2.1. The actuator

The SJA used in this study has a circular cylindrical cavity with a 63.5 mm diameter bi-morph, piezoelectric, bending disk driver positioned opposite the orifice which is contained in a stereolithography (SLA) printed part (figure 5). The SLA orifice part also contains the side and upper wall of the $58.4\pm 0.2\ \textrm {mm}$ diameter, $3.2\pm 0.2\ \textrm {mm}$ tall cavity and was printed in-house with a Formlabs Form 2 3-D printer. A set of precision metric pins and callipers were used to measure the dimensions of the rectangular orifice which was found to be $14.59\pm 0.05\ \textrm {mm}$ long ($l$) by $1.10\pm 0.01\ \textrm {mm}$ wide ($h$) yielding an aspect ratio, $AR=l/h$, of 13 (figure 6). Additionally, the orifice was designed with a $2.59\pm 0.2\ \textrm {mm}$ deep throat which is roughly 2.35 times longer than the orifice is wide. The coordinate system used in this study aligns the $x$-direction with the axis of the jet, the $y$-direction with the short or minor axis of the orifice and the $z$-direction with the long or major axis of the orifice. The $x$, $y$ and $z$-directions will also be referred to as the axial or streamwise, cross-stream and spanwise directions, respectively.

Figure 5.

Cross-section of a typical synthetic jet actuator.

Figure 6.

Schematic of the data coordinate system and orifice dimensions.

The velocity at the throat of a SJA varies with time and has a zero mean. Therefore, a characteristic velocity of the jet was obtained by integrating the jet velocity at the exit of the orifice, $u_o(t)$, over the blowing part of the cycle before averaging over the full cycle period, T (Smith & Glezer Reference Smith and Glezer1998). The result is called the average blowing velocity, $U_o$, and is related to the effective stroke length, $L_o$, and the actuation frequency, $f$, in (2.1).

(2.1)\begin{equation} U_o = \frac{1}{T} \int_{0}^{T/2} u_o (t)\,\textrm{d}t = f\, L_o. \end{equation}

The two common orifice length scales used in related studies are the orifice width, $h$, and the equivalent diameter, $D_{eq}$, which is the diameter of a circular orifice with the same exit area. In order to be consistent with prior work on rectangular orifice synthetic jets, the orifice width, $h$, is used here. The one exception is stroke length, $L_o$, which has been non-dimensionalized by $D_{eq}$ to facilitate comparison with the vortex formation number set out by Gharib, Rambod & Shariff (Reference Gharib, Rambod and Shariff1998). The definitions for Reynolds number ($Re$) and Strouhal number ($St$) that were employed are presented in (2.2) and (2.3), respectively:

(2.2)\begin{gather} Re = \frac{U_o h}{\nu}, \end{gather}

(2.3)\begin{gather} St = \frac{f\,h}{U_o}. \end{gather}

This Strouhal number is effectively the ratio of the actuation frequency to the convective frequency and provides important information about the spacing of coherent structures in the jet and whether a jet is even formed (Holman et al. Reference Holman, Utturkar, Mittal, Smith and Cattafesta2005). Note that non-dimensional stroke length, $L_o/D_{eq}$, is merely the reciprocal of the $St$ when both are computed using the length scale $D_{eq}$.

During operation the actuator was driven by a sinusoidal voltage waveform from an Agilent 33500B function generator which was boosted by a Piezo Masters VP7210 amplifier. The conditions of the jet were controlled by varying the driving signal produced by the waveform generator. Adjusting the frequency of the driving signal directly modified the actuator frequency, while varying the voltage amplitude to the piezoelectric disk altered the actuator stroke length, and therefore the average blowing velocity.

The performance of the actuator was mapped for a range of driving voltages and frequencies using an AA Labs System AN-1003 constant-temperature anemometer system with a Dantec Dynamics $5\ \mathrm {\mu }\mathrm {m}$ miniature straight single element hot-wire probe (Model Number 55P11) positioned at the middle of the orifice with the wire aligned to the major axis and inserted to half the depth of the throat. Peak resonance of the actuator was found to occur at a driving frequency, $f$, of 470 Hz and produced a jet with a peak blowing velocity, $U_p$, of up to $50\ \textrm {m}\,\mathrm {s}^{-1}$. The average blowing speed, $U_o$, of the jet was determined by integrating the blowing phase of the rectified hot-wire data for at least 140 actuation cycles and dividing by the number of cycles. This integration was conducted because, at some operating conditions, the velocity programme, $u_o(t)$, at the jet orifice was slightly broader than a pure sinusoid. As a result, the use of an idealized sinusoidal relationship, $U_o = U_p/{\rm \pi}$, resulted in non-negligible errors in $U_o$.

