An Analytical Model for Synthetic Jet Actuation

Rajnish N. Sharma*

The University of Auckland, Auckland, NEW ZEALAND

An analytical model for synthetic jet actuation based on the laws of fluid dynamics is presented. A synthetic jet actuator consists of a cavity with a driven wall and an orifice. Under actuation, the wall is oscillated resulting in an oscillatory flow through the orifice. In the model, the driven wall is modeled as a single degree of freedom mechanical system, which is pneumatically coupled to the cavity-orifice arrangement acting as a Helmholtz resonator. The latter has been modeled using the unsteady form of the continuity and Bernoulli equations. The model has been validated against experimental data available in the published literature, and very good agreement is obtained between the predicted and measured frequency responses as well as the phase relationships between velocities and pressures. The model and analysis based on it provides valuable insights into the behavior of synthetic jet actuators.

Nomenclature Ao = orifice area Aw = wall area Cd = discharge coefficient CI = inertia coefficient CL = sudden flow contraction loss coefficient CLeff = effective loss coefficient cw = diaphragm damping coefficient ca = added diaphragm damping coefficient cwt = cw + ca

Da = effective acoustic piezo-electric coefficient = ∆V / va when uncoupled from actuator cavity do = orifice diameter dw = diaphragm / wall diameter f = frequency fh = Helmholtz resonance frequency fw = diaphragm / wall natural frequency F = force Fa = force amplitude kh = stiffness of orifice air against cavity kw = diaphragm / wall stiffness kwp = = pneumatic stiffness of diaphragm against sealed actuator cavity oow VPA /2γL = length Lc = characteristic length of the actuator lo = orifice length (thickness) le = effective length of air jet / slug mh = = mass of air jet at the orifice eoa lAρmw = diaphragm / wall mass ma = added diaphragm / wall mass mwt = mw + man = polytropic exponent pi = gauge internal pressure * Senior Lecturer, Department of Mechanical Engineering, The University of Auckland, Private Bag 92019, Auckland, NEW ZEALAND; Member AIAA.

3rd AIAA Flow Control Conference5 - 8 June 2006, San Francisco, California

AIAA 2006-3035

Copyright © 2006 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

pio = steady gauge internal pressure Pi = pi + PoPo = ambient pressure U = orifice flow velocity Uo = steady orifice jet velocity US = speed of sound Uvc = velocity at the region of vena contracta near the orifice Uw = wall velocity V = instantaneous cavity volume Vo = nominal cavity volume ∆V = volume displaced by the diaphragm v = input voltage va = amplitude of input voltage xw = wall displacement φPi-U = phase angle between actuator internal pressure and orifice flow velocity φVw-U = phase angle between diaphragm and orifice flow velocities ε = Uvc / U = velocity ratio γ = specific heat ratio λ = wavelength ρa = ambient air density ρi = internal air density

,iρ = internal air density perturbation

ζw = diaphragm / wall damping ratio ω = 2πf ωh = 2πfh

ωw = 2πfw

ωwp = wtwp mk /

I. Introduction

A ‘synthetic jet’ is a quasi-steady jet of air generated a few diameters from the orifice of a ‘synthetic jet actuator’, as illustrated in Figure

1. Actuation involves the forcing of an oscillatory air-flow through the orifice from a cavity with a driven flexible wall.1 During out-flow, an air jet is formed directly in front of the orifice, accompanied by a vortex ring around the orifice which is shed as out-flow weakens and reverts to in-flow. Since the in-flow part of the oscillatory cycle entrains air from the periphery rather than from in front of the orifice, the flow-field established directly downstream of the orifice is relatively undisturbed leading to the maintenance of a quasi-steady or synthetic jet. The train of vortex rings that are periodically shed are advected away from the orifice and eventually break down in the region of this synthetic jet, thus intermittently transporting momentum to and thereby sustaining the jet.2 A major advantage of the synthetic jet over the ordinary steady jet is that it does not require a continuous supply of fluid from elsewhere for the maintenance of the jet, since the air jet is synthesised from the surrounding fluid. Consequently, the actuator has advantages of simple structure, low cost and easy installation. The synthetic jet has thus attracted much research within the last decade and a comprehensive review will be found in Glezer and Amitay.3 Past investigations into applications of synthetic jets range from thrust vectoring of jet engines 4, through mixing enhancement 5, to external flow boundary layer separation control.6 A number of studies (for example Smith 7), have focussed upon the interaction of synthetic jets with boundary layer cross-flows. Benchiekh et al8 and Amitay et al9 have attempted the control of separation in internal flows, in particular inside a diffuser, with some measure of success. Synthetic jets have also been used to enhance heat and mass transfer.10-11

