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## Material Information- Title:
- Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet Actuators for Active Flow Control
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- 2008
## Subjects- Subjects / Keywords:
- Boundary conditions ( jstor )
Diameters ( jstor ) Diaphragm ( jstor ) Electric potential ( jstor ) Frequency response ( jstor ) Modeling ( jstor ) Natural frequencies ( jstor ) Parametric models ( jstor ) Pies ( jstor ) Velocity ( jstor )
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- 8/8/2002
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PAGE 1 LUMPED ELEMENT MODELING OF PIEZOELECTRIC-DRIVEN SYNTHETIC JET ACTUATORS FOR ACTIVE FLOW CONTROL By QUENTIN GALLAS A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2002 PAGE 2 Copyright 2002 By Quentin Gallas PAGE 3 iii ACKNOWLEDGMENTS Financial support for the research project was provided by a NASA-Langley Research Center Grant and an AFOSR grant. First, I would like to thank my advisor, Dr. Louis N. Cattafesta, for his continual guidance and support, which gave me the motivation and encouragement needed to make this work possible. I would also like to express my gratitude to Dr. Mark Sheplak and Dr. Bhavani Sankar for advising and guiding me with various aspects of the project. I thank all the members of the Interdisciplinary Microsystems group, particularly fellow students Jose Mathew, Ryan Holman, Anurag Kasyap and Steve Horowitz, for their help with my research and their friendship. Finally, special thanks go to my family and friends for always encouraging me to pursue my interests and for making that pursuit possible. PAGE 4 iv TABLE OF CONTENTS Page A C KNO W L E D GMEN T S ................................................................................................. . iii TAB L E O F CONT E N T S .................................................................................................. . iv L I ST O F T A B L ES ............................................................................................................ . vi L I ST OF F I G U RE S .......................................................................................................... . vii ABSTRACT....................................................................................................................... .x C HA P TER 1 I NT R ODU C T I O N .......................................................................................................... . 1 2 L U M P E D E L E M E N T M OD E L I N G ............................................................................. . 4 3 EQ U I V A L E NT C I R C U I T AN A L Y S I S ......................................................................... . 8 3.1 C a se 1: I n c o mp r e ssible Limit 1 0aC aDC C....................................................11 3.2 C a se 2: Ri g id Di a phra g m L imit 0aD aCC C.................................................13 4 MOD E L PARAMET E R ES T I MA T I O N ..................................................................... . 15 5 MOD E L VE R I F I C A T I O N AND P A R AMET E R EXT R A C T I O N ............................. . 22 5.1 P i e z oe l ec t r i c Transduc ti o n ............................................................................... . 23 5.2 Cavi t y Acoustic Compli a nc e ............................................................................ . 25 5.3 Acoustic Mass and Resi s t ance in Orif i c e ......................................................... . 26 6 C O M P A R I S ON B E T W EEN MOD E L & EX P E R I MENT S ........................................ . 28 6.1 L a s e r Doppl e r Vel o cimet r y S y stem ................................................................. . 29 6.2 D a ta A c quisitio n ............................................................................................... . 30 6.3 R e sult s .............................................................................................................. . 31 7 C ON C L U S I O NS A N D FUTU R E W O R K .................................................................. . 35 RE F E RE N C E S ................................................................................................................. . 37 PAGE 5 v APPENDIX A C OM P O S I TE P I E Z O E L E C T R I C D I A P H R A G M C HA R A C TE R I Z A T ION ............ . 40 B S YNTHE T I C J ET A C TUATO R S MEA S U R MENTS ............................................... . 50 C A M P L I F I E R C H A R A C T E R IZ A T I O N ....................................................................... . 61 B I OG R A P H I C A L S K E T CH ........................................................................................... . 63 PAGE 6 vi LIST OF TABLES Table page 5-1: P i e z ocera mi c d i ap h r a g m ch a rac t er i s ti c s .................................................................. . 23 5 2: C a l c ul a t e d lump e d e l e m e n t mod e l p a r a m e t e r s ........................................................ . 24 6-1: Specifications of p i e z oceramic diap h r a g ms. ........................................................... . 28 6-2: Calculated lumped paramet e rs of pie z oc e r a mic diaphr a g ms. ................................. . 29 6-3: Geomet r y of s y nth e tic jet actuators. ........................................................................ . 29 6-4: Uncer t a in t y in ma x imum veloci t y at sel e ct f r equencies for Case I I . ....................... . 31 6-5: Comparison betwe e n lumped element model and e x periment . ............................... . 34 A-1: Pie z oelectric diap h r a g m properties . ........................................................................ . 40 A-2: E x perimental values of natural frequency and dc response of tested pie z oelectric diaph r a g m s ............................................................................... . 41 B 1: Geo m et r y of modu l ar s y nthetic jet act u a tors. ......................................................... . 50 PAGE 7 vii LIST OF FIGURES Figure page 1: Schematic of a pie z oelectri c driven s y nthetic jet . ..................................................... . 1 2: Equivalent circuit r e presen t a tion of a pie z o e lectric-dri v en s y nthetic j e t . ................. . 6 3: Alternative equiva l e nt circuit model . ........................................................................ . 8 4: Variation in velocity profile vs. St = 1, 10, and 30 for oscillatory channel flow in a circul a r duct. .................................................................................................... . 19 4: Ratio of avera g e velocity to centerline velocity vs. St f o r os c ill a to r y c h a nn e l flow in a circu l ar duct. ............................................................................................ . 20 4: Acoustic resistance and reactance normali z ed by steady value vs. S t for os c ill a tory c h a nn e l f low i n a c i r c ular du c t . ............................................................. . 21 5: Assemb l y di a g r a m of modular s y nthetic jet . ........................................................... . 22 5: Comparison between predicted and measured response of pie z oceramic diaphr a g ms to a sinusoidal e x citation volta g e at 1 00 H z . ....................................... . 25 5: Measured acoustic compliance aCC vs. frequency in closed cavity of s y nthetic j e t. ........................................................................................................................... . 26 6: L a s e r Doppl e r Vel o cimeter s y stem setup for Case I . .............................................. . 30 6: Comparison between the lumped element m odel and e x periment for Case I . ........ . 32 6: Comparison between the lumped element m odel and e x periment for Case I I . ...... . 33 A: F r equency response (ma g nitude, phase and coherence) of the center of two sample pie z oceramic di a phr a g ms Case I . (Amplitude input volta g e =1V ) ........... . 42 A: F r equency response (ma g nitude, phase and coherence) of the center of pie z oceramic diaphra g m Case I for different boundary conditions. (Amplitude input volta g e =1V ) ................................................................................ . 