2.2. SPIV set-up and data processing

The SPIV experiments were conducted with the actuator inside a $76\ \textrm {cm} \times 61\ \textrm {cm} \times 61\ \textrm {cm}$ quiescent enclosure with cast acrylic panels on four sides of the box and a door on the fifth side to isolate the experiment from air currents in the room while still allowing optical access. Inside the enclosure the actuator was mounted on a two-axis linear motorized traverse system in an $y$–$z$ configuration that facilitated the precise positioning of the actuator and enabled the jet to be stepped through the laser for multi-planar SPIV measurements. The air was seeded for the SPIV measurements by burning incense inside the enclosure prior to data acquisition. The median diameter of incense smoke particles has been reported to fall between 0.1 and $1 \ \mathrm {\mu }\textrm {m}$ (Chang, Lee & Tseng Reference Chang, Lee and Tseng2007).

A LaVision SPIV system comprising two $2560 \times 2160$ pixel resolution 16-bit dynamic range sCMOS cameras was used to image the flow field. Both cameras were mounted with Nikon Micro-NIKKOR 200 mm $f/4D$ macro lenses with Scheimpflug adapters that allowed the full imaging plane to be brought into sharp focus even though the camera sensor was not parallel to the plane. The cameras were set up on opposite sides of the laser beam in a forward scattering orientation at an angle of $35^{\circ }$ relative to the normal of the measurement plane (figure 7). This arrangement enabled the cameras to capture out of plane particle motion so that the three-component velocity field could be measured. The distance between the synthetic jet orifice and the camera sensors was approximately 80 cm which produced a $31.8\ \textrm {mm} \times 37.6\ \textrm {mm}$ field of view and a magnification of $14.7\ \mathrm {\mu }\textrm {m}\,\textrm {pixel}^{-1}$. Illumination was provided by a Quantel Evergreen dual-pulsed 200 mJ Nd:YAG laser fitted with a focusing element and a $-50\ \textrm {mm}$ focal length cylindrical lens which spread the laser into a vertical sheet. The laser light sheet was focused to a minimum waist above the actuator orifice. The SPIV data were acquired with the laser light sheet parallel with either the $x\text {--}z$ plane which contains the orifice major axis, or the $x\text {--}y$ plane which contains the orifice minor axis (figure 6). The orifice centreline planes, i.e. the $x\text {--}z$ plane and $x\text {--}y$ plane, are referred to as the major axis plane and the minor axis plane, respectively. Note, this orifice-relative terminology will be used irrespective of the orientation of the vortex ring axes for all forthcoming discussions.

Figure 7.

Experimental configuration for SPIV measurements.

To reconstruct the 3-D, three-component velocity field, SPIV data were acquired in one half of the jet on a set of tightly spaced, parallel measurement planes by stepping the jet through the laser light sheet using the two motorized linear traverses. Three to four more sparsely distributed data planes were also acquired in the other half of the jet and compared with their counterpart planes on the opposite side of centreline to confirm symmetry of the periodic structures. The level of symmetry observed in the off-centreline planes is similar to the left–right symmetry in the two centreline planes, the velocity and vorticity fields of which are presented in § 3. Thus the full jet was reconstructed by mirroring the tightly spaced planes onto the opposite side of centreline. This type of multi-planar reconstruction was conducted twice: once with the measurement plane parallel with the minor axis plane and again with the measurement aligned with the major axis plane. The two independent reconstructions not only aided in checking the symmetry of the coherent structures but also made it possible to confirm that the orientation of the measurement plane relative to the vortices did not affect whether a vortex could be resolved or not. Because no substantial differences were seen in the structures detected in the two reconstructions, only the reconstruction with the measurement plane parallel with the major axis plane is presented in this paper. That reconstruction is composed of 22 measurement planes (43 after mirroring) which are distributed every 0.5 mm in the $y$-direction.

A programmable timing unit (PTU) controlled and synchronized the release of the camera shutters and the firing of the lasers in short two-image bursts. The time between the two images captured by the cameras varied from $8\ \mathrm {\mu }\mathrm {s}$ to $15\ \mathrm {\mu }\mathrm {s}$ depending on the speed of the jet for the given test condition. The three-component velocity field was computed in the measurement plane using the DaVis 8.4.0 software from LaVision to find the stereoscopic cross-correlation between the two successive image pairs obtained from the two cameras. Processing was completed in three passes with a single pass utilizing $64 \times 64$ pixel interrogation windows followed by two passes with $32 \times 32$ pixel interrogations windows. All passes used a 50 % interrogation window overlap, and the result after masking and processing was a $154 \times 153$ vector field with a resolution of 4.25 vectors $\mathrm {mm}^{-1}$.

