. (2003a) has been presented and reviewed, and it has been shown that it could be extended to more general device configurations, particularly in terms of orifice geometry and driver configuration. Also, some of their restricting assumption limits could be, if not completely removed, at least greatly attenuated, and this is further analyzed and discussed in the last part of this chapter. But before, a general dimensional analysis of an isolated actuator is carried out in the next section that gives valuable insight on the parameter space and on the system response behavior. Dimensional Analysis Definition and Discussion In the first section of this chapter, the primary output variables of interest have been defined, and specifically the spatial and timeaveraged ejection velocity of the jet V, defined in Eq. 24. It is then interesting to rewrite them in terms of pertinent dimensionless parameters. Using the BuckinghamPi theorem (Buckingham 1914), the dependence of the jet output velocity can be written in terms of nondimensional parameters. The derivation is presented in full in Appendix D and the results are summarized below: Sf os h (w A15) St f= d,,ddkdS (215) Re H d The quantities in the left hand side of the functional are possible choices that the dependent variable V can take. Q /Qd represents the ratio of the volume flow rate of the driver (Qd = WdAV) to the jet volume flow rate of the ejection part. St is the Strouhal number and Re is the jet Reynolds number defined earlier. Notice the close relationship between the jet Reynolds number, the Stokes number and the Strouhal number that were given by Eq. 27 and found again here by manipulation of the nI groups (see Appendix D for details). Therefore, for a given geometric configuration, either the Strouhal or the Reynolds numbers along with the Stokes number could suffice to characterize the jet exit behavior. It is also interesting to view Eq. 27 as the basis for the jet formation criterion defined by Utturkar et al. (2003). Actually, it is intrusive to look at the different physical interpretations that the Strouhal number can take. In the fluid dynamics community, it is usually defined as the ratio of the unsteady to the steady inertia. However, it can also be interpreted as the ratio of 2 length scales or 2 time scales, such that St cod d d St =  V o) L, i V (216) Sft = od o) tOscillation VJ d convection where d/L, is the ratio of a typical length scale d of the orifice to the particle excursion Lo through the orifice. The Strouhal number can also be the ratio of the oscillation time scale to the convective time scale. The physical significance of each term in the RHS of Eq. 215 is described below: * 3o/oH is the ratio of the driving frequency to the Helmholtz frequency OH = co /h'V (see Appendix B for a complete discussion on the definition and derivation of oH ), a measure of the compressibility of the flow inside the cavity. * h/d is the orifice/slot height to diameter aspect ratio. * w/d is the orifice/slot width to diameter aspect ratio. * c0/cod is the ratio of the operating frequency to the natural frequency of the driver. * AV/d3 is the ratio of the displaced volume by the driver to the orifice diameter cubed. * kd = d/A is the ratio of the orifice diameter to the acoustic wavelength. * S = odjv is the Stokes number, the ratio of the orifice diameter to the unsteady boundary layer thickness in the orifice v/co. It is evident that in the case of an isolated ZNMF actuator, the response is strongly dependant on the geometric parameters {co/cH ,h/d,w/d,kd} and the operating conditions ({c/od ,AV/d3 ,S}. In fact, from the functional form described by Eq. 215 and for a given device with fixed dimensions and a given fluid, the actuator output is only dependent on the driver dynamics (od, AV) and the actuation frequency c . Although compressibility effects in the orifice are neglected in this dissertation, it warrants a few lines. Compressibility will occur in the orifice for high Mach number flows and/or for high density flows. If the compressibility of the fluid has to be taken into account, it follows by definition that density must be considered as a new variable. For instance, the pressure is now coupled to the temperature and density through the equation of state. Similarly, the continuity equation is no longer trivial. Also, temperature is important, and one has to reminder that the variation of the thermal conductivity k and dynamic viscosity /u that are transport quantities with temperature may be important. Dimensionless Linear Transfer Function for a Generic Driver Valuable physical insight into the dependence of the device behavior on geometry and material properties is provided by the frequency response of the ZNMF actuator device. In order to obtain an expression of the linear transfer function of the jet output to the input signal to the actuator, the compact nonlinear analytical model given by LEM is used in a similar manner as described and introduced in the previous section, since it was shown to be a valuable design tool. Notice however that the nonlinear part of the model in its present form only confined in the orifice is neglected for simplicity in this analysis. Figure 26 shows a schematic representation of a ZNMF actuator having a generic driver using LEM. This representation enables us to bypass the need of an expression for the acoustic impedance ZD> of the driving transducer, although it lacks its dynamics modeling. IQd Q \ ZC Z ia Figure 26: Schematic representation of a genericdriver ZNMF actuator. In this case, a convenient representation of the transfer function is to normalize the jet volume flow rate by the driver volume flow rate, Q, /Q,, and