Driven Wall / Membrane

do

Synthetic Jet

Vortex Rings

Figure 1. Synthetic jet actuator

Whilst understanding of the evolution and characteristics of synthetic jets are important in the development of its applications, the actuation and optimisation of actuators are equally as important. Mallinson et al12 have shown that the resultant jet shows evidence of the influence of acoustic resonance of the cavity and structural resonance of the

flexible wall. Guy et al13-14 show that the output of the actuator is maximised at the two resonance frequencies. Furthermore, it is suggested that there is an optimum combination of all geometric parameters at which the actuator will operate at its full capacity. Gallas et al15-17 have offered a lumped element model (LEM) based upon an analogy with electrical circuitry, in which the components of a piezo-electrically driven synthetic jet actuator are modelled as elements of an equivalent electrical circuit. The writers have discussed methods of estimation of model parameters and have provided experimental verification of their model. Agreement between the model and experimental data was found to be fair, allowing some measure of optimisation of the actuator. Gallas et al18 have also found that the actuator orifice flow is characterised by linear and non-linear losses depending upon oscillating wall stroke lengths relative to the orifice size; however conclude that further investigation is required to develop correlations for orifice loss coefficients as function of Reynolds number and wall stroke lengths.

While the LEM model15-17 is able to predict orifice flow velocity, it is however not capable of predicting fluctuations in actuator cavity internal pressure and the phase relationships between input forcing, diaphragm displacements, internal pressure and orifice flow velocity. These properties of the actuator are fundamental to the understanding and optimisation of the actuator.

The purpose of this paper is to present an alternative synthetic jet actuation model based on the laws of fluid-dynamics, capable of additionally predicting cavity internal pressure fluctuations and the phase relationships between the different variables. This model is then utilized towards an improved understanding of the performance of the actuator to important parameters.

II. Development of the Model The problem to be considered is the actuation of the cavity wall of a typical synthetic jet actuator through an

input supply voltage, as illustrated in the schematic of Figure 2. The actuator is assumed to interact with quiescent air. We are interested in deriving a relationship between the output of the actuator, namely the orifice flow velocity U(t), and also the actuator cavity internal pressure pi(t), to the driving force F(t) effected through an input voltage v(t). We assume at this stage that F(t) is some known function of v(t)

( ))()( tvgtF = (1)

and the exact form for which will depend upon the characteristics of the actuation mechanism. The oscillating wall could take one of a number of different forms, such as a piezo-electrically driven diaphragm, a mechanically driven piston, or mechanically driven diaphragm. For example, F(t) could be an electromagnetic force generated with varying input voltage v(t) such as that in an audio speaker mechanism.

For simplicity, we consider the flexible wall behaving as a piston where it is deemed to displace as a rigid body in one dimension. An audio speaker displays such a behaviour in the middle to low frequency range (Kinsler et al19; page 410) since the speed of transverse waves is about 500m/s. Consequently, it is reasonable to assume that the speaker cone moves as a unit for frequencies below 500Hz. Similarly, a piezo-electrically driven actuator displays behaviour similar to that of a single degree of freedom mechanical system.15-17 In this way, the driven wall of a synthetic jet actuator is readily modelled as a mechanical mass-damper-stiffness system having a mass mw, damping coefficient cw, and stiffness kw. An added mass term ma might also be applicable for small actuators with lightweight plate-type wall. In addition, an aerodynamic-acoustic damping force could be expected. Under actuation, the gauge pressure inside the cavity p

wa xc &

i(t) (= absolute internal pressure minus atmospheric pressure = Pi(t) – Po) will fluctuate about a zero mean value as the air undergoes compression and expansion. Consequently, an equation governing the dynamics of the flexible membrane may thus be obtained under the action of the electromotive and internal pressure forces as

V(t) = Vo – Aw xw U(t) PoPi(t) = pi(t) + Po xw(t)

F(t)

ww xk

wwt xc &

wwt xm &&

pi(t) Aw

L

Figure 2. Schematic of the synthetic jet actuator model

(2a) wiwwwwtwwt ApFxkxcxm −=++ &&&

in which Aw = πdw2/4 is the area of the flexible wall, xw(t) is its displacement as shown in Figure 2, and