The uncertainties in the $u$, $v$ and $w$ velocity component measurements were also computed using the correlation statistics method in DaVis 8.4.0 (Wieneke Reference Wieneke2015). For consistency with the uncertainty reporting convention used by Wieneke (Reference Wieneke2015), the root-mean-square of the uncertainty across the velocity field for all vectors inside the jet (i.e. in the region where $u(x,y) > 0.01*u_{max}(x)$) was computed. This yielded a single root-mean-square-averaged uncertainty of each velocity component in a given ensemble averaged phase-locked or time-averaged measurement. The uncertainty values at all jet conditions and actuator phases measured are reported as ranges in table 1. The uncertainty in velocity was also propagated to derived quantities, like circulation and volume flux, assuming normally distributed error (Sciacchitano et al. Reference Sciacchitano, Neal, Smith, Warner, Vlachos, Wieneke and Scarano2015) using the method outlined by Sciacchitano & Wieneke (Reference Sciacchitano and Wieneke2016). The propagated uncertainty in circulation is indicated on the relevant plots with uncertainty bars. However, the uncertainty in volume flux was too small for the uncertainty bands to be easily distinguished from the data in the figures and so the uncertainty bands have been omitted. For reference, the fractional uncertainty in the volume flux of the full jet ranged from $\sigma _Q/Q = 0.08\,\%\text { to }1.83\,\%$.

Table 1.

Uncertainty in centreline plane SPIV measurements.

Both time-averaged and phase-locked SPIV data were obtained in this experiment. For time-averaged measurements, 500 image sets were captured throughout the cycle of the actuator, and the average of the vector fields computed from them provided the mean jet characteristics. These measurements were taken by setting the PTU to trigger the SPIV system at a frequency which was not a factor of the actuator frequency to progressively alias the signal. Alternatively, for phase-locked measurements, 400 image sets were taken on the centreline planes, or 200 images sets on the off-centreline planes, at a single phase angle of the jet driving frequency for multiple actuation cycles. Ensemble averaging the vector fields computed from these image sets reveals the periodic structures at that point in the actuation cycle. Phase-locking was achieved by setting the PTU to trigger off a specific point of the voltage waveform from the function generator. Three phases of the actuator, evenly spaced across the cycle, were obtained on each measurement plane for each jet operating condition. The precise set of phase angles used varied for the different jet operating conditions in order to capture the vortex rings at similar points in the bifurcation process.

The phase lag between the driving signal output from the function generator and the fluid dynamic response of the jet was quantified by comparing the transient data from the hot-wire probe inserted into the actuator orifice to the unamplified driving signal. Accordingly, the phase angles, $\phi$, quoted in this paper are corrected to represent the phase of the jet inside the orifice. Therefore, the values of $\phi$ can be approximately interpreted as $\phi =0^{\circ }$ representing the start of the blowing cycle, $\phi =90^{\circ }$ the point of peak blowing, $\phi =180^{\circ }$ the start of the suction phase, and $\phi =270^{\circ }$ the point of peak suction.

Two techniques were used to identify vortices in the SPIV data. In post-processing planar SPIV data the $\mathcal {Q}$-criterion was utilized for vortex identification, where $\mathcal {Q}$ is the second invariant of the velocity gradient tensor. In this technique, vortices are identified as regions of the flow where the magnitude of rotation exceeds the magnitude of strain, which distinguishes the vorticity in the core of a vortex from the vorticity generated by the shear in a boundary layer, for example (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988; Jeong & Hussain Reference Jeong and Hussain1995). Another virtue of the $\mathcal {Q}$-criterion is that it is Galilean invariant. The equation for $\mathcal {Q}$ is

(2.4)\begin{equation} \mathcal{Q} = \tfrac{1}{2}\left(u^2_{i,i}-u_{i,j}u_{j,i}\right)=-\tfrac{1}{2}u_{i,j}u_{j,i}=\tfrac{1}{2}\left(\left\|\boldsymbol{\varOmega}\right\|^2-\|\boldsymbol{\mathsf{S}}\|^2\right), \end{equation}

where $\|\boldsymbol {\varOmega }\|$ and $\|\boldsymbol{\mathsf{S}}\|$ in (2.4) are the Frobenius norms of the rotation rate and strain rate tensors, respectively. For vortex detection, a threshold was applied to the $\mathcal {Q}$ field which was either a percentage of the maximum $\mathcal {Q}$ in the field or a fixed value greater than zero. To identify the location of the vortex axis, the point of maximum vorticity was interpolated from the SPIV grid data in the region selected by the $\mathcal {Q}$-criterion.