43 PAGE 8 viii A: F r equency response (ma g nitude, phase and coherence) of the center of pie z oceramic diaphra g m Case I I for different boundary conditions. (Amplitude input volta g e =1V ) ................................................................................ . 44 A: F r equency response (ma g nitude, phase and coherence) of the center of p i e z o ce r a mi c d i a ph r a g m C a s e III f o r d i f f e r e n t boundary conditions. (Amplitude input volta g e =1V ) ................................................................................ . 45 A: F r equency response of pie z oceramic diaphra g m Case I . (Amplitude input volt a g e = 1V) ............................................................................................................ . 46 A: F r equency response of pie z oceramic diaphra g m Case II . (Amplitude input volt a g e = 1V) ............................................................................................................ . 47 A: F r equency response of pie z oceramic diaphra g m Case III. ( A m p litude input volt a g e = 1V) ............................................................................................................ . 47 A: Comparison between two boundary condition t y pes applied to pie z oceramic diaphr a g m Case I driv e n b y a sinusoidal e x citation volta g e o f 1V at 300 H z . .... . 48 A: Comparison between the response of two sample pie z oceramic diaphra g ms C a s e III d r i v e n b y a s i n u s o i d a l e x c it a ti o n vo lt a g e o f 1 V a t 500 H z . The pie z oceramic of s a mple 1 is not centered on the b r ass diaphr a g m . ........................ . 49 B : L a s e r Doppler Ve l ocimeter s y stem setup f o r Case II h a vi n g cavi t y v olume V3. ....51 B : Cavi t y volume ch a n g e in Ca s e I (model p r e diction ) .............................................. . 52 B : Cavi t y volume ch a n g e in Ca s e II (model pr e diction ) ............................................. . 53 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 1 . ............................ . 54 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 2 . ............................ . 55 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 3 . ............................ . 56 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 4 . ............................ . 57 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 5 . ............................ . 58 B : Comparison between the lumped element model and e x periment for s y nthetic jet having diap h r a g m C a s e I I , cavi t y volume V3, and orifice 6 . ............................ . 59 PAGE 9 ix B : Mean jet velocity response function of input volta g e for Case I I with cavity volume V3 at f=400Hz ............................................................................................ . 60 C: Amplifier g a in lin e a ri t y at constant f r eque n c y . ...................................................... . 61 C: Amplifier g a in fu n c tion of freq u en c y . .................................................................... . 62 PAGE 10 x Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science LUMPED ELEMENT MODELING OF PIEZOELECTRIC-DRIVEN SYNTHETIC JET ACTUATORS FOR ACTIVE FLOW CONTROL By Quentin Gallas August 2002 Chairman: Dr. Louis N. Cattafesta Major Department: Aerospace Engineering, Mechanics, and Engineering Science This thesis presents a lumped element model of a piezoelectric-driven syntheticjet actuator. A synthetic jet, also known as a zero net mass-flux device, uses a vibrating diaphragm to generate an oscillatory flow through a small orifice or slot. In lumped element modeling, the individual components of a synthetic jet are modeled as elements of an equivalent electrical circuit using conjugate power variables. The frequency response function of the circuit is derived to obtain an expression for the volume flow rate through the orifice per applied voltage across the piezoceramic. The circuit is analyzed to provide physical insight into the dependence of the device behavior on geometry and material properties. Methods to estimate the model parameters are discussed along with pertinent model assumptions, and experimental verification is presented of the lumped parameter models. In addition, two prototypical synthetic jet PAGE 11 xi actuators are built and tested. Very good agreement is obtained between the predicted and measured frequency response functions. PAGE 12 1 CHAPTER 1 INTRODUCTION Synthetic jet actuators have been the focus of significant research activity for the past decade [ 1 ]. The interest in synthetic jets is primarily due to their utility in flow control applications, such as separation control, mixing enhancement, etc. [ 2 6 ]. A schematic of a synthetic jet actuator is shown in Figure 1Â–1 . A typical synthetic jet, also known as a zero net mass-flux device, uses a vibrating diaphragm to drive oscillatory flow through a small orifice or slot. Although there is no flow source, a mean jet flow is established a few diameters from the orifice due to the entrained fluid. orifice cavity vibrating piezoceramic composite diaphragm net flow Figure 1Â–1: Schematic of a piezoelectric-driven synthetic jet. PAGE 13 2 In addition to studies that emphasize applications, there are numerous others that have concentrated on the design, visualization, and/or measurements of synthetic jets [ 7 10 ]. Several computational studies also have focused on fundamental aspects of these devices [ 11 14 ]. Crook and Wood [ 9 ] emphasize the importance of understanding the scaling and operational characteristics of a synthetic jet. Clearly, this information is required to design an appropriate device for a particular application. In addition, feedback control applications require the actuator transfer function that relates the input voltage to the output property of interest (e.g., volumetric flow rate, momentum, etc.) in the control system. The analysis and design of coupled-energy domain transducer systems are commonly performed using lumped element models [ 15 17 ]. The main assumption employed in lumped element modeling (LEM) is that the characteristic length scales of the governing physical phenomena are much larger than the largest geometric dimension. For example, in an acoustic system, the acoustic wavelength must be significantly larger than the device itself. If this assumption holds, then the temporal and spatial variations can be decoupled, and the governing partial differential equations for the distributed system can be Â“lumpedÂ” into a set of coupled ordinary differential equations [ 17 ]. This approach provides a simple method to estimate the low-frequency dynamic response of a system for design and control-system implementation. The purpose of this thesis is to rigorously study the application of LEM to piezoelectric-driven synthetic jet actuators. To the authorÂ’s knowledge, this represents the first application of LEM to piezoelectric-driven synthetic jets. McCormick [ 18 ] PAGE 14 3 employed LEM to a speaker-driven synthetic jet, which is an electromagnetic transducer, to relate the voltage input to the output volume velocity of the actuator. Rathnasingham and Breuer [ 19 ] were the first to develop a low-order model of a synthetic jet, using a control-volume model for the flow and an empirical model for the structural dynamics of the diaphragm. In this thesis, the various lumped elements for each component of a synthetic jet are theoretically developed. The resulting equivalent circuit is then analyzed to understand the effects of geometry and material properties on important design parameters, such as resonance frequency and volume displacement per applied voltage. The model assumptions and limitations are discussed, along with the results of experiments designed to assess the validity of this modeling approach. PAGE 15 4 CHAPTER 2 LUMPED ELEMENT MODELING In LEM, the coupling between the various energy domains is realized by using equivalent two-port models of the physical system. An equivalent circuit model is constructed by lumping the distributed energy storage and dissipation into ideal generalized one-port circuit elements. In an electroacoustic system, differential pressure and voltage are effort variables, while current and volumetric flow rate are flow variables. In this thesis, an impedance analogy is employed, in which elements that share a common effort are connected in parallel, while those sharing a common flow are connected in series. For a synthetic jet, three different energy domains are