obtain an expression via the current/flow divider shown in Figure 26, QJ (s) Z1c /sC, Qd (s) Zac + Zo 1/sCc + Ro + sMo 1 (217) CaMaM 1 + Ro s+s2 CacM o Mao assuming that the acoustic orifice impedance Zo = Ro +Mo, only contains the linear resistance R, and the radiation mass Mo1d is neglected or added to MIO. Knowing that the Helmholtz resonator frequency of the actuator is defined by S, (218) H = acMo and the damping ratio of the system by r= R (219) 2 r)Cva ( ) by substituting in Eqs. 218 and 219, Eq. 217 can then be rewritten as 48 Q (s) Q Ws) 02 (220) Qd (S) S2 + 2C(HS + H2 This is a secondorder system whose performance is set by the resonator Helmholtz frequency. Figure 27 below shows the effect of the damping coefficient ; on the frequency response of Q /Qd, where for ; <1 the system is said to be underdamped, and for ; > 1 the system is overdamped. The damping coefficient controls the amplitude of the resonance peak, allowing the system to yield more or less response at the Helmoholtz frequency. 40 0 20 60 1I I I 1 1 1 1 1 10 100 10 0 .......... ... .. ..... .. " 50 =01 I I I I I I I =0.5 S100 I II I I I I I 1 60 L^ 101 100 10 0 ............................................. ..... ........... I ..... =0.01. . 200 Figure 27: Bode diagram of the second order system given by Eq. 220, for different damping ratio. Since the expression of aH differs from the orifice geometry, two different cases are examined and summarized in Table 22. The definitions can be found in Appendices o 100    f"..  = 1 B, C, and D. The damping coefficient is found from the following arrangement (shown for the case of a circular orifice, but one can similarly arrive at the same result for a rectangular slot) D:' 150o   . ^ " "i ..\  200 L^ rectangular slot) 1 8,h 2 z(d 2)4 (4ph) j76Vh vh v l L; X 642/2zh V 3V/ d/o6) pc 16p (221) ' V 0) 2 3rd2C2 Id4w)2 I/4 01 "=12 0)H S2 (222) Table 22: Dimensional parameters for circular and rectangular orifices Circular orifice Rectangular slot Qd (m3/s) JOdAV JOdAV (, (rad/s) 3r(d 2)2 c 5w(d2) 4hV 3hV V V C, (s2.m4/kg) 2 pCo pc, 8/ph 3/ h R, (kg/m4.s) 8h4 3 z (d 2) 2w (d 2)3 4ph 3ph MAN (kg/m4) 3(d )2 S3r (d 2)2 5w(d 2) 1 C (o 1 o 1 = R2v r 12 2 5 2 (A M mH S H S Notice that the damping coefficient has the same fundamental expression whether the orifice is circular or rectangular, the difference being incorporated in a multiplicative constant. Substituting these results into Eq. 220 and replacing the Laplace variable s = jo) yields the final form for a generic driver and a generic orifice Q Q(). (223) a e i, that is, Clearly, the advantage of nondimensionalizing the jet volume flow rate by the driver flow rate allows us to isolate the driver dynamics from the main response, thereby decoupling the effect of the various device components from each other. Eq. 223 is an important result in predicting the linear system response in terms of the nondimensional parameters co/cod, Ho/WoH and S as a function of the driver performance. It yields such interesting results that actually a thorough analysis of Eq. 223 is provided in details in Chapter 5 where the reader is referred to for completeness. To summarize, this section has provided a dimensional analysis of an isolated ZNMF actuator. A compact expression, in terms of the principal dimensionless parameters, has been found for the nondimensional transfer function that relates the output to the input of the actuator. Most importantly, such an expression was derived regardless of the orifice geometry and regardless of the driver configuration. Actually, as an example, a piezoelectricdriven ZNMF actuator exhausting into a quiescent medium is also considered in Appendix E where the idea is to find the same general expression as derived above in Eq. 215 for a generic ZNMF device, but starting from the specific and already known transfer function of a piezoelectricdriven synthetic jet actuator as given in Gallas et al. (2003a). Appendix E presents the full assumptions and derivation of the nondimensionalization and the derivation of the linear transfer function for this case. Next, with this knowledge gained, the modeling issues presented earlier in the introduction chapter and at the beginning of this chapter are further considered. Modeling Issues Cavity Effect The cavity plays an important role in the actuator performance. Intuitively, an actuator having a large cavity may not act in a similar fashion to one having a very small cavity. As mentioned above, the cavity of a ZNMF actuator permits the compression and expansion of fluid. It is more obvious when looking at the equivalent circuit of a ZNMF device (see Figure 21 for instance), where the flow produced by the driver is split into two branches: one for the cavity where the fluid undergoes successive compression and expansion cycles, the other one for the orifice neck where the fluid is alternatively ejected and ingested. The question arises as to when, if ever, an incompressible assumption is valid. The definition of the cavity incompressibility limit is actually twofold. First, from the equivalent circuit perspective, a high cavity impedance will prevent the flow from going into the cavity branch, thereby allowing maximum flow into and out of the orifice neck, thus maximizing the jet output. Or from another point of view, the incompressible limit occurs for a stiff cavity, hence for zero compliance in the cavity, which should yield to Q /Qd 1. On the other hand, from a computational point of view, it is rather essential to know whether the flow inside the cavity can be considered as incompressible, the computation cost being quite different between a compressible and an incompressible solver. Actually, because of its