The swirling strength criterion was used to visualize the vortices in the 3-D reconstructed jet. This technique examines the magnitude of the imaginary part of the velocity gradient tensor eigenvalues, denoted $\lambda _{ci}$ in (2.5) (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999).

(2.5)\begin{equation} \boldsymbol{\nabla} \boldsymbol{u} = [\boldsymbol{\nu_r} \enspace \boldsymbol{\nu_{cr}} \enspace \boldsymbol{\nu_{ci}}]\begin{bmatrix} \lambda_r & 0 & 0\\ 0 & \lambda_{cr} & \lambda_{ci}\\ 0 & -\lambda_{ci} & \lambda_{cr}\end{bmatrix}[\boldsymbol{\nu_r} \enspace \boldsymbol{\nu_{cr}} \enspace \boldsymbol{\nu_{ci}}]^{\textrm{T}}. \end{equation}

When the eigenvalues of the velocity gradient tensor include a complex conjugate pair, then $\lambda _{ci}$ is non-zero and swirling motions are present in the flow. Setting a threshold greater than zero on $\lambda _{ci}$ selects only regions of the flow where the strength of the swirling motion is higher than the threshold. The swirling strength criterion is also Galilean invariant and was found to be slightly more capable than the $\mathcal {Q}$-criterion at detecting weak secondary structures without also accentuating noise in the experimental data.

Isosurfaces of phase-locked $\lambda _{ci}$ and $|\boldsymbol {\omega }|$ are presented in several figures in §§ 3 and 4 to demonstrate the 3-D shape of the deforming vortex rings. The isosurfaces are plotted at a level which is a percentage of either the maximum value of the quantity throughout the whole flow field or the maximum observed within a particular generation of the vortex ring, i.e. the local maximum. Different isosurfaces levels were experimented with to determine which values most effectively highlight the flow features to be discussed. For example, isosurfaces at 20 %–50 % $\lambda _{ci,max}$ delineated the most recently formed vortex in its entirety, 5 %–20 % $\lambda _{ci,max}$ captured two generations of vortex rings and values of 3 %–4 % $\lambda _{ci,max}$ were effective in visualizing secondary structures.

Identifying vortex locations in instantaneous PIV measurement fields – as opposed to the average field of a phase-locked measurement – is challenging due to measurement noise and turbulence, which is why it is usually not attempted. However, notable success was achieved in this study by computing the derivatives of velocity needed to find $\mathcal {Q}$ using a spatially filtered differentiation technique. This technique utilizes a circular stencil with a diameter roughly the size of the vortices one wishes to detect, which filters out noise and emphasizes sufficiently large regions of coherent swirling motion. It is conceptually similar to the normalized angular momentum computation described by Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) for vortex identification. For example, the spatially filtered angular velocity at a point can be scaled to obtain vorticity via (2.6), where $\boldsymbol {x_p}$ is the position were vorticity is being computed, $\boldsymbol {x_i}$ are the adjacent data points within a prescribed radius from the point $\boldsymbol {x_p}$, and $\boldsymbol {n}$ is the normal of the measurement plane.

(2.6)\begin{equation} \boldsymbol{\omega}(\boldsymbol{x_p})=\frac{2}{N} \sum_{i=1}^{N} \frac{(\boldsymbol{x_i}-\boldsymbol{x_p}) \times (\boldsymbol{V_i}-\boldsymbol{V_p}) \boldsymbol{\cdot} \boldsymbol{n}}{|\boldsymbol{x_i}-\boldsymbol{x_p}|^2}. \end{equation}

While the resultant vorticity or $\mathcal {Q}$ fields are attenuated somewhat by this process, they are very useful for identifying regions where a vortex exists. In all other circumstances, spatial derivatives were computed using a $3 \times 3$ kernel method similar in concept to the one outlined by Raffel et al. (Reference Raffel, Willert, Wereley and Kompenhans2007) for computing vorticity and strain. Vorticity is plotted throughout this paper using red to signify positive vorticity and blue negative vorticity according to a right-handed coordinate system, regardless of whether the rotation appears to be clockwise (CW) or counter-clockwise (CCW) in the planar view presented.

2.3. Experimental conditions

SPIV data were obtained for eight synthetic jet conditions in this experiment, covering a range of $Re$ and $St$. A 3-D reconstruction of the resolution used in this study, for three phases of the actuator and the mean jet, generated a large amount of data. Therefore, the 3-D reconstruction was only performed at one jet condition. For all other cases, data were obtained on the centreline major axis and minor axis planes only.