involved, electrical, mechanical, and fluidic/acoustic. The electromechanical actuator consists of an axisymmetric PZT patch bonded to a clamped metal diaphragm. The composite diaphragm is driven into motion via an applied ac voltage. The primary purpose of the piezoelectric composite diaphragm is to produce large volume displacements in order to force fluid into and out of the cavity, which represents a conversion from the mechanical to the acoustic/fluidic energy domain. Consequently, the frequency range of the analysis is limited from dc to somewhat beyond the fundamental frequency of the composite diaphragm but less than the natural frequency of any higher-order modes [ 20 ]. Linear composite plate theory is used to obtain the short-circuit pressure-deflection characteristics. Then, the diaphragm is lumped into an equivalent acoustic mass and acoustic compliance. The former represents PAGE 16 5 stored kinetic energy, and the latter models stored potential energy. Similarly, the electromechanical transduction characteristics are determined by the unloaded or Â“freeÂ” voltage-deflection characteristics. The piezoelectric electromechanical coupling is lumped into an effective acoustic piezoelectric coefficient. In general, the cavity contains a compressible gas that stores potential energy and is therefore modeled as an acoustic compliance. Viscous effects in the orifice dissipate a portion of the kinetic energy stored in the motion of the oscillating fluid mass. Therefore, there will be an effective acoustic mass and acoustic resistance associated with the orifice neck. Flow through the orifice produces losses associated with the discharge of flow from the jet exit. An acoustic radiation impedance must also be added if the fluid is ejected into a semi-infinite medium. The equivalent circuit representation for the synthetic jet is shown in Figure 2Â–1 . In the notation below, the first subscript denotes the domain (e.g., Â“ a Â” for acoustic and Â“ e Â” for electric), and the second subscript describes the element (e.g., Â“ D Â” for diaphragm). In the electrical domain, ebC is the blocked electrical capacitance of the piezoelectric diaphragm. The term blocked is used since it is the impedance seen by the source when the diaphragm motion is prevented. Although not shown here, a resistor can be introduced in series or in parallel with ebC to represent the dielectric loss in the piezoceramic (see Rossi [ 17 ], p. 358). PAGE 17 6 1:aP I MaD Q RaNMaNMaRadRaO QcQout Ceb CaD CaC I1 Vac electrical domain acoustic/fluidic domain RaDI-I1 Figure 2Â–1: Equivalent circuit representation of a piezoelectric-driven synthetic jet. In the acoustic domain, aDC and aD M are the short-circuit acoustic compliance and mass of the piezoceramic composite diaphragm, respectively, and aD R is the structural damping in the acoustic domain. Also, a radiation impedance must be included, if the back-side of the diaphragm is radiating into an open medium. aCC is the acoustic compliance of the cavity, while aN R and aN M are the acoustic resistance and mass of the fluid in the neck, respectively. Finally, aO R is the nonlinear resistance associated with the orifice discharge, and aRadM is the acoustic radiation mass of the orifice. In this thesis, it is assumed that the synthetic jet exhausts into a semi-infinite ambient air medium, and that the diaphragm is not subject to a mean differential pressure. If necessary, a vent channel can be used to equilibrate the mean static pressure across the diaphragm, in a manner similar to a microphone [ 21 ]. For simplicity, it is assumed that there is no grazing flow, and compressibility effects in the orifice are neglected. PAGE 18 7 The structure of the equivalent circuit is explained as follows. An ac voltage acV is applied across the piezoceramic to create an effective acoustic pressure that drives the diaphragm into motion. This represents a conversion from the electrical to the acoustic domain (by-passing the mechanical domain via integration to produce a volume displacement) and is accounted for via a transformer possessing a turns ratio a with units of [ P aV]. An ideal transformer converts energy from one domain to another without losses and obeys the relations 1and .aac a P IQV {1} In addition, a transformer converts an electrical impedance e Z to an acoustic impedance a Z via 22 1 acaa e aaaP P VZ Q Z IQ . {2} The motion of the diaphragm can either compress the fluid in the cavity or can eject/ingest fluid through the orifice. Physically, this is represented as a volume velocity divider, coutQQQ . The goal of the design is to maximize the magnitude of the volume flow rate through the orifice per applied voltage outacQV. PAGE 19 8 CHAPTER 3 EQUIVALENT CIRCUIT ANALYSIS Before estimating the lumped parameters defined above, it is instructive to analyze the equivalent circuit to obtain the frequency response function outacQsVs, where s j. Using Eq. {2} , the transformer can be eliminated by converting each of the acoustic impedances to their electrical equivalent. The result is depicted in Figure 3Â– 1 , where 2 2 211 (), 11 (), and ()eDaDaD aaD eC aaC aNaOaNaRad eO aZsRsM sC Zs sC RRsMM Zs !" #$ %& !" #$ %& '( )* {3} are the electric impedance of the diaphragm, cavity, and orifice, respectively. Ceb aQ ZeC I-aQa( Q-Qout)aQout ZeO ZeD I Vac Figure 3Â–1: Alternative equivalent circuit model. PAGE 20 9 Substituting in the expressions for , , and eDeCeO Z ZZ and grouping powers of s in the numerator and denominator results in 432 43211out aaD acQs Cs Vsasasasas , {4} where 1 2 3 4, , , and .aDaOaNaDaCaOaN aDaRadaNaDaCaRadaNaCaDaDaOaN aCaDaDaOaNaRadaNaD aCaDaDaRadaNaCRRRCRR aCMMMCMMCCRRR aCCMRRMMR aCCMMM '( )* {5} Although this expression is somewhat complicated, it reveals some important features without having to estimate any of the parameters in Eq. {5} . For the purpose of this discussion, these parameters can be thought of as constants, although in reality some are likely to exhibit frequency and amplitude dependence (i.e., due to nonlinear effects). For a dc voltage (0s), the volume velocity is zero. At low frequencies (0s), the volume velocity is proportional to aacdV since the transduction factor is defined as aaaDdC , where ad is an effective acoustic piezoelectric coefficient defined below Â— see Eq. {22} . This result emphasizes the need to optimize the design of the piezoceramic composite diaphragm [ 20 ] and also indicates that outQ and acV are 90 out of phase at low frequencies. At high frequencies ( s ), Eq. {4} becomes 3 outa acaCaDaDaNaRadQd VCCMMMs . {6} PAGE 21 10 The output therefore decreases at a rate of 60 dB/decade and is inversely proportional to the product of the masses and compliances in the system. The denominator in Eq. {4} is a 4th order polynomial in s , indicating two resonance frequencies. A compact analytical expression for the two resonance frequencies is desirable. The first and second resonance frequencies, 1 f and 2 f , are controlled by, but are not identical to, the short-circuit piezoelectric diaphragm natural frequency, 11 2D aDaDf M C , {7} and the Helmholtz resonator frequency, 11 2H aNaRadaCf M MC , {8} with the constraint that 12DH f fff . {9} The measure of the coupling of a system is a function of the mass or compliance ratio [ 15 ]. Assuming small damping, equating the denominator in Eq. {4} with a 4thorder system and using the definition given in Eq. {7} and Eq. {8} yields the following relationship between the natural frequencies: 2222 21 2 DH aNaRad HaDffff MM fM !" #$ %&, {10} where is defined as the ratio of the orifice masses to the diaphragm mass. Similarly, a compliance ratio can be defined as PAGE 22 11 2222 21 2 DH aD DaCffff C fC . {11} Given the compliance ratio, which is usually easier to estimate, one can obtain the following quadratic formula for 2 i f : 2222210DHDHffff '( )*. {12} The two roots of Eq. {9} are the square of the natural frequencies of the synthetic jet, 2 1 f and 2 2 f . In the next section, two important cases are examined to gain physical insight into the behavior of the device under different limiting conditions. 