(2b) awwt mmm += awwt ccc +=

For a circular piston vibrating at cyclic frequency f or radian frequency ω = 2πf, the added mass and damping terms are (Kinsler et al19)

( ) ωρ /1 yXUAm Swaa = and ( )yRUAc Swaa 1ρ= (2b)

where

(2c) Sw Udy /ω=

⋅⋅⋅−⋅⋅+⋅−== )753/()53/(3/)/4(/2 11 yyyyyyX πH 22523

( ) ( ) 226242

(2d) ( ) ( ) ( )

(2e) ⋅⋅⋅−⋅⋅⋅+⋅⋅−⋅=−= )8642/()642/()42/(/21 11 yyyyyJyR

Equation (2) may be re-written as

(3) wtwiwtwwwwww mApmFxxx //2 2 −=++ ωως &&&

in which

wwtwtw kmc 2/=ς (4)

and

wwtww fmk πω 2/ == (5)

are the damping ratio and the natural frequency respectively, of the flexible wall. These parameters will depend upon the structural characteristics of the wall, and the manner in which it is supported.

If the wall is now oscillated, then the air inside the cavity will alternately undergo compression and expansion. This will be accompanied by an oscillatory flow through the orifice. The internal pressure in the cavity can be considered to be uniform (except close to the opening) at any given instant in time provided the wavelength λ of the pressure oscillations is much greater than a typical dimension Lc of the actuator cavity i.e. provided that the frequency for the biggest standing wave of quarter-wavelength is well beyond the operating frequency range of the actuator. This requirement (see for example Hall20) can be expressed as

(6) cSS LfUU >>=≡ //2 ωπλ

where US = γ Po /ρa is the speed of sound. For air at standard conditions, US is approximately 343 m/s and the criteria for the assumption of uniform internal pressure for three typical situations are summarised in Table 1. It is clear that for a relatively large bench-top model, this assumption may be justified for frequencies of actuation less than 680 Hz, while that for a miniature scale model,

Table 1 Criteria for uniform internal pressure Description Length scale Lc Criteria Bench-top < 0.50 m f < 680 Hz Pocket < 0.05 m f < 6.8 kHz Miniature < 0.01 m f < 34 kHz

for frequencies up to well over 30 kHz. When the above assumption is justified, then the instantaneous density of the bulk of the internal air can be expressed as the sum of a mean component and a fluctuating component as

(7) ,ρρρ += iai

If the amplitude of oscillations of the flexible wall are much smaller than the length of the actuator i.e. if |xw| << L, then the physical volume of the cavity

(8) wwo xAVV −=

will suffer relatively small fluctuations ∆V = Awxw as compared with the nominal volume Vo i.e. ∆V << Vo. Since air can flow out of and in through the orifice as well, then << i.e. density changes are relatively small as compared with its mean value. When this holds, application of the compressible form of the continuity equation to the air contained within gives

,iρ aρ

(9) UAxAdtdVdtVddtdm oawwaioii ρρρρ −=−≈= &)/(/)(/

where U is the air velocity in the plane of the orifice. The bulk behaviour of the air within the cavity may be assumed to follow a polytropic process. Assuming that

the initial internal pressure equalled the ambient pressure Po, we can express the relationship between instantaneous internal pressure and instantaneous density as

(10) nao

nii PP ρρ // =

in which n is the polytropic exponent, which is equal to 1 for an isothermal process and equal to the specific heat ratio γ for an isentropic process. Based on previous studies21-24, we can assume that air undergoes isentropic contractions and expansions in the cavity. Thus

( ) ( )aoii PdtdPdtd ργρ //// ≈ (11)

Since

(12) dtdpdtPpddtdP ioii //)(/ =+=

the continuity equation (Equation (9)) can now be re-written as

( ) ( ) UAxAPdtdpV owwoio −=− &γ// (13)

Consider now an outflow part of the cycle as illustrated in Figure 3a. The unsteady form of the Bernoulli equation may be applied between a point far enough upstream of the orifice, inside the cavity where the flow velocity is negligible (i.e. much smaller than the orifice flow velocity), to a point at the vena-contracta where the velocity may be expressed as

(14) UU vc ε=

In Equation (14), ε is a multiplier which may vary according to the characteristic and strength of the vena-contracta. At the vena-contracta, the pressure is ambient. Thus