The jet conditions tested are summarized in table 2 along with the case naming convention. To study the influence of varying $St$ at a constant $Re$, the frequency of actuation was varied from 280 Hz to 715 Hz while the average blowing velocity was held constant. Conversely, to vary $Re$ while holding $St$ constant, both the actuator frequency and average blowing velocity had to be adjusted concurrently. Five Strouhal numbers (cases St1–St5) and four Reynolds numbers (cases Re1–Re4) were tested. The sweep in $Re$ and $St$ shared one common point which was chosen as the condition for the 3-D reconstruction (case Re3/St3).

Table 2.

Synthetic jet conditions tested.

Several characteristics of the early vortex rings produced by the actuator were extracted from the SPIV data at the various conditions tested. One such parameter was the initial circulation strength of the vortex ring. By the definitions used in this study, which are typical for reporting synthetic jet flows, the vortex circulation strength varied with both $Re$ and $St$. For an ejection of fluid from an orifice with an arbitrary velocity programme, $u_o(t)$, the generated circulation is

(2.7)\begin{equation} \varGamma = \int_{0}^{T/2} \tfrac{1}{2} u_o^2 (t)\,\textrm{d}t. \end{equation}

Essentially, the circulation production depends on the strength of the blowing, its duration and its programme. Glezer (Reference Glezer1988) showed that the initial circulation of laminar vortex rings produced by different velocity programmes collapse to a straight line relation of the form

(2.8)\begin{equation} \frac{\varGamma}{\nu} = C_1P\frac{U_o L_o}{2\nu}, \end{equation}

where $C_{1}$ is a fit constant and $P$ is the programme factor. Utilizing (2.1), (2.2) and (2.3), it can be shown that the left-hand side of (2.8) relates to $Re$ and $St$ as follows (note factors of two absorbed into the constant $C_{2}$):

(2.9)\begin{equation} C_1\frac{U_o L_o}{2\nu}=C_2\frac{U_o h}{\nu}\frac{U_o T}{h}=C_2Re\frac{1}{St}. \end{equation}

The peak circulation of the vortex rings formed in the synthetic jet was computed for all cases by integrating the SPIV field data around circuits defined using the contour of vorticity at 5 % of the maximum vorticity experienced for that jet condition. These data are presented in figure 8 using the non-dimensional form represented by (2.9). The circulation in the minor axis plane (black symbols) is lower than the major axis plane (grey symbols) due to a suspected branch in the vortex tube discussed in § 4.1. The data follow a linear trend (dashed lines) and confirm that the initial vortex ring circulation increases with $Re$ but decreases with $St$.

Figure 8.

Non-dimensional maximum circulation, $\varGamma /\nu$, of the vortex ring at formation for the various $Re$ and $St$ tested. Black symbols represent circulation computed in the minor axis plane while grey symbols represent the major axis plane.

The shape of a pseudo-elliptic vortex ring, specifically the initial $AR$, has a strong influence on whether bifurcation occurs and how it progresses (e.g. bifurcation vs trifurcation). Accordingly, an approximate initial vortex ring $AR$ was computed for the various conditions tested by measuring the distance of separation between the centres of the counter-rotating vortices in the major and minor centreline planes before axis switching deformations started. The results vary from $AR=7.9$ to $8.4$ but had an average of $AR=8.2$. Trends in $AR$ with changes in $Re$ and $St$ were too small relative to the uncertainty of the method to draw any conclusions. What can be stated with confidence is that the $AR$ of the vortex rings for all conditions tested exceeded the critical $AR$ for bifurcation reported in the literature.