3.1 Case 1: Incompressible Limit 1 0aC aDC C Assuming that the fluid is an isentropic, ideal gas, the acoustic cavity compliance, is obtained from the cavity volume 0V , gas density 0 , and the isentropic speed of sound 0c via 0 2 00 aCV C c . {13} In practice, 0aCaDCC is achieved by minimizing the cavity volume or operating in a liquid medium. Since the coefficients 3a and 4a in Eq. {5} are both proportional to aCC , the synthetic jet transfer function reduces to the 2nd-order system, PAGE 23 12 2 2 1 221a out acd s Qs a Vs a ss aa , {14} where the prime denotes ia in the limit of 0aCaDCC . The denominator in Eq. {14} is written in the form of a canonical 2nd-order system, 222nnss . By inspection, the natural frequency and damping ratio for the incompressible case are given by 1 1incomp aNaRad aDaD aDMM CM M!" #$ %& {15} and 1 2aD incompaOaNaD aNaRadaDC RRR MMM . {16} If aDaNaRadMMM!, then the natural frequency of the synthetic jet actuator equals that of the diaphragm. At resonance, the response is proportional to the effective acoustic piezoelectric coefficient and is limited by the resistances of the circuit and the acoustic compliance of the diaphragm 1outa acaDaOaNaDQd VCRRR . {17} PAGE 24 13 3.2 Case 2: Rigid Diaphragm Limit 0aD aCC C As described in Prasad et al. [ 20 ], the size of the piezoceramic patch is not negligible compared to the metal diaphragm for high actuation performance. Therefore, the piezoceramic composite diaphragm cannot accurately be modeled as a homogeneous circular plate. Nonetheless, assuming that the diaphragm is clamped, the acoustic compliance of a homogeneous clamped circular plate provides insight into the scaling behavior of the diaphragm 62 31 16aDa C E h , {18} where a is the radius, E is the elastic modulus, is PoissonÂ’s ratio, and h is the thickness. From Eq. {18} , aDC decreases with decreasing the ratio ah and increasing elastic modulus. As in the previous case, the coefficients 3a and 4a in Eq. {5} are zero, and the synthetic jet transfer function reduces to a 2nd-order system. The limit 0aDaCCC leads to the following expressions for the natural frequency, damping ratio, and response at resonance: 1stiff aNaRadaC M MC , {19} 1 2aC stiffaOaN aRadaNC RR MM , {20} PAGE 25 14 and 1outa acaCaOaNQd VCRR . {21} In this case, the natural frequency of the jet corresponds to the resonant frequency of the Helmholtz resonator. At resonance, the response is limited by the orifice flow resistances and the cavity compliance. By comparing with Eq. {17} , the resonant response differs for these cases by the ratio of the acoustic compliances, and by the contribution of the acoustic resistance of the diaphragm in the incompressible case. PAGE 26 15 CHAPTER 4 MODEL PARAMETER ESTIMATION In this section, the methods and assumptions used to estimate each of the quantities in Eq. {5} are outlined. The interested reader is referred to Merhaut [ 16 ], Rossi [ 17 ], and Beranek [ 22 ] for details of the methodology. The details of the two-port model for the piezoceramic plate can be found in Prasad et al. [ 20 ]. The piezoelectric diaphragm vibrates in response to both an applied ac voltage and oscillatory differential pressure according to the relation aacaDQjdVCP, {22} where 0aac PdQjV is defined as the effective acoustic piezoelectric coefficient that relates the volume velocity Q of the diaphragm to the applied voltage acV, and 00acacaD VVCQjPVolumeP is the short-circuit acoustic compliance that relates an applied differential pressure to the volume displacement of the diaphragm. The vertical deflection wr due to an applied differential pressure is lumped into an equivalent acoustic mass aD M by equating the lumped kinetic energy of the vibrating diaphragm to the total kinetic energy using 2 2 01 2 22a aDr M Qwrrdr +" , {23} PAGE 27 16 where r is the distributed mass per unit area, Q is the net volume velocity of the diaphragm, a is the radius of the diaphragm, and wrjwr " is the distributed vertical velocity. All of these parameters are calculated via linear composite piezoelectric plate theory (see Prasad et al. [ 20 ] for details). The acoustic resistance aD R represents the losses due to damping effects in the diaphragm and is given by 2aD aD aDM R C, {24} where is the damping coefficient that is determined experimentally. The blocked electrical capacitance ebC in Figure 2Â–1 is related to the free electrical capacitance of the piezoceramic /efppCAh by 2 211a ebefef efasDd CCC CC!"#$ #$ %&, {25} where 2 is the electroacoustic coupling factor, is the dielectric constant, p A is the piezoceramic area, and p h is the thickness of the piezoceramic patch. However, ebC does not appear in Eq. {4} and is therefore not required for the present analysis. However, it should be noted that the piezoceramic dielectric properties play a major role in determining the electric power requirements of the actuator. The acoustic impedance of the cavity is given by [ 23 ] 00cotaCckD Z jS, {26} PAGE 28 17 where D is the depth, 0kc is the acoustic wavenumber, and S is the cross-sectional area of the cavity. Since 1kD#, the Maclaurin series is truncated after the first term to yield [ 23 ] 22 0000 01aC aCcc Z j SDjVjC . {27} At low frequencies, the acoustic resistance of the neck is obtained assuming fullydeveloped laminar pipe flow in the neck of length L and radius 0a (see Beranek [ 22 ] p. 135) 4 08out aN outP L R Qa , {28} where is the viscosity of the fluid and outQ is the volume flow rate produced by the differential pressure out P . Using the same assumption of fully-developed pipe flow, the acoustic mass in the neck is obtained by integrating the distributed kinetic energy and equating it to the lumped kinetic energy in the acoustic domain, 02 2 22 00 0 011 12 22a aNoutr L urdrMQ a '( !" ,#$ ,%& )*+, {29} where 0u is the centerline velocity which is related to the volume flow rate by 2 002outQua. The solution of Eq {29} yields the effective acoustic mass, 0 2 04 3aN L M a . {30} PAGE 29 18 The acoustic radiation mass aRadM can be modeled for 00.5 ka as a piston in an infinite baffle if the circular orifice is mounted in a plate that is much larger in extent than the orifice (see Beranek [ 22 ], p. 124) 0 2 08 3aRadM a . {31} The acoustic resistance associated with the discharge from the orifice can be approximated by modeling the orifice as a generalized Bernoulli flow meter [ 18 , 24 ], 00 224 0011 22DDout aO K uKQ R aa , {32} where u is the mean velocity, and 1DKO is a nondimensional loss coefficient that is a function of orifice geometry, Reynolds number, and frequency. Note that aO R is a function of the volume flow rate outQ through the orifice and thereby represents a nonlinear resistance, necessitating an iterative solution of Eq. {4} . At higher frequencies, the velocity profile in the orifice is modeled as flow in a circular duct driven by an oscillating pressure gradient. The solution is given in White [ 25 ] as 2 0 2 0 0 0,1jt outr Jj P urtje L a Jj ./ !" 00 #$ #$ 00 %& 12 !" 00 #$ 00 #$ %& 34 , {33} where 0 J is a Bessel function of zero order and is the kinematic viscosity. The velocity u is proportional to the pressure gradient and inversely proportional to 0 . PAGE 30 19 Furthermore, the velocity profile is characterized by the Stokes number 2 0Sta , as shown in Figure 4Â–1 . In the limit of 0St, the velocity profile asymptotes to Poiseuille flow. As St increases, the thickness of the Stokes layers decreases below 0a, leading to an inviscid core surrounded by a viscous annular region. Figure 4Â–1: Variation in velocity profile vs. St = 1, 10, and 30 for oscillatory channel flow in a circular duct. Figure 4Â–2 shows that the ratio of the average velocity to the centerline velocity, which is 0.5 for Poiseuille flow, is strongly dependant on the Stokes number. PAGE 31 20 St 0.11101001000 uavg/u0 0.5 0.6 0.7 0.8 0.9 1.0 Figure 4Â–2: Ratio of average velocity to centerline velocity vs. St for oscillatory channel flow in a circular duct. The acoustic impedance of the orifice under these assumptions is determined by directly integrating the velocity profile to obtain outQ as a function of out P . In this case, the real (resistive) and imaginary (reactive) parts of the acoustic impedance are functions of the Stokes number out aNaNaNaNaN outP Z RjXRjM Q . {34} The results shown in Figure 4Â–3 reveal that, at low frequencies, the acoustic resistance asymptotes to the steady value given in Eq. {28} and increases gradually with frequency. However, the acoustic mass is approximately constant with frequency. The PAGE 32 21 data in Figure 4Â–3 are used to provide frequency-dependent estimates for the acoustic resistance and mass in Eq. {4} . Figure 4Â–3: Acoustic resistance and reactance normalized by steady value vs. St for oscillatory channel flow in a circular duct. PAGE 33 22 CHAPTER 5 MODEL VERIFICATION AND PARAMETER EXTRACTION