( )dtdUlUCUPP eaaLvcaoi /2212

21 ρρρ +++= (15a)

U(t)Pi(t) Po U(t) Pi(t) Po

(a) Out-flow (b) In-flow

Figure 3. Orifice flow situations

which is readily re-written as

( )dtdUlUCp eaaLi /)( 2212 ρρε ++= (15b)

For an in-flow part of the cycle as shown in Figure 3b, and assuming loss and discharge coefficients are similar to those during out-flow, we similarly obtain

( )dtdUlUCp eaaLi /)( 2212 ρρε ++−= (16)

Noting from Figure 2 that U is positive for out-flow, and negative for in-flow, Equations (15) and (16) may be combined into a single equation

( )dtdUlUUCp eaaLi /)( 212 ρρε ++= (17)

In these equations, ε is the ratio of the velocity at the location of a nominal vena-contracta to that in the plane of the orifice, CL is a loss-coefficient that defines the losses through the orifice to the point of the vena-contracta, and le is an effective length used to define the inertia of the air jet or slug moving through the orifice.

Under steady conditions, the velocity ratio ε can be shown to be given by

Ld

CC

−+−

= 42

42 1 ββε (18)

in which Cd is the orifice discharge coefficient for flow through the orifice of area Ao, and β = do / dw = √(Ao / Aw). For a small β ratio, the steady value for Cd is approximately 0.6. However, when conditions are unsteady as in the problem under consideration, Cd might vary cyclically. Unfortunately, detailed data for such situations is not available, although various writers20-21 have shown that the value of 0.6 works well under certain unsteady conditions, and it could be as high as 0.88. It might be argued that as the orifice flow velocity varies cyclically from zero to a maximum (and thus the orifice flow Reynolds number varies from zero through low values to some maximum value), the characteristics of the flow near the orifice varies cyclically as well. At low velocities, attached potential flow will be present and in this case there will be no vortex rings and thus no vena-contracta and the losses will also be very small. Under such conditions, the outside point of consideration (which would otherwise be the location of a vena-contrcta) would have a flow velocity lower than that at the plane of the orifice i.e. Uvc < U since the streamlines would be divergent. Consequently ε < 1. As the orifice flow velocity increases, flow separation will set in and a vortex ring will begin to form accompanied by a contraction of flow resulting in Uvc > U and therefore ε > 1 can be expected. However, as the orifice flow velocity increases further, the vortex ring will become stronger

and the contraction of flow will become severe, and in the extreme case will approach the steady flow situation whence Cd ≈ 0.6, and ε will then be given by Equation (18). For small β ratio, it can be shown using Equation (18) that the steady state value for the velocity ratio ε is approximately 1.55.

Values for the effective length may be estimated using data available in the acoustics literature, see for example Hall20, or from studies on building internal pressure dynamics induced through dominant openings22-24. The effective length of the air jet / slug at the orifice is a sum of the actual orifice length lo and end corrections quantified using an inertia coefficient CI, such that

oIoe ACll += (19)

Loss-coefficient data for steady flow situations are readily available from texts on Fluid Mechanics, however the orifice flow under consideration is time varying for which data is scarce. Vickery23 concluded from his studies on building cavity excitation through orifice type openings, that in spite of the unsteady nature of the orifice flow, a constant value reflective of the steady flow situation never the less yields acceptable results. Drawing on this, and based on a severe sudden contraction (see for example White25), it might be reasonable to assume that a loss coefficient given by

( )2142.0 β−=LC (20)

may be applicable during out-flow and in-flow respectively. While the above parameter estimates might be reasonable under certain conditions, a more rigorous approach may be taken along the lines suggested in reference15-

17 which is based on information in reference 26. Equations (13) and (17) may now be combined. First however, given that there are indeed three ill-defined

parameters, namely le (or CI), CL and ε, and that the velocity ratio ε appears in the damping or loss term only, we introduce an effective loss coefficient CLeff, such that

(21) 2ε+= LLeff CC

If the orifice flow velocity U is eliminated, then an equation governing the dynamics of internal pressure is obtained,

( ) woowihwoowiwoowieooLeffoi xVPApxVPApxVPAplPACVp &&&&&&&& )/()/()/()2/( 2 γωγγγ =+−−+ (22a)

wwwpwihwwwpwiwwwpwieooLeffoi xAmpxAmpxAmplPACVp &&&&&&&& )/()/()/()2/( ωωωωγ =+−−+ (22b) ( ) 2222

