3.1. Fluids
The synthetic jet actuator frequency response was established by determining the maximum exit velocity of the jet at its centerline, for frequencies from
. To do this, a frequency sweep with
increments near the Helmholtz frequency and
away from it, was performed.
Figure 13 shows the frequency response of the SJA used in the current work. Frequencies near the mechanical resonance frequency of the diaphragm (
) were not investigated as the hotwire experiments found no significant peak in that range. This is likely due to the geometry of the SJA. Many studies utilize SJA cavities with "pancake" shapes, meaning they are short with an aspect ratio on the order of 10 or higher [19,25–28]. These configurations yield significant peaks in jet velocity near the mechanical resonance frequency of the diaphragm because the volume of fluid displaced by the diaphragm is significant compared to the volume of the cavity. The current study considers an SJA with an aspect ratio of
.
Figure 13 also includes the normalized hotwire velocity measurements from Feero et al. [29]. It can be seen that the current work captures the location of the velocity peak associated with the Helmholtz frequency closely. The current (multiphysics) work predicts that the SJA performance (center line velocity) is maximized when the actuation frequency is
. This peak, associated with the Helmholtz frequency, is
from the theoretical Helmholtz frequency (
), a
deviation from the experimentally determined Helmholtz frequency (
), and
from the velocity peak in the hotwire experiments (
) [29]. The magnitude of the maximum centerline velocity predicted by the multiphysics model showed reasonable agreement with the experimental results. The multiphysics model predicted a highest centerline velocity of
, which was an over-prediction of
. The centerline velocity hot wire measurements were taken
downstream of the orifice, whereas the computational results were taken at the exit plane of the orifice. Furthermore, hot wire measurements cannot capture flow direction, and the computational model assumes perfect electrostatic conditions (no power loss). All of these factors may play a role in the over-prediction of the maximum jet velocity. These results are summarized in
Table 6.
Another parameter of interest when analyzing synthetic jet actuators is the radial profile of axial jet velocity. During expulsion, a "top-hat" velocity profile is the optimal case. This profile indicates that velocity is constant across the width of the orifice [9]. Feero et al. [29] performed hot wire measurements across the diameter (from
to
) to measure this radial profile for diferent phases of operation of the SJA. Ziade et al. [22] performed a numerical study with the same geometry using a second-order scheme and also plotted their radial profiles of axial jet velocity. The results of Feero et al., Ziade et al., and the current work can be seen in
Figure 14. The diaphragm actuation frequency was
. The velocities are normalized by their respective maximum centerline jet velocity. It should be noted that there is significant uncertainty in the hot wire measurements near the orifice edges. Again, some minor discrepancies exist due to measurement techniques. The hot wire measurements were taken
downstream of the orifice, but the current work used the orifice exit plane to record the velocity profiles.
The overall agreement between the current model and the experiments is excellent. The multiphysics model was able to capture the shape of all profiles accurately despite the velocity profile during expulsion, (
)
2, deviating significantly from the ideal "top-hat" shape. The velocity profile during the first half of expulsion is representative of fully developed flow in a circular duct which is driven by an oscillating pressure gradient [29,44]. Fully developed flow is expected in this case, as the nozzle length to diameter ratio is relatively large. This type of flow is characterized by the Stokes number (
S), which is approximately 22 for this case. White [44] also states that as
S reaches about 20, the average velocity reaches
of the centerline velocity and approaches
as
S approaches infinity [29]. The ingestion portion of the cycle also shows excellent agreement with the hot wire measurements and other computational work. The flow during ingestion is a more typical "top-hat" shape, as expected.
Figure 15 and
Figure 16 show the velocity and vorticity magnitude contours, respectively, of the SJA at 6 different phase angles in one cycle. The contours are from the
cycle in a simulation of the SJA operating at an actuation frequency of
.
The beginning of expulsion (
) shows the early stage of the formation of a vortex ring at the orifice exit. The fluid is just exiting the orifice and has begun to curl over the edges, creating an area of circulation, as shown by the vorticity contours (
Figure 16a). During this time, vortices inside the cavity from the previous cycle have detached from the nozzle entrance and are travelling back toward the diaphragm, and vortices outside the cavity permeate downstream. The vortex from two cycles prior has impacted the diaphragm and has begun to dissipate along its surface. The development of the vortex ring is depicted nicely during the next phases of expulsion (
Figure 16b and
16c). These contours show a clear picture of "vortex roll-up." The velocity in the nozzle also begins to decrease, as the ingestion phase is set to begin shortly. As ingestion beigns (
), the vortex ring outside the orifice forms fully, as fluid rushes into the orifice behind it. The vorticity contour (
Figure 16d) shows that a coherent structure has formed, proving that the vortex detaches and is not ingested back into the cavity. Simultaneously, the formation of a vortex ring begins inside the cavity. At the nozzle exit, where fluid travels around the orifice edge, zones of recirculation are noted. The rest of the ingestion cycle (
) sees the completion of the vortex ring inside the cavity and the arrival of the previous vortex ring at the diaphragm.