A modular piezoelectric-driven synthetic jet was constructed, as shown in Figure 5Â–1 , to perform a series of experiments to test the validity of the lumped element model parameters. The modular design permits a systematic variation of the cavity volume, orifice diameter and length, and piezoelectric diaphragm diameter and thickness. In addition, an access hole is provided for a microphone to monitor the fluctuating pressure inside the cavity. orifice plate diaphragm mount body plate top plate access hole for microphone clamp plate Figure 5Â–1: Assembly diagram of modular synthetic jet. PAGE 34 23 5.1 Piezoelectric Transduction The first experiment tested the linear composite plate theory that provides estimates for aDC, aD M , and ad. This was accomplished by measuring the velocity of the clamped vibrating diaphragm (excited by acV) using a scanning laser vibrometer (Polytec Model PSV-200) and integrating the velocity in the frequency domain to obtain displacement. The clamped circular diaphragm was removed from the synthetic jet apparatus and mounted on an optical table. The test was also performed in a vacuum chamber to eliminate fluid loading effects. Table 5-1 shows the characteristics of the device tested and Table 5-2 shows the parameters predicted by the lumped element model and defined in CHAPTER 4 . Table 5-1: Piezoceramic diaphragm characteristics Shim (Brass) Elastic Modulus P a 8.9631010 PoissonÂ’s Ratio 0.324 Density 3kgm 8700 Thickness mm 0.2012 Diameter mm 23 Piezoceramic (PZT-5A) Elastic Modulus P a 6.31010 PoissonÂ’s Ratio 0.31 Density 3kgm 7700 Thickness mm 0.2322 Diameter mm 20.0 Relative Dielectric Constant 1750 31d mV -1.7510-10 efC F 2.095 10-8 PAGE 35 24 Table 5-2: Calculated lumped element model parameters Lumped Element Model Parameters a P aV 139.3 aD M 4kgm 13538 aDC 24 s mkg 1.491110-13 ad 3mV)2.077110-11 0.117 ebC F 20.95 Short circuit natural frequency (Hz) 3542.4 Center deflection ( mV) 0.1146 Figure 5Â–2 shows a comparison between the predicted and measured mode shape of the piezoceramic diaphragm to a sinusoidal excitation voltage at f = 100 Hz for quasi-static case. The agreement between theory and experiment is excellent. The measured natural frequency of the diaphragm was 3505 Hz, while the computed shortcircuit resonance frequency from Eq. {7} was within 1%. Figure 5Â–2 includes random uncertainty estimates estimated from the measured response at 100 Hz, as outlined in Bendat and Piersol [ 26 ]. Key assumptions of the composite plate model, discussed in detail by Prasad et al. [ 20 ], include a linear response, negligible bond-layer thickness, full coverage of the piezoceramic surface by a metal electrode, and an ideal clamped boundary condition. Any violation of these assumptions will obviously degrade the accuracy of the model. PAGE 36 25 Figure 5Â–2: Comparison between predicted and measured response of piezoceramic diaphragms to a sinusoidal excitation voltage at 100 Hz. 5.2 Cavity Acoustic Compliance The value of the cavity acoustic compliance aCC is obtained from Eq. {13} . The cavity volume can be calculated from the geometry. To test the theory, the orifice was replaced with a solid cap to provide a closed cavity and all leaks were carefully minimized. The piezoceramic was then driven with a nominal 1 V amplitude sinusoid, and the displacement of the vibrating diaphragm was measured with a laser displacement sensor (Micro-Epsilon Model ILD2000-10). A 1/8 in. BrÃ¼el & KjÃ¦r (B&K) type 4138 PAGE 37 26 condenser microphone with B&K type 2669 preamplifier measured the fluctuating pressure in the closed cavity. The frequency response function between the pressure and displacement signal at the diaphragm center 0w was used to measure 0 P w at several frequencies and calculate aCC, as shown in Figure 5Â–3 . Using the average measured value of 9.89 M Pam and the measured mode shape, the cavity volume was determined to be 2.5310-6 3m 12%. This nominal value is less than 1% of the cavity volume calculated from the geometry. Figure 5Â–3: Measured acoustic compliance aCC vs. frequency in closed cavity of synthetic jet. 5.3 Acoustic Mass and Resistance in Orifice The flow in the neck of the orifice is modeled via Eq. {33} as a steady, fullydeveloped laminar flow driven by an oscillatory pressure gradient in a circular duct of radius 0a and length L . The data in Figure 4Â–3 are used to provide frequency dependent estimates for the acoustic resistance and mass in the lumped element model. Also, it has PAGE 38 27 been verified that this model is in good agreement with the semi-empirical model given by Ingard [ 27 ] for the impedance of a perforated plate. It should also be noted that, depending on the aspect ratio 0 L a of the orifice, the fully developed assumption may not be valid. Only for large values of 0 L a is the fullydeveloped assumption expected to be reasonable. For small values of 0 L a, the orifice dump loss given in Eq. {32} is expected to dominate. The models discussed in this section are simple and neglect potentially significant issues, such as nonlinear effects due to large amplitude pressure oscillations in the cavity [ 28 ] and transition to turbulent flow and compressibility effects in the orifice. Grazing flow effects, which are relevant when the synthetic jet interacts with a boundary layer, have also been ignored [ 29 ]. PAGE 39 28 CHAPTER 6 COMPARISON BETWEEN MODEL & EXPERIMENTS In this section, the lumped element model is used to predict the frequency response of two synthetic jet actuators. The dimensions and material properties of the piezoceramic diaphragms are summarized in Table 6-1 , and the pertinent lumped element parameters are provided in Table 6-2 . Table 6-3 lists the geometry of both synthetic jets. Table 6-1: Specifications of piezoceramic diaphragms. Shim (Brass) Case I Case II Elastic Modulus P a 8.9631010 PoissonÂ’s Ratio 0.324 Density 3kgm 8700 Thickness mm 0.20 0.10 Diameter mm 23.5 37 Piezoceramic (PZT-5A) Elastic Modulus P a 6.31010 PoissonÂ’s Ratio 0.31 Density 3kgm 7700 Thickness mm 0.11 0.10 Diameter mm 20.0 25.0 Relative Dielectric Constant 1750 31d mV -1.7510-10 efC F 4.42 10-87.45310-8 PAGE 40 29 Table 6-2: Calculated lumped parameters of piezoceramic diaphragms. Quantity Case I Case II aDC 24 s mkg6.5310-132.9310-11 aD M 4kgm 8.15103 2.25103 a P aV 88.6 16.06 Table 6-3: Geometry of synthetic jet actuators. Cavity: Case I Case II Volume 0V 3m 2.5010-6 5.5010-6 Orifice: Radius 0a mm 0.825 0.42 Length L mm 1.65 0.84 6.1 Laser Doppler Velocimetry System A Dantec, fiber-optic, two-component, Laser Doppler Velocimetry (LDV) system was used to measure the magnitude of the peak centerline velocity produced by a synthetic jet for varying frequency. Theatrical fog fluid was used to produce particles with a specified mean diameter of 0.35 m. Figure 6Â–1 shows a schematic of the synthetic jet device and the relative size of the probe volume. A 400 mm focal length, 60 mm diameter lens was used to create a probe volume size of 0.194 mm x 0.194 mm x 4.095 mm with a beam half-angle of 2.718. Because of the beam half-angle and the thickness of the synthetic jet device, the minimum distance from the orifice that velocity data could be acquired was 0.3 mm. PAGE 41 30 Top View Side View 2a0 = 1.65 mm 12.7 mm L = 1.65 mm beams measuring the vertical velocity component beams measuring the horizontal velocity component 2.718Â° orifice cavity orifice 3.81 mm 2.718Â° Figure 6Â–1: Laser Doppler Velocimeter system setup for Case I. 6.2 Data Acquisition Velocity data were acquired phase-locked with the input actuation sinusoidal signal. As shown in Figure 6Â–1 , the probe volume length extended well beyond the orifice. Therefore, the data contained many points with near zero velocity. Fortunately, these data do not affect the envelope of the phase-averaged velocity over one cycle. The maximum value was extracted from the phase-averaged velocity measurements over a range of input frequencies with 25 acVV . To verify repeatability of the data, several frequencies were chosen and velocity data was acquired over a period of several days for Case II. Table 6-4 summarizes the repeatability of the velocity measurements for several PAGE 42 31 frequencies. While the uncertainty exceeds 10% near resonance, the data are sufficiently accurate to test the validity of the lumped element model. Table 6-4: Uncertainty in maximum velocity at select frequencies for Case II. Frequency H z Uncertainty in maxUf ms 200 0.3 500 0.7 