importance in numerical simulations and relevance in the physical understanding of a ZNMF actuator, Chapter 5 is entirely dedicated to the question of the cavity modeling. The reader is therefore referred to Chapter 5 for a thorough investigation on the role of the cavity in a ZNMF actuator. Orifice Effect The orifice is one of the major components of a ZNMF actuator device. Its shape will greatly contributes in the actuator response, and knowledge of the nature of the flow at the orifice exit is determinant in predicting the system response. The LEM technique presented earlier was shown to be a satisfactory tool in this way, but has still fundamental limitations, especially in the expression of the orifice nonlinear loss coefficient Kd. Similarly, the existing lumped element model is employed in the frequency domain. Because of the oscillatory nature of the actuator response, it may also be instructive to study the response of ZNMF actuator in the time domain. Lumped element modeling in the time domain The LEM technique presented above and used throughout this work identifies a transfer function in the Laplace domain, consequently in the frequency domain as well by assuming s = c + jo) > j) Note that this variable substitution is only correct when an input function g(t) is absolutely integrable, that is if it satisfies g (t)t < 00, (224) i.e., the signal must be causal and that the system is stable conditions that are always met in this work. For a given transfer function of the system (ZNMF actuator) relating the output (jet velocity) to the input (driver signal) in the frequency domain, it could therefore be of interest to gain some insight from the time domain response. Referring to Figure 26 and Eq. 217, the equation of motion for the ZNMF actuator is given by Qj (Z + ZC) = QdZ, c, (225) where again Za= /jcoCc is the acoustic impedance of the cavity, and Zao =RaOhn +Raonl(QJ)+jiwMoo is the acoustic orifice impedance. The orifice mass Moo includes the contributions from the radiation and inertia, while the orifice resistances are distinguished between the linear terms Rohn = Ra (viscous losses) and nonlinear R on = f(Q ("dump loss") defined by Eq. 214. Also, Qj = y,S, is the jet volume flow rate, Qd = ydSd is the volume velocity generated by the driver, and y, and Yd are, respectively, the fluid particle displacement at the orifice and the vibrating driver displacement. Notice that yJ can take positive or negative values, which corresponds respectively to the time of expulsion and ingestion during a cycle, as seen in Figure 28. Therefore, since the nonlinear resistance is associated to the time of discharge and considering the coefficient Kd as a constant independent of Q,, it takes the form 0.5KdpQj 0.5Kdp RaOn s2 y = A yn .j (226) n Sn A Y B C +yJ Y, max ingestion xQ + expulsion starts Q C APc 0 time d exptl sion    stitrts .  .max Yd ingestion Figure 28: Coordinate system and sign convention definition in a ZNMF actuator. A) Schematic of coordinate system. B) Circuit representation. C) Cycle for the jet velocity. The following expression for the equation of motion of a fluid particle can then be easily derived 1 S Sj +Ron +Rohn + joMo = d Yd,. (227) But since frequency and time domain are related through jo > d/dt and 1/jo > dt, and assuming a sinusoidal motion for the source term, i.e. Yd = W sin(o(), with WO corresponding to the driver centerline amplitude, then the equation of motion in the time domain is written as S S S + S ,4A1, A + S, Roon jV + SMooy = d Wo sin(cot), (228) or by rearranging the terms, MAoyj,+ +An Y +Ro~jn~, + y, = Wo sin (0t). (229) aC aCSn Similarly, the pressure API across the orifice can be derived from continuity, AP = QjZZ = (Qd Q ) Z. (230) Thus, substituting in Eq. 230 and rearranging yields A = QdZac QZac = SYd Sn, (231) and finally the pressure drop takes the following expression S S SdW0 sin(ct) S,y A = Wosin(mot) y,= d ) nyj (232) C C"C Cac To validate this temporal approach of the lumped element model, three test cases are now considered having three different orifice shapes to also gain insight into the orifice geometric effects. First, the response of a ZNMF actuator having a simple straight rectangular orifice shape and a high aspect ratio hid is viewed, and that corresponds to Case 1 in the NASA LaRC workshop (CFDVal 2004), as shown in Figure 29. Then, Case 2 of the same workshop (CFDVal 2004) is considered since the orifice of this ZNMF actuator has a rounded beveled shape (D/d = 2, see Figure 210 for geometric definition) and an aspect ratio less than unity, where high values of the orifice discharge coefficient are expected. The actuator geometry is shown in Figure 210. A third example is taken from the results provided by Choudhari et al. (1999), in which they perform a numerical simulation of flow past Helmholtz resonators for acoustic liners, with the orifice aspect ratio h/d equal to unity. h diaphragm ozone 2 moving diaphragm located at slade ofvity 1 ZOnrl 3 Zone 1not visible Figure 29: Geometry of the piezoelectricdriven ZNMF actuator from Case 1 (CFDVal 2004). d =1.27mm, d/D=0.59, h/d =10.6, w/d=28, f = 445Hz. (Reproduced with permission) d_ K Plate Surface (Flow side) h ) D _'I'I Caviy_  Actuator Figure 210: Geometry of the pistondriven ZNMF actuator from Case 2 (CFDVal 2004). d = 6.35mm, d/D = 0.5, h/d = 0.68, f = 150Hz. (Reproduced with permission) Because of their special orifice shape, pipe theory was used to model the dimensionless "dump loss" coefficient Kd in the acoustic orifice impedance for Case 1 and Case 2 (CFDVal 2004). From pipe theory (White 1979), the dump loss coefficient for the orifice is Kd D(1,ICD) (233) with / = d/D is the ratio of the exit to the entrance