Figure 15e and
15f show the velocity contours of fluid which was ejected from the cavity during this cycle and now travels downstream. This summary explains qualitatively how a synthetic jet actuator works, with imagery to illustrate its inner workings.
Van Buren et al. [45] found that a phase shift between the diaphragm displacement and jet velocity existed, indicating that the flow was compressible within the cavity and/or nozzle. This phenomenon occurs as a product of solving the Navier-Stokes equations for channel flow with an oscillating pressure gradient. Despite low fluid velocities relative to supersonic flows, several studies have reported that compressibility should be considered when modelling synthetic jet actuators. As indicated earlier, Gallas [10] stated that compressibility effects must be considered when the actuation frequency is
. Sharma [23] reported that a phase shift (and thus flow compressiblity), began to form near the Helmholtz frequency, and grew until the diaphragm displacement and orifice velocity were completely out of phase. As the current work focuses on actuation frequencies near the Helmholtz frequency, compressible flow was used.
Figure 17 shows the oscillations of the diaphragm velocity and displacement at its center, and jet velocity at the exit of the nozzle in response to a
actuation frequency. Clearly, a phase shift exists between the variables, indicating that compressibility must be accounted for to accurately model SJAs. At an actuation frequency of
, a phase shift between the diaphragm displacement and jet velocity is approximately
.
When compressibility is introduced, temperature effects become relevant as well. Van Buren et al. [45] discussed this briefly in their work on high-speed and momentum SJAs where they found that an increase in temperature was observed in the nozzle due to fluid compressibility. Temperature variation inside the nozzle was seen in the current work, but is negligible because fluid velocities are relatively low.
3.2. Acoustics
As with fluid behaviour, the noise generated by synthetic jet actuators is highly frequency dependent. As is the case with most performance variables, the noise generated by SJAs is maximized when the actuation frequency matches the Helmholtz frequency, or the mechanical resonance of the diaphragm. Arafa et al. [7] showed that at higher harmonics, noise generation can be lower, at minimal cost to the jet’s performance. For the purposes of the current work, however, only frequencies near the Helmholtz frequency are considered.
Figure 18 shows the maximum centerline acoustic pressure at the orifice exit, and at the center of the diaphragm’s surface. The noise generation for both locations is maximized at the Helmholtz frequency, as expected. Interestingly, the position of the peak acoustic pressure is at
, which is only
from the theoretical value of
. This was the expected behaviour of the jet velocity, which peaked at an actuation frequency of
.
The form of the acoustic signal is also important to understand. Mallionson et al. [46] reported that the mechanical resonance of the diaphragm and the acoustic resonance of the cavity influence the jet plume formed by SJAs. Further, numerical work by Wang et al. [4] found that the sound generated by SJAs can be broken down into two monopole components. These monopoles are structure-borne noise (caused by the flexing of the diaphragm and its interaction with the fluid), and fluid-borne noise (related to the resonant frequency of the cavity). They found that cancellation exists between these monopoles at low frequencies when their signals are out of phase.
When modelling pressure acoustics without thermoviscous effects, the viscous- and thermal-related losses are not accounted for. Initial trials for the current work modelled the acoustics of SJAs with linear elastic behaviour (lossless). While this is not physically realistic, it does present an interesting discussion. When the viscous and thermal losses are ignored, the transient acoustic response develops a beat frequency - a consequence of the cancellation of the structure-borne and fluid-borne sound monopoles.
Figure 19 and
Figure 20 show the transient acoustic response from exciting the synthetic jet actuator at
and
, respectively, in the absence of thermoviscous losses. This is a beat frequency and a result of alternate addition and cancellation of sources which are out of phase. In this case,
is approximately
from the Helmholtz frequency of the cavity. Therefore, the two sound sources (diaphragm and cavity) interfere and partially cancel each other approximately every
. Similarly,
is approximately
from the Helmholtz frequency, meaning the signals cancel each other approximately every
. This is illustrated further by
Figure 21 and
Figure 22, which are discrete Fourier transforms of the transient acoustic signals.