800 8.5 6.3 Results As shown in Table 6-3 , Case I uses a smaller piezoceramic diaphragm and cavity volume and a larger orifice than Case II. However, both devices use an orifice with the same aspect ratio 02La and are driven with acV = 25 V amplitude sinusoids. Figure 6Â–2 and Figure 6Â–3 show the comparison between the experiment and the model developed above. The lumped element model uses Eq. {33} and Figure 4Â–3 to model the frequency dependence of the velocity profile to determine the relationship between maxu and outQ, which is obtained via Figure 4Â–2 . The agreement between the model and experiment in Figure 6Â–2 is satisfactory. The main difference between the two models proposed occurs at the maximum velocity peaks where the resistance terms dominate the system response. The model accurately predicts the two resonance frequencies using Eq. {12} . PAGE 43 32 Figure 6Â–2: Comparison between the lumped element model and experiment for Case I. Case II in Figure 6Â–3 corresponds to a case with a single dominant peak that provides jet velocities in excess of 60 ms. The excitation voltage amplitude of 25 V corresponds to approximately 35% of the coercive or breakdown electric field strength (~1.18 Vm ) of the piezoceramic. The lumped element model accurately predicts the resonance frequency and maximum velocity and also possesses the proper shape of the frequency response function. It is important to note that two resonance frequencies are predicted by the lumped element model, one near 350 Hz and the other near 900 Hz. However, the lower peak is damped in Figure 6Â–3 due to the frequency dependent nonlinear orifice resistance term aOR. An inspection of Eq. {4} and {5} reveals that PAGE 44 33 this term has a larger effect at lower frequencies. Gallas et al. [ 30 ] mistakenly argued that this case corresponds to the situation when a single peak is obtained in the frequency response function 12DH f fff . Table 6-5 summaries the comparison between experiments and the analytical analysis developed above. Figure 6Â–3: Comparison between the lumped element model and experiment for Case II. Finally, the peak in the experimental results for Case II, which occurs around 1200 Hz, corresponds to the second resonance frequency of the piezoelectric diaphragm (deducted from experimental tests). The lumped element model is not valid at frequencies corresponding to higher-order vibration modes. As a result, the peak near 1200 Hz is not captured by the model. PAGE 45 34 Table 6-5: Comparison between lumped element model and experiment. Quantity Source Case I Case II D f H zEq. {7} 2180 620 H f H zEq. {8} 940 470 Eq. {11} 0.036 0.735 1 f H z Eq. {12} 920 330 2 f H z Eq. {12} 2230 882 1 f H z experiment 970 N/A 2 f H z experiment 2120 850 PAGE 46 35 CHAPTER 7 CONCLUSIONS AND FUTURE WORK A lumped element model of a piezoelectric-driven synthetic jet actuator has been developed and compared with experiment. LEM provides a compact analytical model and valuable physical insight into the dependence of the device behavior on geometry and material properties. The model reveals that a synthetic jet is a 4th-order coupled oscillator. One oscillator is a Helmholtz resonator, and the second is the piezoelectric diaphragm. Simple arguments reveal two important special cases corresponding to single oscillators. One case occurs when the fluid is incompressible, while the second case is similar to that of a rigid piston and occurs when the acoustic compliance of the piezoelectric diaphragm is small compared to that of the cavity. In this case, the synthetic jet acts like a driven Helmholtz resonator. For any case with low damping, a simple formula was obtained to estimate the two natural frequencies of the synthetic jet as a function of the Helmholtz and diaphragm natural frequencies and the compliance ratio . 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 results indicate that the linear composite plate theory is accurate when the model assumptions are realized. Similarly, the cavity acoustic compliance model was validated. The details of the flow in the orifice require further careful study. It is this region that dictates the acoustic mass and resistance in the neck. PAGE 47 36 Accurate knowledge of the acoustic mass is required to determine the Helmholtz frequency of the synthetic jet, while the resistance limits the achievable velocities near resonance. The model was applied to two prototypical synthetic jets and found to provide very good agreement with the measured performance. The results reveal the power and shortcomings of the model in its present form. The flow in the vicinity of the orifice must be studied further in order to obtain better quantitative estimates of the losses and velocity profile characteristics. Furthermore, the loss coefficient D K in Eq. {32} is treated as a constant but is probably a function of the orifice Reynolds number, Stokes number, and geometry. Furthermore, grazing flow effects remain to be studied in a rigorous fashion. In future work, additional parameters will be varied in the model and accompanying experiments to yield optimal design rules for the synthetic jet. Future testing of the piezoelectric diaphragm will assess the severity of nonlinear effects when the excitation amplitude is increased. For the orifice, emphasis will be placed on the ratio of the orifice length to the hole radius, 0 L a. This variable will be systematically varied in concert with the other important nondimensional parameters, such as the orifice Reynolds and the Stokes numbers. Additional velocity measurements with improved spatial resolution will also be performed to map out the spatial variations in the synthetic jet velocity field. Finally, an optimization of the overall device efficiency will be investigated. PAGE 48 37 REFERENCES 1. Smith, B. L. and Glezer, A., Â“The Formation and Evolution of Synthetic Jets,Â” Physics of Fluids, Vol. 10, No. 9, pp. 2281-2297, 1998. 2. Amitay, M., Smith, B. L., and Glezer, A., Â“Aerodynamic Flow Control Using Synthetic Jet Technology,Â” AIAA Paper 98-0208, Jan. 1998. 3. Smith, D. R., Amitay, M., Kibens, V., Parekh, D. E., and Glezer, A., Â“Modification of Lifting Body Aerodynamics Using Synthetic Jet Actuators,Â” AIAA Paper 98-0209, Jan. 1998. 4. Chen, Y., Liang, S., Aung, K., Glezer, A., Jagoda, J., Â“Enhanced Mixing in a Simulated Combustor Using Synthetic Jet Actuators,Â” AIAA Paper 99-0449, Jan. 1999. 5. Honohan, A. M., Amitay, M., and Glezer, A., Â“Aerodynamic Control Using Synthetic Jets,Â” AIAA Paper 2000-2401, June 2000. 6. Chatlynne, E., Rumigny, N., Amitay, M., and Glezer, A., Â“Virtual Aero-Shaping of a Clark-Y Airfoil Using Synthetic Jet Actuators,Â” AIAA Paper 2001-0732, Jan. 2001. 7. Crook, A., Sadri, A. M., and Wood, N. J., Â“The Development and Implementation of Synthetic Jets for the Control of Separated Flow,Â” AIAA Paper 99-3176, July 1999. 8. Chen, F.-J., Yao, C., Beeler, G. B., Bryant, R. G., and Fox, R. L., Â“Development of Synthetic Jet Actuators for Active Flow Control at NASA Langley,Â” AIAA Paper 2000-2405, June 2000. 9. Crook, A. and Wood, N. J., Â“Measurements and Visualizations of Synthetic Jets,Â” AIAA Paper 2001-0145, Jan. 2001. 10. Gilarranz, J. L. and Rediniotis, O. K., Â“Compact, High-Power Synthetic Jet Actuators for Flow Separation Control,Â” AIAA Paper 2001-0737, Jan. 2001. 11. Kral, L. D., Donovan, J. F., Cain, A. B., and Cary, A. W., Â“Numerical Simulation of Synthetic Jet Actuators,Â” AIAA Paper 97-1824, June 1997. PAGE 49 38 12. Rizzetta, D. P., Visbal, M. R., and Stanek, M. J., Â“Numerical Investigation of Synthetic Jet Flowfields,Â” AIAA 98-2910, June 1998. 13. Mallinson, S. G., Reizes, J. A., Hong, G., and Haga, H., Â“The Operation and Application of Synthetic Jet Actuators,Â” AIAA Paper 2000-2402, June 2000. 14. Utturkar, Y., Mittal, R., Rampunggoon, P., and Cattafesta, L., Â“Sensitivity of Synthetic Jets to the Design of the Jet Cavity,Â” AIAA Paper 2002-0124, Jan. 2002. 15. Fischer, F. A., Fundamentals of Electroacoustics, Interscience Publishers, Inc., New York, NY, 1955, Chapter III and XI. 16. Merhaut, J., Theory of Electroacoustics, McGraw-Hill, Inc., New York, NY, 1981, Chapter 6. 17. Rossi, M., Acoustics and Electroacoustics, Artech House, Norwood, MA, pp. 245-373, 1988. 18. McCormick D. C, Â“Boundary Layer Separation Control with Directed Synthetic Jets,Â” AIAA Paper 2000-0519, Jan. 2000. 19. Rathnasingham, R. and Breuer, K. S., Â“Coupled Fluid-Structural Characteristics of Actuators for Flow Control,Â” AIAA Journal, Vol. 35, No. 5, pp. 832-837, May 1997. 