The size of the vortex core may also influence the progression of vortex ring bifurcation (Cheng et al. Reference Cheng, Lou and Lim2016). To quantify the vortex size, the circulation was found by integrating velocity along concentric circular paths of increasing radius centred on the vortex axis. Integration circuits were prevented from overlapping when two vortices of opposite sign were close to each other. The resulting $\varGamma$ vs $r$ distribution was then fit with an Oseen–Lamb vortex profile (figure 9a). From the Oseen–Lamb profile fit, the core radius, $R_c$, was extracted as the distance between the vortex axis and the point where the average tangential velocity, $u_{\theta }$ reached a peak (figure 9b). Using this technique, the initial vortex core radius was found to vary from $R_c/h= 0.62$ to $0.79$ in the major axis plane and $R_c/h= 0.73$ to $0.86$ in the minor axis plane for all conditions tested. This calculation was performed on the individual frames that constitute the time-averaged data sets for a vortex height of $x/h=3.5$, which was the approximate point where the vortex was fully formed, i.e. had stopped accumulating additional circulation. The slightly higher values reported for $R_c$ in the minor axis plane are at least partially due to the asymmetry in the distorted cores of the proximate minor plane vortices. Any trends with increasing $Re$ or $St$ were again too small relative to the uncertainty in the calculation technique to allow for analysis. However, it is apparent from the data that the vortex rings produced at all conditions had wide cores relative to the ring's overall dimensions. Note that uncertainty bars have been omitted from figure 9 because the uncertainty was too low for the bars to be distinguishable from the data symbols. Furthermore, the diamond symbols representing the phased-locked SPIV data in figure 9 have the same spacing as the vectors in the measured velocity field and demonstrate that the vortex is well resolved in the measurements.

Figure 9.

Vortex profile from phase-locked data ($\blacklozenge$) with best fit to Oseen–Lamb vortex () and computed core radius () showing $(a)$ normalized circulation, $\varGamma /\nu$, and $(b)$ normalized tangential velocity, $u_{\theta }/U_o$, from minor axis plane of case Re3/St3, $\phi =90^{\circ }$.

4.1. Vortex formation bridges

The first structures of interest form with the vortex ring during the blowing phase. They start as regions of diffuse, counter-rotating cross-stream vorticity ($\omega _y$) inside the vortex ring, i.e. between the primary vortex pair, and are bisected by the major axis plane (figure 10a). As the vortex ring propagates away from the orifice and contracts due to self-induction in the major axis plane, these cross-stream structures advect upwards and inwards towards the jet axis. The cross-stream vorticity is at first concentrated by the convergence of the vortex ring and is then rolled up and over the stronger primary vortex pair. This induced motion is visible at $\phi =210^{\circ }$ in figure 10(c) where the structures appear as arcs in the ensemble average frame due to variability in their instantaneous positions around the stronger primary vortex (i.e. jitter). As the cross-stream structures are compressed against the primary vortices, their vorticity annihilates and they essentially disappear by the time they have revolved to the outside of the jet (figure 10e). In the time-averaged data, these structures appear as streamwise streaks of vorticity inside the jet (figure 13). They were observed for all conditions tested, although the number of counter-rotating sets increased with $St$.

In the domain where the cross-stream vorticity exists, the flow is subject to an elevated strain rate due to the deformations of the adjacent vortex ring. Therefore, $\mathcal {Q}$ and $\lambda _{ci}$ criteria fail to capture the full extent of these structures and tend to misidentify the location of their centroids. To visualize them it is thus preferable to plot isosurfaces of vorticity magnitude, $|\omega |$, as presented in figure 23. At this phase, the cross-stream vorticity appears to be part of a bridge which is connected to and spans the vortex ring. Although these structures are not well identified by the $\lambda _{ci}$ criterion, in figure 18(b), the points where they connect to the vortex ring are visible as lumps projecting from the vortex surface. When circulation was computed around the early vortex ring using integration circuits which excluded the secondary cross-stream vorticity, the circulation was found to be variable around the vortex ring (figure 8). More specifically, the circulation at the ends of the vortex ring in the major axis plane was higher than the circulation at the middle of the vortex ring in the minor axis plane. However, when the integration circuits were redrawn to also encompass the secondary vorticity, the circulation was found to be remarkably consistent around the ring (figure 14).

Figure 23.

Isosurface of $|\boldsymbol {\omega }|$ at 15 % of $|\boldsymbol {\omega }|_{max}$ for case Re3/St3 at $\phi =210^{\circ }$: $(a)$ axonometric projection, $(b)$ side view and $(c)$ top view.

The 3-D topology and circulation data suggest that the cross-stream vorticity may be a vortex tube branch off of the primary vortex ring. Like the bridge vortices formed during bifurcation, these too appear to arise due to vorticity reconnection. The vortex rings generated by the SJA are in direct contact across the minor axis. By $\phi =300^{\circ }$ the vortex cores in the minor axis plane vorticity contours (figure 10d) and the $\lambda _{ci}$ isosurface (figure 21b) are also elongated in the streamwise direction. This proximity and core distortion in anti-parallel vortices establishes a contact zone across which vorticity reconnection is known to occur. Vorticity annihilation by viscous cross-diffusion reduces the circulation of the primary structure at the contact zone while vorticity reconnection feeds bridges that contain the redistributed vorticity. This continues until axis switching of the vortex ring separates the vortices.