Figure 21 has two large distinct peaks, corresponding to the actuation frequency (
) and the Helmholtz frequency (
). Conversely,
Figure 22 has only one distinct peak. This is because the figure’s sensitivity is too low to display the actuation and Helmholtz frequencies separately. Furthermore, because these peaks are much closer together, they superimpose to generate more noise. The proceeding peaks shown in the Fourier transforms correspond to twice the actuation frequency (
) and the mechanical resonance frequency of the diaphragm.
These findings, while interesting ignore thermoviscous losses which take place in the nozzle and cavity. When thermoviscous losses were accounted for, the acoustic behaviour did not display the same beat frequency.
Figure 23 and
Figure 24 show the transient response, and discrete Fourier transform of the response when the diaphragm is excited at
and thermoviscous losses are incorporated. In the transient response it is evident that the beat frequency behaviour has disappeared. The transient response reaches a steady state condition after about 12 cycles. The Fourier transform supports this, as the Helmholtz frequency peak has been almost entirely damped out. The only prominent peak is associated with the actuation frequency.
The same conclusions can be drawn regarding the SJA driven with an actuation frequency of
(
Figure 25 and
Figure 26). The only prominent peak that remains when thermoviscous losses are incorporated is associated with the actuation frequency. The peak associated with the Helmholtz frequency is almost entirely dampened out.
The current work is also capable of calculating the acoustic pressure in the farfield. The setup of the perfectly matched layer and exterior boundary condition make it possible to extrapolate to a point in space outside the computational domain to determine noise levels anywhere. The ability to study noise generation is an invaluable addition to the literature concerning SJAs. For designs to be viable in the real world, noise generation must be minimized. This work shows the importance of considering thermal and viscous losses when modeling and performing experiments. When performing experiments, extra care should be taken to standardize fluid properties and environmental conditions to prevent external viscous and thermal effects from impacting acoustic behavior. Furthermore, because the thermal and viscous boundary layers depend on frequency, these effects can help explain the acoustic response across a broad frequency range. For example, this explains the difference in dampening at the Helmholtz and mechanical resonant peaks.
3.3. Diaphragm
Many studies use an oscillating uniform velocity boundary condition in lieu of a moving wall for simplicity and computational efficiency [3,15,16,46]. Others, wishing to improve accuracy have modelled the diaphragm with an equation to describe the oscillating velocity profile [17,20]. Modelling the diaphragm as a moving wall with a profile defined by an equation has also been done with some success [19]. Some researchers choose to omit the cavity and/or nozzle altogether to simplify computations and focus on the exterior flow [47–49]. Some researchers been employing a more realistic multiphysics approach which incorporates piezoelectricity and structural mechanics [25–28].
Figure 27 shows a comparison between the modelling methods that have been used, including the displacement profile determined through the current multiphysics modelling. All of these profiles assume that the deflection is static, not operating in an active SJA. Apart from the clamped boundary condition, the second-order polynomial captures the deflection well [19], but does not account for the flexing the diaphragm undergoes while the SJA is in operation. The theoretical solution from [18] does not perform well for plates which have multiple layers of different diameters.
When a static voltage is applied to a THUNDER TH-5C actuator, it deflects the same amount in each direction. However, when operating in an SJA in the presence of a flow field, the pressure in the SJA cavity varies sinusoidally and causes the diaphragm to flex. The force that this fluid pressure puts on the diaphragm as the vortex rings within the cavity impact its surface causes the diaphragm to deflect further downward (away from the cavity) than upward (toward the cavity). The impact that the fluid has on the diaphragm deflection is not insignificant. When operating near the Helmholtz frequency, the difference between the upper maximum deflection and lower maximum deflection is about
.
Figure 28 shows the deflection of the SJA when operating at
, normalized by maximum upward deflection. This shows that approximating a diaphragm with an equation ignores the compliance of the diaphragm. This also indicates that the cavity compliance cannot be assumed to be zero, as some research suggests. The total volume of fluid displaced during an expulsion cycle when capturing the diaphragm shape using multiphysics and accounting for cavity compliance can be determined and compared against the approximations used by [17,19]. When normalized by peak diaphragm center displacement, the current work predicts the fluid volume displaced to be
and
% more than found by [17,19]. For these reasons, modelling the diaphragm with multiphysics provides a more realistic representation of the boundary condition than a velocity profile or a moving wall.