orifice diameter, and with the discharge coefficient taking the form CD = 0.9975 6.53(//Re)o5, (234) for a beveled shape, Re being the Reynolds number based on the orifice exit diameter d. For each case, the Reynolds number given by the experimental data provided in the workshop (CFDVal 2004) is used in Eq. 234, although it should be rigorously implemented in a converging loop since this variable is usually not known beforehand. For Case 1, it was found that Kd = 0.884, while for Case 2, Kd = 0.989. This is to be compared with the value Kd = 1 that is used in Gallas et al. (2003a). Notice though that Eq. 234 is specifically defined for high Reynolds number, which may not always be the case. Similarly, Eqs. 233 and 234 only account for the expulsion part of the cycle. During the ingestion part the flow sees an inversedd" orifice shape, hence the discharge coefficient should take a different form. How to account for the oscillatory behavior on the orifice shape, i.e. to separate the expulsion to the ingestion phase for the flow discharge, is investigated in the next chapters of this dissertation. Yet, these results validate the approach used and provide valuable insight into the nonlinear behavior. The nonlinear ODE that describes the motion of the fluid particle at the orifice, Eq. 229, is numerically integrated using a 4th order RungeKutta method with zero initial conditions for y, (0) = y (0) = 0. The integration is carried out until a steadystate is reached. The jet orifice velocity, pressure drop across the orifice via Eq. 232, and the driver displacement are shown in Figure 211 for Case 1. All quantities exhibit sinusoidal behavior, and it can be seen that the cavity pressure is in phase with the driver displacement, while the jet orifice velocity lags the driver displacement by 900. Once the pressure reaches its maximum (maximum compression, the fluid cavity starts to expand), the fluid is ingested from the orifice, then reaches its maximum ingestion when the cavity pressure is zero and finally, as the fluid inside the cavity starts to be compressed, the fluid is ejected from the orifice. 1 ..... driver displacement S pressure drop 0.8 /  jet orifice velocity 0.   0.4  7) S0.6 . * 0.84 \ 1 phase Figure 211: Time signals of the jet orifice velocity, pressure across the orifice, and c= 0.2 i I I driver displacement during one cycle for Case 1. The quantities are normalized by their respective magnitudes for comparison. 0.2 \ 1 '' 0 0.4 q \ T \\ 1 / n ,\ '// 0.8 \  0 45 90 135 180 225 270 315 360 phase Figure 211: Time signals of the jet orifice velocity, pressure across the orifice, and driver displacement during one cycle for Case 1. The quantities are normalized by their respective magnitudes for comparison. The other test case response, namely Case 2, is plotted in Figure 212, where the jet orifice displacement and velocity, pressure drop across the orifice, and the driver displacement are shown for both the a) linear and the b) nonlinear solutions of the equation of motion Eqs. 229 and 232. The linear solution is obtained by setting Ron = 0 and is performed to verify the physics of the device behavior and thus confirm the modeling approach used. The linear solution in Figure 212A shows that the pressure inside the cavity (which equals the pressure drop across the orifice) and the driver motion are almost out of phase. All quantities exhibit sinusoidal behavior. The jet orifice velocity y, lags the cavity pressure for both the linear and the nonlinear solution. Figure 212B shows the effect of the nonlinearity of the orifice resistance. Its main effect is to shift the pressure signal such that the fluid particle velocity and the cavity pressure are out of phase. Also, those two signals exhibit obvious nonlinear behavior due to the nonlinear orifice resistance. Linear Solution  driver displacement Non linear Solution 1 \ ."".  pressure drop 1 /  ,/ \ ,\ jet orifice velocity / \ / 0.8 08 0.8  S I / I \ I I I I S0.6/ / \ 0.4  02 ,0.2 0 L  I 0.2 \ \  0.2   S \ \ S\displacement during one cycle for Case 2. A) Linear solution. B) Nonlinear \ /  0.2 S04 \ / '04  \'A // \ 0.6  __.0.6  i \ / /\ 0 45 90 135 180 225 270 315 360 0 45 90 135 180 225 270 315 360 phase phase Figure 212: Time signals of the jet orifice velocity, pressure across the orifice and driver displacement during one cycle for Case 2. A) Linear solution. B) Nonlinear solution. The quantities are normalized by their respective magnitudes for comparison. Then, Figure 213 shows the numerical results from Choudhari et al. (1999), with their notation reproduced, where the reference signal shown corresponds to that measured at the computational boundary where the acoustic forcing is applied, and the xaxis in the plot is normalized by the period T of the incident wave. Notice that they used a perforate plate having a porosity a equal to 5%. In a similar trend as for the previous case, the pressure drop and jet orifice velocity exhibit distinct nonlinearities in their time signals. From Figure 213A, it is seen that the pressure perturbations at each end of the orifice are almost out of phase, while in Figure 213B, the velocities at different locations in the orifice are in phase with each other. Also, it appears that the pressure and velocity perturbations have about a 900 phase difference, similar to Case 1 above. .... . T UT A) Disturbance pressure p/pco B) Streamwise velocity perturbation u/co Figure 213: Numerical results of the time signals for A) pressure drop and B) velocity perturbation at selected locations along the resonator orifice. The subscripts i, c, and e refer to the orifice opening towards the impedance tube (exterior), the orifice center, and the orifice