Bifurcation or trifurcation of vortex rings with sufficiently large $AR$ shortly after formation was predicted in simulations by Kiya et al. (Reference Kiya, Toyoda, Ishii, Kitamura and Ohe1992) and Menon & Soo (Reference Menon and Soo2004). The present results may therefore represent a marginal case where vorticity reconnection occurs but is insufficiently complete to cause the vortex ring to bifurcate or cease axis switching. Kiya et al. (Reference Kiya, Toyoda, Ishii, Kitamura and Ohe1992) indicated this may have been the case in their complementary experiment with an $AR=8.3$ pseudo-elliptic vortex ring. Therefore, we call these new regions of vorticity ‘formation bridges’ to differentiate them from the bridge vortices which occur after the vortex ring has switched axes. As the vortex ring undergoes axis switching deformations, these formation bridges are stretched and then annihilated against the tops of the primary vortices. Similar dynamics is presented by Kida et al. (Reference Kida, Takaoka and Hussain1991a) in figure 4(b) (v–viii).

Secondary structures, which are potentially related to the formation bridges outlined here, have been observed in similar studies. Amitay & Cannelle (Reference Amitay and Cannelle2006) detected counter-rotating sets of cross-stream vorticity inside the major axis of rectangular orifice synthetic jets with $AR=50.8, 76.2$ and $101.6$. They found that, as stroke length increased, the number of structures decreased. This is equivalent to our finding that increased $St$ led to more of the secondary structures. Amitay & Cannelle (Reference Amitay and Cannelle2006) also found that, as $AR$ of the jet was increased, so did the number of structures. Van Buren et al. (Reference Van Buren, Whalen and Amitay2014a, figure 14) supports the same conclusion for $AR=6, 12$ and $18$ orifice synthetic jets. The dependence of the number of counter-rotating cross-stream vorticity sets on stroke length and orifice $AR$ may both be related to azimuthal waves in the vortex ring. When jet conditions or vortex ring geometry allow for higher modes of azimuthal fluctuations, the vortex ring could make contact in more places along the major axis, and so more sets of formation bridges would be generated.

4.2. Vortex pinch-off and secondary vortex rings

At the lowest $St$ tested, $St = 0.039$ (St1), a unique secondary structured formed below the vortex ring which was also associated with the formation process of the vortex. At these conditions a significant amount of unincorporated vorticity remained in the shear layer upstream of the vortex ring above the orifice. As the primary vortex ring advected upwards, instability in the shear layer led to the formation of a secondary vortex ring (labelled SG3 in figure 24).

Figure 24.

Contours of normalized out-of-plane vorticity for case St1 in the $(a{,}c{,}e)$ major axis and $(b{,}d{,}\,f)$ minor axis planes for phases $(a{,}b)$$\phi =110^{\circ }$, $(c{,}d)$$\phi =230^{\circ }$ and $(e{,}\,f)$$\phi =350^{\circ }$. Black contour lines drawn at select levels to highlight structures (, $\omega <0$; , $\omega >0$).

Vortex rings form when the shear layer produced by fluid expulsion through an orifice or nozzle rolls up into a spiral of concentrated vorticity. As more turns of the spiral accumulate, the vortex ring grows in diameter and strength (Didden Reference Didden1979). This process cannot continue indefinitely, however. Gharib et al. (Reference Gharib, Rambod and Shariff1998) determined that the growth of circular vortex rings reaches a critical limit beyond which the vortex ring cannot accumulate additional circulation. Physically, this happens because, as the vortex circulation increases so does the self-induced velocity of the vortex ring. At the limiting condition, the speed of the vortex ring exceeds the speed of the shear layer and so the ring outruns the shear layer and pinches off from it. Gharib et al. (Reference Gharib, Rambod and Shariff1998) showed that the non-dimensional formation time or formation number for a circular vortex ring was $L/D\approx 4$. The vorticity produced by the piston stroke, after $L/D=4$ has been exceeded, remains in a detached tail behind the pinched-off vortex ring. Conversely, experimental studies and computational analysis of 2-D or infinite vortex pairs have shown that either these vortices do not have a critical formation number or that this number is significantly higher than $L/D=4$ (Afanasyev Reference Afanasyev2006; Pedrizzetti Reference Pedrizzetti2010). The computational analysis by Domenichi (Reference Domenichi2011) predicted that elongated vortex rings with curved ends and a rectilinear mid-section could exhibit both behaviours at different points along their circumference, i.e. pinching off on the curved ends but not on the rectilinear mid-section. However, O'Farrell & Dabiri (Reference O'Farrell and Dabiri2012) found experimentally that oval and elliptical vortex rings pinched off everywhere at the same time, with $L/D\approx 3-4$.