However, if we chose to eliminate internal pressure pi then an equation governing the dynamics of the mean orifice flow velocity is obtained,

( ) ( ) woeaowheLeff xVlPAUUUlCU &&&& )/(/ 2 ργω =++ (23a)

( ) whowheLeff xAAUUUlCU // (23b) ( ) &&&& 22 ωω =++

In Equations (20-21),

)/()/()()/()/(/2 2oeoSoeaoooeaooohhhh VlAUVlPAAlVPAmkf ===== ργργπω (24)

is the Helmholtz resonance frequency of the cavity with the orifice; and

woowwwpwp mVPAmk /)/(/ 2γω == (25)

is the natural frequency of the pneumatic spring made of the oscillatory wall of mass mw against a sealed cavity volume Vo. Equations (22) and (23) are readily recognised as those for a Helmholtz acoustic resonator, but with pneumatic coupling to the dynamics of the oscillating wall. The synthetic jet actuator may therefore be viewed as a coupled mechanical (dynamic wall) – Helmholtz resonator system with two degrees of freedom. If we take this view, then a number of deductions are possible. The actuation system will have two resonance modes, and it is possible to maximise the output velocity U if the actuator is driven near the resonance frequencies. It might also be possible to ‘tune’ the actuator so that the Helmholtz and wall resonance frequencies (fh and fw) are similar, thus giving an even larger output. In this regard, the geometrical properties of the actuator are important in two ways. Firstly, they directly determine the Helmholtz resonance frequency (see Equation 24), and secondly, they have an influence on the damping of the coupled system. Examining Equations (22-24) reveals that a decrease in actuator cavity volume Vo will increase fh (which will be of significant benefit if fh < fw) and will also decrease the damping. The same effects are also obtained with increase in size of the orifice (Ao), however, while the energy efficiency of the system will be improved, the output velocity will decrease which may not be desirable from the point of view of the evolution of the synthetic jet.

For very low frequencies, or rather for the wall moving steadily at a velocity U=wx& w (positive inwards and negative outwards from cavity), the corresponding steady values for the orifice velocity Uo and the internal pressure pio can be obtained from Equations (22) and (23) by setting time derivatives of U and pi to zero, giving

( ) wowo UAAU /= (26)

ooaLeffio 2 UUCp ρ1×= (27)

These are expected results that can also be obtained from application of continuity and energy equations for the steady flow situation.

The coupled Equations (1), (3), (13) and (22-23) constitute the synthetic jet actuator model, which relates the input forcing voltage v(t) to the oscillating wall displacement xw(t), cavity internal pressure pi(t), and the actuator output or orifice velocity U(t). Since Equations (22-23) are non-linear, a numerical solution scheme has to be employed in order to solve for these variables, for a given forcing voltage v(t), or oscillating force F(t).

III. Model Validation For validation of the present model, we consider the two piezoelectric-driven synthetic jet actuators (I and II)

studied by Gallas et al15-17, for which physical and experimental data are available and which are summarized here in Table 2. Since input to the present model is a sinusoidal force on the diaphragm, the amplitude of this force is estimated using data from references 15-17 in Table 2. Using the effective acoustic piezoelectric coefficient Da, a volume displacement magnitude ∆V is first calculated for the given forcing voltage amplitude. By equating this to an equivalent volume displacement ∆V = Awxw in the present model, a displacement amplitude results which combined with the wall stiffness yields the force amplitudes of 0.574N and 0.401N for cases I and II respectively.

Figures 4a and 4b compare the maximum orifice velocity predicted by the present model with those from the experiment and LEM model of Gallas et al15-17, for cases I and II respectively. Figure 4a shows that there is good agreement between the present model and experiment, except in the mid-frequency region where the model over predicts the maximum velocities somewhat. This is similar to the prediction of the LEM model. The LEM model on the other hand under predicts the first resonant peak, which clearly relates to the choice of a higher loss coefficient of 1.0 as compared with a value of 0.42 + 0.36 = 0.78 used in the present study. In Figure 4b for case II, agreement between the present model and experiment is good again. In this case, while the model over predicts the maximum velocities somewhat near the highly damped first resonance, it nevertheless gives a good prediction of the maximum velocity at the resonant peak at 860Hz, with CLeff = 0.78. The LEM model with a loss coefficient of 1.0 over predicts the maximum velocity in this case. That this is the case is not surprising as the loss and other orifice coefficients utilised in the present and past studies are taken as constant values, whereas the oscillatory nature of the orifice flow would mean that these are in fact not constant. Clearly, further studies are required to obtain accurate representations for these coefficients. Not withstanding this, the agreement between the present model and experiment is never the less good, which allows the new model to be used in the analysis of the performance of the actuator to important parameters.