opening towards the backing cavity, respectively. d= 2.54mm, h/d= 1, f = 566Hz, = 0.05. (Reproduced with permission from Choudhari et al. 1999) Clearly, the orifice shape does have a significant impact on the nonlinear signal distortion in the orifice region. It should be noted that the actuation frequency and amplitude are also important, as discussed in Choudhari et al. (1999), and mentioned in the introduction chapter where Ingard and Ising (1967) and later Seifert et al. (1999) showed that for low actuation amplitude the pressure fluctuations and the velocity scale as u' p'/pco whereas for high amplitude u' ~ p/p. However, it still emphasizes the need to accurately model the orifice discharge coefficient in terms of the flow conditions. As mentioned before, also of interest is the fullydeveloped assumption for the flow inside the orifice. Clearly, while Case 1 (CFDVal 2004) has an orifice geometry that justifies such an approximation, it seems quite doubtful for Case 2 (CFDVal 2004) and perhaps the Helmholtz resonator geometry from Choudhari et al. (1999). It is expected that a developing region exists at the orifice opening ends, where a different relationship relates the pressure drop and the fluid velocity, the velocity being now dependant on the longitudinal location inside the orifice. In this regard, the next subsection provides more details on this entrance region. Finally, another orifice issue that may not be negligible is the radius of curvature at the exit plane. In fact, the formation and subsequent shedding of the vortex ring (pair) at the orifice (slot) exit relies on the curvature of the exit plane. Sharp edges facilitate the formation and rollup of the vortices, due to a local higher pressure difference, while smooth edges having a large radius of curvature lessen the formation of vortices at the exit plane, as shown in the recent work by Smith and Swift (2003b) who experimentally studied the losses in an oscillatory flow through a rounded slot. This parameter, R/d, may enter in the present nondimensional analysis for completeness, although it is omitted in this dissertation. Loss mechanism In this subsection, an attempt is made to physically describe the flow mechanism inside the orifice. The flow inside the orifice is by nature unsteady and is exhibiting complex behavior as demonstrated in the literature review. One approach to understand the nature of the flow physics is to consider known simpler cases. First it is instructive to consider the simpler case of steady flow through a pipe where losses arise due to different mechanisms. In any undergraduate fluid mechanics textbook, these losses are characterized as "major" losses in the fully developed flow region and "minor" losses associated with entrance and exit effects, etc. For laminar flow, the pressure drop Ap in the fullydeveloped region is linearly proportional to the volume flow rate Q or average spatial velocity VJ, while the nonlinear minor pressure losses are proportional to the 2 dynamic pressure 0.5pVj Similarly, for the case of unsteady, laminar, fullydeveloped, flow driven by an oscillatory pressure gradient, the complex flow impedance, Ap/Q, can be determined analytically and decomposed into linear resistance and reactive components as already discussed above. Unfortunately, no such solution is yet available for the nonlinear, and perhaps dominant, losses associated with entrance and exit effects. It then appears that the orifice flow can be characterize by three dominant regions, as shown schematically in Figure 214, where the first region is dominated by the entrance flow, then follows a linear or fullydeveloped region away from the orifice ends, to finally include an exit region. Notice that this is for one half of the total period, but by assuming a symmetric orifice the flow will undergo a similar development as it reverses. Also shown schematically in Figure 214 are the pathlines or particle excursions for three different running conditions. The first one corresponds to the case where the stroke length is much smaller than the orifice height (L0 << h) recall that the stroke length is simply related to the Strouhal number via Eq. 27. In this case it is expected that the flow inside the orifice may easily reach a fullydeveloped state, thus having losses dominated by the "major" linear viscous loss rather than the nonlinear "minor" ones associated with the entrance and exit regions. A second case occurs when the stroke length is this time much larger than the orifice height (L0 > h). In this scenario, the losses are now expected to be largely dominated by the minor nonlinear losses due to entrance and exit effects, the entrance region basically extending all the way through the orifice length. Finally, in the case where the stroke length and orifice height have the same order of magnitude (L0 h), the linear losses due to the fullydeveloped region should compete with the nonlinear losses from the entrance and exit effects. Notice that here, "fully developed" means that there exists a region within the orifice away from either exit, where the velocity profile at a given phase during the cycle is not a function of axial position y. exit & entrance losses viscous loss (fullydeveloped flow) Lo> h "/" "" h"" L << h ..... ...