20. Prasad, S., Horowitz, S., Gallas, Q., Sankar, B., Cattafesta, L., and Sheplak, M., Â“Two-Port Electroacoustic Model of an Axisymmetric Piezoelectric Composite Plate,Â” AIAA Paper 2002-1365, April 2002. 21. Sheplak, M., Schmidt, M.A., and Breuer, K.S., Â“A Wafer-Bonded, Silicon Nitride Membrane Microphone with Dielectrically-Isolated, Single-Crystal Silicon Piezoresistors,Â” Technical Digest, Solid-State Sensor and Actuator Workshop, pp. 23-26, Hilton Head, SC, June 1998. 22. Beranek, L. L. Acoustics, Acoustic Society of America, Woodbury, NY, pp. 4777, 116-143, 1993. 23. Blackstock, D. T., Fundamentals of Physical Acoustics, John Wiley & Sons, Inc., New York, NY, p. 145, 2000. 24. White, F. M., Fluid Mechanics, McGraw-Hill, Inc., New York, NY, pp. 377-379, 1979. 25. White, F. M., Viscous Flow, McGraw-Hill, Inc., New York, NY, pp. 143-148, 1974. PAGE 50 39 26. Bendat, J. S., and Piersol, A. G., Random Data: Analysis and Measurements Procedures, 3rd Edition, John Wiley & Sons, Inc., New York, NY, p. 341, 2000. 27. Ingard, K. U., Notes on Sound Absorption Technology, Poughkeepsie, NY: Noise Control Foundation, 1994. 28. Ingard, U., Â“Acoustic Nonlinearity of an Orifice,Â” Journal of Acoustic Society of America, Vol. 42, No. 1, pp. 6-17, 1967. 29. Mittal, R., Rampuggoon, P., Udaykumar, H. S., Â“Interaction of a Synthetic Jet with a Flat Plate Boundary Layer,Â” AIAA Paper 2001-2773, June 2001. 30. Gallas, Q., Mathew, J., Kaysap, A., Holman, R., Carroll, B., Nishida, T., Sheplak, M., and Cattafesta, L., Â“Lumped Element Modeling of Piezoelectric-Driven Synthetic Jet Actuators,Â” AIAA Paper 2002-0125, January 2002. PAGE 51 40 APPENDIX A COMPOSITE PIEZOELECTRIC DIAPHRAGM CHARACTERIZATION The limitations of the assumptions discussed in Section 5.1 are outlined. Three different piezoelectric diaphragms have been tested. Case I and Case II correspond to the devices used in CHAPTER 6 , while Case III corres ponds to the device tested in CHAPTER 5 . Their specifications are summarized below in Table A-1 . The set up described in Section 5.1 has been used. Table A-1: Piezoelectric diaphragms properties. Shim (Brass) Case I Case II Case III Elastic Modulus P a 8.9631010 PoissonÂ’s Ratio 0.324 Density 3kgm 8700 Thickness mm 0.20 0.100 0.201 Diameter mm 23.5 37 23 Piezoceramic (PZT-5A) Elastic Modulus P a 6.31010 PoissonÂ’s Ratio 0.31 Density 3kgm 7700 Thickness mm 0.11 0.102 0.232 Diameter mm 20.0 25.0 20.0 Relative Dielectric Constant 1750 31d mV -1.7510-10 efC F 4.42 10-87.45310-8 2.095 10-8 PAGE 52 41 Figure AÂ–1 shows the plot of frequency response function magnitude, phase and coherence of 2 sample piezoceramic diaphragms Case I, while Figure AÂ–2 , Figure AÂ–3 and Figure AÂ–4 , respectively, show the plot of frequency response function magnitude, phase and coherence of piezoceramic diaphragm Case I, piezoceramic diaphragm Case II and piezoceramic diaphragm Case III. It can be observed that each sample piezoeramic diaphragm is unique in the sense that their natural frequencies occur for different values. Also, the type of boundary condition used is important since it may change the magnitude of the diaphragm deflection and the natural frequency. The values of the natural frequency and dc response for each case are tabulated in Table A-2 . Table A-2: Experimental values of natural frequency and dc response of tested piezoelectric diaphragms Natural frequency ( Hz ) dc response ( mV) Case I Sample 1 ( Figure AÂ–1 , Figure AÂ–8 ) 2003.9 0.418 Sample 2 ( Figure AÂ–1 , Figure AÂ–8 ) 1996.1 0.446 Clamped BC ( Figure AÂ–2 , Figure AÂ–5 ) 1996.1 0.446 Compliant BC ( Figure AÂ–2 ) 2445.3 0.268 Case II Clamped BC ( Figure AÂ–3 , Figure AÂ–6 ) 762.5 0.718 Compliant BC ( Figure AÂ–3 ) 770.0 0.664 Sample 1 ( Figure AÂ–9 ) 3381.2 0.131 Sample 2 ( Figure AÂ–9 ) 3387.5 0.133 Clamped BC ( Figure AÂ–4 , Figure AÂ–7 ) 3381.2 0.131 Case III Compliant BC ( Figure AÂ–4 ) 3400.0 0.112 PAGE 53 42 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E 0 4 Magnitude (m/V) Sample 1 Sample 2 -200 -150 -100 -50 0 50 100 150 200 020004000600080001000012000Phase (degrees) 0 0.2 0.4 0.6 0.8 1 020004000600080001000012000Frequency (Hz)Coherence Figure AÂ–1: Frequency response (magnitude, phase and coherence) of the center of two sample piezoceramic diaphragms Case I. (Amplitude input voltage=1V) PAGE 54 43 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 020004000600080001000012000Magnitude (m/V) Clamped BC Compliant BC -200 -150 -100 -50 0 50 100 150 200 020004000600080001000012000Phase (degrees) 0 0.2 0.4 0.6 0.8 1 020004000600080001000012000Frequency (Hz)Coherence Figure AÂ–2: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm Case I for different boundary conditions. (Amplitude input voltage=1V) PAGE 55 44 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 0500100015002000Magnitude (m/V) Clamped BC Compliant BC -200 -150 -100 -50 0 50 100 150 200 0500100015002000Phase (degrees) 0 0.2 0.4 0.6 0.8 1 0500100015002000Frequency (Hz)Coherence Figure AÂ–3: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm Case II for different boundary conditions. (Amplitude input voltage=1V) PAGE 56 45 1.E-08 1.E-07 1.E-06 1.E-05 010002000300040005000600070008000900010000 M agn it u d e ( m /V) Clamped BC Compliant BC -200 -150 -100 -50 0 50 100 150 200 010002000300040005000600070008000900010000Phase (degrees) 0 0.2 0.4 0.6 0.8 1 010002000300040005000600070008000900010000Frequency (Hz)Coherence Figure AÂ–4: Frequency response (magnitude, phase and coherence) of the center of piezoceramic diaphragm Case III for different boundary conditions. (Amplitude input voltage=1V) PAGE 57 46 Figure AÂ–5 , Figure AÂ–6 and Figure AÂ–7 respectively show the effectiveness of the clamp for piezoceramic diaphragms Â– Case I, Case II and Case III. The magnitude of the center deflection of the piezoceramic diaphragm and of a point very near the clamp edge are plotted, along with the noise floor of the laser vibrometer. It can be observed that a perfect compliant boundary condition is not easy to achieve. 1.00E-11 1.00E-10 1.00E-09 1.00E-08 1.00E-07 1.00E-06 1.00E-05 1.00E-04 0100020003000400050006000 Frequency (Hz)Magnitude (m/V) Center Deflection Noise Floor Clamp Deflection Figure AÂ–5: Frequency response of piezoceramic diaphragm Case I. (Amplitude input voltage=1V) PAGE 58 47 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 0500100015002000 Frequency (Hz)Magnitude (m/V) Center Deflection Noise Floor Clamp Delfection Figure AÂ–6: Frequency response of piezoceramic diaphragm Case II. (Amplitude input voltage=1V) 1.E-12 1.E-11 1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 010002000300040005000 Frequency (Hz)Magnitude (m/V) Center Deflection Noise Floor Clamp Deflection Figure AÂ–7: Frequency response of piezoceramic diaphragm Case III. (Amplitude input voltage=1V) PAGE 59 48 Figure AÂ–8 and Figure AÂ–9 show the mode shape of piezoceramic diaphragms Â– Case I and Case III respectively, driven by a sinusoidal signal with amp litude of 1V. Figure AÂ–8 enhances the boundary condition problem at low frequency, while Figure AÂ– 9 shows the effect of asymmetry of the ceramic layer on top of the brass diaphragm. 