0

5

10

15

20

25

30

35

Um

ax, m

/s

F

Figu

calculatwithin athe newGallas eand theincompthe phaaround qualitat

A ffrequenpeak. Tother aresonan

Table 2 Properties of piezoelectric-driven synthetic jet actuators studied by Gallas et al15-17

I II Brass Shim Density (kg/m3) 8700 8700 Thickness (mm) 0.20 0.10 Diameter (mm) 23.5 37.0 Piezoceramic Density (kg/m3) 7700 7700 Thickness (mm) 0.11 0.10 Diameter (mm) 20.5 25.0 Diaphragm Compliance (s2.m4/kg) 6.53×10-13 2.23×10-11

Acoustic mass (kg/m4) 8.15×103 2.43×103

Acoustic piezoelectric coefficient da (m3/V) 5.53×10-11 4.32×10-10

Mechanical damping ratio 0.03 0.03 Natural frequency (Hz) 2114 632 Cavity Volume (m3) 2.5 × 10-6 5.0 × 10-6

Equivalent cylindrical diameter (mm) 23.5 37.0 Equivalent cylindrical length (mm) 5.76 4.65 Orifice Diameter (mm) 1.65 0.84 Length (mm) 1.65 0.84 Effective loss coefficient 0.78 0.78 Inertia coefficient 0.705 0.86 Helmholtz frequency (Hz) 977 473 Forcing Voltage amplitude (V) 25 25 Force amplitude (N) 0.574 0.401

0 500 1000 1500 2000 2500 3000Frequency, Hz

Experiment Gallas [17]LEM Gallas [17]Present Model

(a)

0

10

20

30

40

50

60

70

80

0 500 1000 1500Frequency, Hz

Um

ax, m

/s

Experiment Gallas [17]LEM Gallas [17]Present Model

(b)

igure 4. Comparisons between the model, experiment and the LEM17, for (a) Case I, (b) Case II

re 5a displays the maximum diaphragm displacement and velocity, internal pressure and the orifice velocity ed using the present model for case I. It shows that significant fluctuations in internal pressure are obtained synthetic jet actuator, and which are most enhanced at the two resonant frequencies. Further validation of model is provided from a comparison of the calculated phase angles with experimentally measured data of t al.18 At the very low frequencies, the model shows that the phase angle between the diaphragm velocity

orifice flow velocity, φVw-U , is near 0° i.e. the two velocities are very nearly in phase, indicating the flow is ressible. Near the first resonant peak however, the flow becomes compressible as indicated by an increase in se angle φVw-U. Similarly, the phase angle between the internal pressure and the orifice velocity, φPi-U is 90° at the low frequencies, which however increases as the first resonant peak is approached. These are ively and quantitatively in agreement with the measurements of Gallas et al.18

urther observation in Figures 5a and 5b is that while the phase angle φPi-U is approximately zero at low cies, it however is approximately 180° at the higher frequencies, and in particular around the second resonant his clearly implies that the diaphragm and orifice flow movements are in phase or in harmony with each t the low frequencies, and in particular at frequencies nearer the first resonance mode. At the second ce mode however, φPi-U ≈ 180° clearly implies that the diaphragm and orifice flow movements are anti-

phase. This behaviour is very similar to that of a two-degree of freedom mechanical system, consisting of two masses, two springs, and two dash-pots, connected in series, although in the present problem, compressibility and non-linear effects have altered the behaviour slightly.

1E-61E-51E-41E-31E-21E-11E+01E+11E+21E+31E+4

0 500 1000 1500 2000 2500 3000Frequency, Hz

Internal pressure, Pa

Orifice velocity, m/s

Diaphragm velocity, m/s

Diaphragm displacement, m

(a)

-90

0

90

180

270

0 500 1000 1500 2000 2500 3000Frequency, Hz

Phas

e an

gle,

deg

rees

Orifice velocity - internal pressure

Orifice velocity - diaphragm velocity

(b)

Figure 5. (a) Calculated maximum diaphragm displacement and velocity, internal pressure and the orifice velocity for Case I. (b) Calculated phase angles for Case I.