~4 < Figure 214 Schematic of the different flow regions inside a ZNMF actuator orifice. Figure 214: Schematic of the different flow regions inside a ZNMF actuator orifice. Thus to refine the existing lumped element model presented above that uses the frequencydependent analytical solution for the linear resistance, the impedance of the nonlinear losses associated with the entrance and exit regions should be extracted. However, the relative importance and scaling of the linear and nonlinear components versus the governing dimensionless parameters is unknown and remains a critical obstacle for designers of ZNMF actuators at this stage. To achieve such a goal i.e., to improve the current understanding of the orifice flow physics and consequently to improve the accuracy of loworder models, a careful experimental investigation is conducted and the extracted results are presented in the subsequent chapters. DrivingTransducer Effect Most of the numerical simulations impose a moving boundary condition in order to model the kinematics of the ZNMF driver that generates the oscillating jet in the orifice neck. However, this approach does not capture the driver dynamics and in most instances, crude models of the mode shape are employed (Rizzetta et al. 1999; Orkwis and Filz 2005). Although this might not be critical if the actuator is driven far from any resonance frequency, the information provided by the driver is relevant from a design perspective, with the frequency response (magnitude and phase) dictating the overall performance of the system and thus its desirable application. The approach used in this dissertation is to decouple the dynamics of the driver from the rest of the device via the analysis of a dimensionless transfer function. Hence, accurate component models can be sought that will provide useful information on the overall behavior of the actuator. In this regard, LEM has been shown to be a suitable solution, as discussed below, for any type of drive configuration, i.e. pistonlike diaphragm, piezoelectric diaphragm, etc. Figure 11 shows the three most common driving mechanisms that are employed in ZNMF actuators, namely an oscillating diaphragm (usually a piezoelectric patch mounted on one side of a metallic shim and driven by an ac voltage), a piston mounted in the cavity (using an electromagnetic shaker, a camshaft, etc.), or a loudspeaker enclosed in the cavity (an electrodynamic voicecoil transducer). In addition to the driver dynamics, the characteristics of most interest are the volume displaced by the driver AV at the actuation frequency f Hence, the driver volumetric flow rate can simply be defined by Qd= j(27rf)AV. (235) It has been shown that this compact expression is useful in the nondimensional analysis performed earlier. However, in order to obtain the full dynamics of the actuator response, the LHS of Eq. 235 must also be known. Only then do the compact analytical expressions derived in the previous section reveal their usefulness. Each of the three types of possible ZNMF actuator drivers are discussed below via LEM, since the analysis and design of coupleddomain transducer systems are commonly performed using lumped element models (Fisher 1955; Merhault 1981; Rossi 1988). I.e., in addition to the driver acoustic impedance Z, that is shown in Figure 26 and Figure 215, the transduction factor and the blocked electrical impedance CeB must be explicitly given. I aQd 1: CaD MaD RaDd + r a+ Vac eB oaV P Figure 215: Equivalent twoport circuit representation of piezoelectric transduction. First, consider the case of a piezoelectric diaphragm driver. Recently (Gallas et al. 2003a, 2003b), the author successfully implemented a twoport model for the piezoceramic plate (Prasad et al. 2002) in the analysis, modeling and optimization of an isolated ZNMF actuator. As shown in Figure 215, the impedance of the composite plate was modeled in the acoustic domain as a series representation of an equivalent acoustic mass M ,D, a shortcircuit acoustic compliance CaD (that relates an applied differential pressure to the volume displacement of the diaphragm) and an acoustic resistance RD (that represents the losses due to mechanical damping effects in the diaphragm). Similarly, a radiation acoustic mass can be added if needed. The conversion from electrical to acoustic domain is performed via an ideal transformer possessing a turns ratio a that converts energy from the electrical domain to the acoustic domain without losses. Figure 21 shows the twoport circuit representation implemented in a ZNMF actuator. 0, MaD and CaD are calculated via linear composite plate theory (see Prasad et al. 2002 for details). Notice that the acoustic resistance RD given by ROD =2D (236) SCaD is the only empirically determined parameter in this model, since the damping coefficient 4D is experimentally determined. The problem in finding a nonempirical expression for the diaphragm damping coefficient (for instance by using the known quality factor) comes mostly from the actual implementation of the driver in the device. A perfect clamped boundary condition is assumed, and deviation from this boundary condition and the problem of high tolerance/uncertainties between the manufactured piezoceramic diaphragms can degrade the accuracy of the model. Nonetheless, the dynamics of the driver are well captured by this model and were successfully implemented in previous studies (Gallas et al. 2003a, 2003b; CFDValCase 1 2004). Consider next an acoustic speaker that drives a ZNMF actuator. Similar to a piezoelectric diaphragm, a simple circuit representation can be made. McCormick (2002) has already performed such an analysis, as shown in Figure 216. The speaker is actually a moving voice coil that creates acoustic pressure fluctuations inside the cavity. Its principle is simple. It is usually composed of a permanent magnet, a voice coil and a diaphragm attached to it. When an ac current flowing through the voice coil changes direction, the coil's polar orientation reverses, thereby changing the magnetic forces between the voice coil and the permanent magnet, and then the diaphragm attached