0.0E+00 5.0E-08 1.0E-07 1.5E-07 2.0E-07 2.5E-07 3.0E-07 3.5E-07 4.0E-07 4.5E-07 -1-0.500.51normalized radiusMagnitude (m/V) Clamped BC Compliant BC Figure AÂ–8: Comparison between two boundary condition types applied to piezoceramic diaphragm Case I driven by a sinusoidal excitation voltage of 1V at 300 Hz. PAGE 60 49 0.0E+00 2.0E-08 4.0E-08 6.0E-08 8.0E-08 1.0E-07 1.2E-07 1.4E-07 -1-0.500.51 normalized radiusMagnitude (m/V) Sample 1 Sample 2 Figure AÂ–9: Comparison between the response of two sample piezoceramic diaphragms Case III driven by a sinusoidal excitation voltage of 1V at 500 Hz. The piezoceramic of sample 1 is not centered on the brass diaphragm. It can be observed that it is difficult to achieve a perfect boundary condition, and that each device responds differently from the others. The frequencies at which the natural frequency peaks occur and the deflection magnitude at DC value are affected. The model described in Section 5.1 needs therefore to be consciously and carefully used. PAGE 61 50 APPENDIX B SYNTHETIC JET ACTUATORS MEASURMENTS As shown in Figure 5Â–1 , a modular synthetic jet has been built in such a manner that one can vary different fundamental aspects of the device, as the cavity volume and the aspect ratio of the orifice. The available geometries are summarized below in Table B-1 . Note that Case I and Case II are those previously discussed in CHAPTER 6 . Table B-1: Geometry of modular synthetic jet actuators. Cavity: V0 V1 (Case I) V2 (Case II) V3 Volume 0V 3m 4.4510-6 2.5010-6 5.0010-6 7.1110-6 Orifice: 1 (Case II)2 (Case I)3 4 5 6 Radius0a mm 0.84/2 1.65/2 3/2 1/2 1/2 3/2 Length L mm 0.84 1.65 1 3 5 5 The magnitude of the peak centerline velocity produced by a synthetic jet for varying frequency was measured by the LDV system discussed in Section 6.1 . For the results below, a 120 mm focal length, 60 mm diameter lens was used to create a probe volume size of 0.059 mm x 0.058 mm x 0.372 mm with a beam half-angle of 8.997. The probe volume is therefore much smaller than the one used previously in CHAPTER 6 and is better able to capture the peak centerline velocity of the synthesized jet. Because PAGE 62 51 of the much larger beam half-angle and the thickness of the synthetic jet device, the velocity data were acquired at least one and an half diameter above the orifice, while the theoretical model predicts the mean velocity at the exit of the orifice. Below in Figure B1 is a schematic of the LDV set up to measure velocity field of Case II having cavity volume V3. Side view of the probe volume over the orifice for Case II with cavity volume V3 2a0 = 0.84 mm 14.2 mm L = 1.65 mm Beams measuring the vertical velocity component Cavity Orifice 4.6mm Distance from probe volume to orifice surface is ~1.5 orifice diameters LDV Lens Characteristics Focal Length = 120.0mm 8.997Â° Figure B-1: Laser Doppler Velocimeter system setup for Case II having cavity volume V3. Figure B-2 and Figure B-3 are comparisons between model predictions to look at the effect on the sole change in cavity volume in the overall response of the device. They use cavity volume data provided in Table B-1 . PAGE 63 52 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 30 Frequency (Hz) Maximum Velocity (m/s) for an applied voltage (50Vpp) V0 V1 V0 > V1 0 500 1000 1500 2000 2500 3000 0 5 10 15 20 25 30 Frequency (Hz) Maximum Velocity (m/s) for an applied voltage (50Vpp) V0 V1 V0 > V1 Figure B-2: Cavity volume change in Case I (model prediction) PAGE 64 53 0 500 1000 1500 0 10 20 30 40 50 60 70 Frequency (Hz) Maximum Velocity (m/s) for an applied voltage (50 Vpp) V2 < V3 V3 V2 Figure B-3: Cavity volume change in Case II (model prediction) Below are presented frequency response of synthetic jet actuators for various orifice configurations. Velocity data were acquired phase-locked with the input actuation sinusoidal signal, and the maximum velocity value was extracted for each frequency. All configurations are driven by a 25 Vac amplitude sinusoidal signal. Figure B-4 , Figure B5 , Figure B-6 , Figure B-7 , Figure B-8 and Figure B-9 compare experimental data with the model prediction. The data provided in Table B-1 have been used. All device tested used the same diaphragm (piezoceramic diaphragm Â– CaseII) and only the orifice cap was removed and changed from the apparatus. Thus, a value of the damping coefficient was chosen to match the first experiment and was kept the same for all cases. PAGE 65 54 0 500 1000 1500 0 10 20 30 40 50 60 70 Frequency (Hz) Maximum Velocity (m/s) Figure B-4: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 1. (5 5 experiment, model) 0.028 02 La PAGE 66 55 0 500 1000 1500 0 5 10 15 20 25 30 35 40 45 50 Frequency (Hz) Maximum Velocity (m/s) Figure B-5: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 2. (5 5 experiment, model) 0.028 02 La PAGE 67 56 0 500 1000 1500 0 5 10 15 20 25 30 35 Frequency (Hz) Maximum Velocity (m/s) Figure B-6: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 3. (5 5 experiment, model) 0.028 023 La PAGE 68 57 0 500 1000 1500 0 10 20 30 40 50 60 70 Frequency (Hz) Maximum Velocity (m/s) Figure B-7: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 4. (5 5 experiment, model) 0.028 06 La PAGE 69 58 0 500 1000 1500 0 10 20 30 40 50 60 70 Frequency (Hz) Maximum Velocity (m/s) Figure B-8: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 5. (5 5 experiment, model) 0.028 010 La PAGE 70 59 0 500 1000 1500 0 5 10 15 20 25 30 35 Frequency (Hz) Maximum Velocity (m/s) Figure B-9: Comparison between the lumped element model and experiment for synthetic jet having diaphragm Â– Case II, cavity volume V3, and orifice 6. (5 5 experiment, model) It can be observed that, although the velocity magnitude does not always match with the experiments, the expected global shape of the frequency response does match. One possibility to explain this discrepancy comes from that the experimental results are obtained with a probe volume far from the orifice while the model predicts the velocity magnitude right at the orifice. And in this set up the probe volume is much smaller than the one used in Section 6.1 , therefore mostly capturing velocity points with finite velocity while in Section 6.1 the data contained many points having near zero velocity. Another cause of discrepancy between experimental data and model prediction may come from the orifice model used, which is accurate for fully-developed orifice flow. Clearly, for 0.028 0103 La PAGE 71 60 some of the orifice used (as orifice 3 and orifice 6), the flow does not have the time to become fully-developed. Better model prediction of the flow inside the orifice is therefore needed. Response of a synthetic jet actuator function of the input voltage at a single frequency is shown in Figure B-10 . The synthetic jet used has diaphragm Case II, cavity volume V3, and orifice 1. It is driven by a sinusoidal signal at 400 Hz, and phaselocked to the input signal. The working frequency of 400 Hz has been chosen since the device is expected to respond without much fluctuation around this frequency. 0 10 20 30 40 50 60 70 020406080100120140160180200220240260VppMean velocity (m/s) Figure B-10: Mean jet velocity response function of input voltage for Case II with cavity volume V3 at 400 f Hz It should be noted that while before experiments were conducted for a single input voltage of 50 Volt peak-to-peak, the above plot shows that at 400 Hz one can expect a velocity of almost 60 m/s at the maximum voltage tolerated by the ceramic. The last four points on the plot indicate that the piezoceramic might have been depolarized. PAGE 72 61 APPENDIX C AMPLIFIER CHARACTERIZATION A PCB amplifier, model 790A06, was used to drive the synthetic jet actuators. A function generator (HP 33120A) supplied the input voltage signal, and the output voltage signal was measured via an oscilloscope. The gain of the amplifier was set to 30 and the plots below show the severity of the non-linearity of the amplifier gain, first for a specific frequency ( F i g ure C ), then function of the frequency ( F i g ure Câ€“ 2 ). 0 50 100 150 200 250 300 012345678Amplifier input voltage (Vpp)Ampli f ier output voltage ( Vpp ) exp. 2 (400 Hz) exp. 1 (600 Hz) f(x)=30x F i g ure C: Amplifier g a in linearity at constant frequenc y . PAGE 73 62 In F i g ure C , e x p. 1 and e x p. 2 correspond to different devices (two sample piezoelectric diaphragms case II) driven at two different frequencies. The last four points of exp. 2 occurred when the piezoceramic had been depolarized. 27 27.5 28 28.5 29 29.5 30 30.5 31 0500100015002000 Frequency (Hz)Gain (Vout / Vin) F i g ure Câ€“ 2 : Amplifier g a in function of frequenc y . It can be observed that this amplifier has a constant gain when used at a single frequency, but the frequency response of its gain presents some fluctuations. PAGE 74 63 BIOGRAPHICAL SKETCH Quentin Gallas was born on September 28th, 1977, in Orange, located in the south of France. He graduated from lÂ’Ecole des Pupilles de lÂ’Air, Grenoble, France, in 1995, specialized in Sciences (major in mathematics and physics). He entered the UniversitÃ© de Versailles--St Quentin-en-Yvelines in Versailles, and earned his undergraduate degree in mathematics, informatics and science applications in June 1998. He moved to Lyon and earned in fall 2001 the degree of Engineering from the Institute of Sciences and Techniques of Lyon, majoring in mechanics. While finishing his third year of mechanical engineering studies in Lyon, he entered the University of Florida in spring 2001 with a graduate research assistantship and is currently pursuing his master's degree in aerospace engineering. He intends to continue in the field of fluid dynamics and experimentation for his doctoral degree. |