IV. Further Analysis Using the Model The model was used to analyze the performance of the actuator to geometrical and physical variables. As an

example, the influence of cavity volume on the maximum orifice flow velocity for Case I is shown in Figure 6a. The trends are similar to the LEM results of Gallas et al18, showing decrease in Helmholtz resonance and thus the first resonance frequency with increasing volume. This is accompanied by decreased response as the first and second resonance peaks become smaller with larger volume. A solution was also obtained for 21% of the nominal volume for case I, at which the Helmholtz resonance frequency is exactly equal to the diaphragm resonance frequency of 2114 Hz. While the maximum output velocity in this situation is slightly lower than that obtained for a 50% volume, a broad band response is nevertheless evident. The actuator output is relatively large over a wide frequency range, and such a condition may be desirable where the operating conditions might drift, or are not exactly determinable during design.

The influence of the diaphragm resonance frequency, physically achieved by adding ‘mass’, on the actuator output is shown in Figure 6b. Two solutions were obtained, corresponding to increased diaphragm mass relative to Case I (see Table 2), one of which was ‘tuned’ so that the diaphragm natural frequency exactly equaled the Helmholtz resonance frequency of 977 Hz. Again, the actuator output is seen to be maximized when the two resonance frequencies are nearly equal.

0

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50

0 500 1000 1500 2000 2500 3000Frequency, Hz

Um

ax, m

/s

200% Volume100% Volume50% Volume21% Volume (fh=fw)

(a)

0

10

20

30

40

50

0 500 1000 1500 2000 2500 3000Frequency, Hz

Um

ax, m

/s

fw = 2114 Hzfw = fh = 977 Hzfw = 552 Hz

(b)

Figure 6. Model predictions for (a) varying cavity volume; (b) varying diaphragm mass (Case I)

V. Conclusions An analytical model for a synthetic jet actuator based on the laws of fluid dynamics has been developed and

validated against experimental data in the published literature. The model is able to predict the performance of the actuator, defined by the orifice flow velocity, to important physical and geometrical properties, and inputs. It is also able to predict actuator cavity pressure variations and the phase relationships between the actuator diaphragm movements and the orifice velocity. The model and analysis based on it provides valuable insights into the behavior of synthetic jet actuators.

VI. Acknowledgements The grant support from the Research Committee of the University of Auckland is gratefully acknowledged.

VII. References 1James RD, Jacobs JW, Glezer A. 1996. A round turbulent jet produced by an oscillating diaphragm. Phys. Fluids 8:2484-95 2Smith BL, Glezer A. 1998 The formation and evolution of synthetic jets. Phys. Fluids 31:2282-97 3Glezer A, Amitay M. 2002 Synthetic jets. Annu. Rev. Fluid Mech. 34:503-29 4Smith, D. &Glezer, A., Vectoring and small-scale motions effected in free shear flows using synthetic jet actuators, AIAA

98-0209, 1998. 5Davis, S. A. & Glezer, A., Mixing control of fuel jets using synthetic jet technology, AIAA 99-0447, 1999. 6Crook, A., Sadri, A. M. & Wood, N. J., The development and implementation of synthetic jets for the control of separated

flow, AIAA 99-3176, 1999. 7Smith DR. 2002 Interaction of a synthetic jet with a crossflow boundary layer. AIAA Journal 40:2277-88 8Benchiekh M, Bera J-C, Michard M, Sunyach M. 2000 Contróle par jet pulse de l’écoulement dans un divergent court à

grand angle. Mécanique des fluids/Fluid mechanics:C. R. Acad. Sci. Paris, t.328, Série II b, 749-56 9Amitay, M., Pitt, D., Kibens, V., Parekh, D. and Glezer, A., Control of internal flow separation using synthetic jet actuators,

AIAA 2000-0903, 2000. 10Amon, C. H., Murthy, J., Yao, S. C., Narumanchi, S., Wu, C. & Hsieh, C., MEMS enabled thermal management of high

heat flux devices (EDIFICE), Experimental Thermal and Fluid Science (25) 231-242, 2001. 11Tranicěk Z., Tesař V., 2003 Annular synthetic jet used for impinging flow mass transfer. Int. Jour. Heat & Mass Transfer

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