to the coil moves and back and forth. This vibrates the air in front of the speaker, creating sound waves. (Speaker corpliance)a (Speaker resistance) Moving voice coil M R U R =(BL)2/R aD Ca R4 U actuator c  (Coil resistance)a T Cs (Speaker + air rass),a M cBL c U RcSd N M C A A B Cavity/NeckDynamics Figure 216: Speakerdriven ZNMF actuator. A) Physical arrangement. B) Equivalent circuit model representation obtained using lumped elements used in McCormick (2000). BL is the voice coil force constant (= magnetic flux x coil length) As represented in Figure 216B, the acoustic impedance ZaD of the driver is modeled via acoustic resistances (from the coil and the speaker) mounted in series with acoustic masses (speaker plus air) and compliances (from the speaker). The main issues concerning such an arrangement are, first, the practical deployment of the speaker to drive the ZNMF actuator in a desired frequency range. Also, a loudspeaker creates pressure fluctuations whose characteristics (amplitude and frequency) depend on the speaker dynamics. For example, if the speaker is mounted in a large cavity enclosure (whose size is greater than the acoustic wavelength), it might excite the acoustic modes of the cavity, thereby resulting in threedimensionality of the flow in the slot. Sorifice .vent sealing membrane channe cavity r bottom T cavityv shaker Figure 217: Schematic of a shakerdriven ZNMF actuator, showing the vent channel between the two sealed cavities. Finally, consider a pistonlike driver. It could be operated either mechanically, for instance by a camshaft or by other mechanical means, or by using an electromagnetic shaker. Here, we turn our attention to the latter application. An electromagnetic piston usually consists of a moving voice coil shaft that drives a rigid piston plate and, in essence, follows the same concept as presented above for the case of a voice coil loudspeaker. Although the previous discussion on the LEM representation remains the same here, the major difference comes from the nature of the piston itself. In fact, while the top face of the piston is facing the cavity of the ZNMF actuator, another cavity on the opposite side of the piston is present, as shown in Figure 217. This cavity may or may not be vented to the other cavity. If sealed, when the ZNMF device is running at a specific condition, an additional pressure load is created on the piston plate to account for the static pressure difference between the cavities that may deteriorate the nominal transducer performance. To alleviate this effect, the ZNMF cavity and the bottom cavity could be vented together, in a similar manner to that employed for a microphone design. Also, this bottom cavity should be added in series with the ZNMF cavity (since they share the same common flow) in the circuit representation of the actuator that is shown in Figure 218. el ectFedTnamic" dcorpng (BL):1 Qd Qj i IId j^ V\ i electrical \ v d Qj source acoustic impedance of the ZNMF cavity, Zac bot is the acoustic impedance of the bottom cavity, and ZaVet is the acoustic impedance of the vent channel. Even though tools are available using lumped element modeling, the ZNMF actuator driver must be modeled with care, especially when deployed in a physical apparatus. However, once the driver dynamics have been successfully modeled, its Chapter 7. Now that some insight has been gained on the dynamics of a ZNMF actuator in still  air, a test matrix is constructed to carefully investigate both experimentally and numerically the unresolved features of these types of devices, especially on refining the nonlinear loss coefficient of the orifice. nonlinear loss coefficient of the orifice. Test Matrix A significant database forms the basis of a test matrix that includes direct numerical simulations and experimental results. The test matrix is comprised of various test actuator configurations that are examined to ultimately assess the accuracy of the developed reducedorder models over a wide range of operating conditions. The goal is to test various actuator configurations in order to cover a wide range of operating conditions, in a quiescent medium, by varying the key dimensionless parameters extracted in the above dimensional analysis. Available numerical simulations are used along with experimental data performed in the Fluid Mechanics Laboratory at the University of Florida on a single piezoelectricdriven ZNMF device exhausting in still air. Table 23 describes the test matrix. The first six cases are direct numerical simulations (DNS) from the George Washington University under the supervision of Prof Mittal. They use a 2D DNS simulation whose methodology is detailed in Appendix F. Case 8 comes from the first test case of the NASA LaRC workshop (CFDVal 2004). Then, Case 9 to Case 72 are experimental test cases performed at the University of Florida for axisymmetric piezoelectricdriven ZNMF actuators. The experimental setup is described in details in Chapter 3, and the results are systematically analyzed and studied in Chapter 4, Chapter 5, and Chapter 6. Table 23: Test matrix for ZNMF actuator in quiescent medium d h w V Case Type f(Hz) w/d S Re St f f/fd Jet (mm) (mm) (mm3 1 CFD 0.38 1 1 oo 800 25.0 262 2.4 0.13 X 2 CFD 0.38 1 2 oo 800 25.0 262 2.4 0.15 X 3 CFD 0.06 1 0.68 oo 360 10.0 262 0.4 0.01 J 4 CFD 0.20 0.1 0.1 oo 800 5.0 63.6 0.4 0.00 J 5 CFD 0.80 0.1 0.1 oo 800 10.0 255 0.4 0.01 J 6 CFD 1.99 0.1 0.1 oo 800 15.8 477 0.5 0.03 J 7 CFD 1.99 0.1 0.1 oo 800 15.8 636 0.4 0.03 J 8 exp/cfd 446 1.27 13.5 28 7549 17.1 861 0.3 2.65 0.99 J 9 exp. 39 1.9 1.8 7109 7.6 8.79 6.6 0.06 0.06 X Case 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 Type exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exp. exo. f(Hz) d (mm) 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 39 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 780 1.9 1200 1.9 39 2.98 39 2.98 39 2.98 500 2.98 500 2.98 780 2.98 780 2.98 780 2.98 780 2.98 780 2.98 39 2.96 39 2.96 39 2.96 39 2.96 780 2.96 780 2.96 780 2.96 780 2.96 780 2.96 h (mm) 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.8 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 1.05 4.99 4.99 4.99 4.99 4.99 4.99 4.99 4.99 4.99 V w/d wd (mm3)  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109  7109 S 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 7.6 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 34.0 42.1 11.9 11.9 11.9 42.6 42.6 53.2 53.2 53.2 53.2 53.2 11.8 11.8 11.8 11.8 52.9 52.9 52.9 52.9 52.9 Re 12.0 22.6 33.2 39.8 46.5 52.5 59.7 66.0 73.7 81.6 88.2 192 242 374 513 637 750 825 930 1131 1120 1200 1264 1510 1589 1683 1774 1842 1876 2755 90.8 40.6 47.3 63.4 1959 2615 109 254 571 1439 2022 29.8 43.0 55.7 71.9 125 318 867 2059 3039 St f/fH f/fd Jet 4.8 0.06 0.06 J 2.5 0.06 0.06 J 1.7 0.06 0.06 J 1.4 0.06 0.06 J 1.2 0.06 0.06 J 1.1 0.06 0.06 J 1.0 0.06 0.06 J 0.9 0.06 0.06 J 0.8 0.06 0.06 J 0.7 0.06 0.06 J 0.6 0.06 0.06 J 6.0 1.24 1.23 X 4.8 1.24 1.23 J 3.1 1.24 1.23 J 2.2 1.24 1.23 J 1.8 1.24 1.23 J 1.5 1.24 1.23 J 1.4 1.24 1.23 J 1.2 1.24 1.23 J 1.1 1.24 1.23 J 1.0 1.24 1.23 J 1.0 1.24 1.23 J 0.9 1.24 1.23 J 0.8 1.24 1.23 J 0.7 1.24 1.23 J 0.7 1.24 1.23 J 0.6 1.24 1.23 J 0.6 1.24 1.23 J 0.6 1.24 1.23 J 0.4 1.24 1.23 J 19.5 1.91 1.90 X 3.49 0.04 0.06 J 2.99 0.04 0.06 J 2.23 0.04 0.06 J 0.93 0.55 0.79 J 0.69 0.55 0.79 J 26.0 0.86 1.23 X 11.2 0.86 1.23 X 4.96 0.86 1.23 J 1.97 0.86 1.23 J 1.40 0.86 1.23 J 4.69 0.06 0.06 J 3.25 0.06 0.06 J 2.51 0.06 0.06 J 1.94 0.06 0.06 J 22.3 1.25 1.23 X 8.79 1.25 1.23 X 3.22 1.25 1.23 J 1.36 1.25 1.23 J 0.92 1.25 1.23 J d h V Case Type f(Hz) () ( w/d S(m3 Re St f/f, f/fd Jet (mm) (mm) (mm3) 60 exp. 39 1.0 5.0 7109 4.0 132 0.12 0.16 0.06 J 61 exp. 39 1.0 5.0 7109 4.0 157 0.10 0.16 0.06 J 62 exp. 39 1.0 5.0 7109 4.0 205 0.08 0.16 0.06 J 63 exp. 500 1.0 5.0 7109 14.3 286 0.72 2.10 0.79 J 64 exp. 500 1.0 5.0 7109 14.3 461 0.44 2.10 0.79 J 65 exp. 730 1.0 5.0 7109 17.3 269 1.11 3.07 1.16 J 66 exp. 730 1.0 5.0 7109 17.3 611 0.49 3.07 1.16 J 67 exp. 730 1.0 5.0 7109 17.3 893 0.33 3.07 1.16 J 68 exp. 730 1.0 5.0 7109 17.3 1081 0.28 3.07 1.16 J 69 exp. 730 1.0 5.0 7109 17.3 1361 0.22 3.07 1.16 J 70 exp. 39 0.98 0.92 7109 3.9 49.6 0.31 0.09 0.06 J 71 exp. 39 0.98 0.92 7109 3.9 112 0.14 0.09 0.06 J 72 exp. 39 0.98 0.92 7109 3.9 179 0.09 0.09 0.06 J To conclude this chapter, the existing lumped element model from Gallas et al. (2003a) has been presented and reviewed, and it has been shown that it could be extended to more general device configurations, particularly in terms of orifice geometry and driver configuration. Then, a dimensional analysis of an isolated ZNMF actuator was performed. A compact expression, in terms of the principal dimensionless parameters, was found for the nondimensional linear transfer function that relates the output to the input of the actuator, regardless of the orifice geometry and of the driver configuration. Next, some modeling issues have been investigated for the different components of a ZNMF actuator. Specifically, the LEM technique has been used in the time domain to yield some insight on the orifice shape effect, and a physical description on the associated orifice losses has been provided. Finally, since one of the goals of this research is to develop a refined loworder model, which is presented in Chapter 6 and that builds on the results presented in the subsequent chapters, a significant database forms the basis of a test matrix that is comprised of direct numerical simulations and experimental results. CHAPTER 3 EXPERIMENTAL SETUP This chapter provides the details on the design and the specifications of the ZNMF devices used in the experimental study. Descriptions of the cavity pressure, driver deflection, and actuator exit velocity measurements are provided, along with the dynamic data acquisition system employed. Then, the data reduction process is presented with some general results. A description of the Fourier series decomposition applied to the phaselocked, ensemble average time signals is presented next. Finally, a description of the flow visualization technique employed to determine if a synthetic jet is formed is then provided. Experimental Setup In this dissertation, two different experiments are performed. The first one, referred to as Test 1, is used in the orifice flow analysis presented in Chapter 4 and the corresponding test cases are listed in Table 23. The second test, Test 2, is used in the cavity compressibility analysis (presented in Chapter 5). Test 1 consists of phaselocked measurements of the velocity profile at the orifice, cavity pressure, and diaphragm deflection, and the device uses a large diaphragm and has an axisymmetric straight orifice. On the other hand, in Test 2 only the frequency response of the centerline velocity and driver displacement are acquired, and the device uses a small diaphragm and the orifice is a rectangular slot. However, since the two tests share the same equipment and basic setup and Test 1 requires additional equipment, only Test 1 is detailed below. Top View displacement sensor 3 component traverse I // Figure 31: Schematic of the experimental setup for phaselocked cavity pressure, diaphragm deflection and offaxis, twocomponent LDV measurements. diaphragm LV WL cavity (V) U/ top plate body plate diaphragm clamp plate mount Figure 32: Exploded view of the modular piezoelectricdriven ZNMF actuator used in the experimental test. 