The case with the highest stroke length in this study, St1, had an $L_o/D_{eq}=6.2$ which exceeds the formation number proposed by Gharib et al. (Reference Gharib, Rambod and Shariff1998). The more moderate St2 had an $L_o/D_{eq}=4.2$, only slightly above the limit, while the remaining six cases were well below $L_o/D_{eq}=4$ (table 2). Figure 24 depicts the vorticity field in the major and minor axis planes for case St1. During the blowing phase, a relatively strong shear layer can be seen upstream of the developing G3 vortices (figure 24a,b). In the minor axis plane the shear layer is still attached to the vortex, however, in the major axis plane it is already apparent that the vortex ring has started pinching off. Earlier pinch-off in the major axis plane is to be expected because the tightly curved ends of the elongated vortex rings will travel downstream faster than the rectilinear mid-section. During the next phase, instability causes the now separate shear layer to develop into a secondary vortex ring denoted SG3 (figure 24c,d). These results can be compared with Re3/St3 ($L_o/D_{eq} = 3.2$), which represents a typical case where there is no secondary vortex ring formation (figure 10).

As the primary vortices in case St1 converge in the major axis plane, the secondary vortices trail behind them and converge at a similar rate (figure 24c,e). In the minor axis plane, however, as the primary vortices spread apart, they induce the secondary structures upwards and between them (figure 24d,f). Therefore, parts of the secondary structure appear to partially pass through the primary structure leading to a highly distorted shape. This partial pass-through of the secondary vortex ring also appears to have been captured in flow visualizations by Adhikari (Reference Adhikari2009).

The circulation in G3 and SG3 was computed in the major axis plane at $\phi =230^{\circ }$ around manually defined integration circuits with the saddle point in the vorticity contours between the primary and secondary vortices used to demarcate where one vortex ended and the other began. The results revealed that 66.8 % of the total circulation generated in the flow field by that actuator stroke was contained within the primary vortex ring or, in other words, G3 was twice as strong as SG3. In theory, the vortex ring will pinch off at a formation number of $L/D=4$ so for the case St1, which had a stroke length of $L_o/D_{eq}=6.2, 64.5\%$ of the total circulation generated should accumulate in the primary vortex ring. This is remarkably close to the results obtained from the SPIV measurements.

Interestingly, the secondary vortices appear to be weaker in the minor axis plane than in the major (compare figures 24c and 24d). This is not due to the variable pinch-off predicted computationally by Domenichi (Reference Domenichi2011) because this trend is opposite the theory that elongated vortex rings accumulate more circulation in rectilinear sections. Instead, the cause is vorticity reconnection brought about by the collision of opposite sides of the secondary vortex ring as it is squeezed through the primary ring in that plane. Evidence of this vorticity reconnection in the secondary vortex ring can be seen in the final measurement phase where secondary bridge vortices (SB3) appear in the expected location (figure 24e). In the same figure, it is also noteworthy that the primary vortex, G3, now appears to have approximately equivalent strength to SG3 despite G3 being much stronger in the prior phase. The culprit is again vorticity reconnection; however, this time it occurs in the primary vortex ring and in the opposite direction. The resulting bridge vortices (B3) can be found in figure 24(f).

In the minor axis plane at $\phi =350^{\circ }$ a discreet secondary vortex SG3 is no longer obvious in the phase-locked data (figure 24f). Instead, the streamwise streak of vorticity suggests variability in the position of SG3 between the instantaneous measurement frames that make up the phase-locked ensemble average. What is apparent is that the secondary vortex ring is in a similar downstream location as the colliding region of the primary vortex ring. The interaction between the primary and secondary vortex rings has important implications for vortex ring bifurcation at these conditions, as will be discussed further in § 5.

A similar region of pinched-off vorticity was also identified in the vorticity field of case St2 during the blowing phase. However, the secondary structure was difficult to precisely locate in the other two phase-locked data sets due to its low strength. Regardless, the results presented in § 5 suggest that even in this marginal case the pinched-off vorticity may weakly interact with the primary vortex ring in some manner.

The conditions which produced vortex pinch-off in this study are not extreme, in terms of typical SJA operation. Indeed, evidence of this behaviour is common, if not always discussed, in the other publications of synthetic jet data. For example, vortex pinch-off can be observed in the vorticity fields presented by Amitay & Cannelle (Reference Amitay and Cannelle2006) (figures 8(c) and 8(f) where $L_o/h=49.4$ is equivalent to $L_o/D_{eq}=6.14$) and Van Buren, Whalen & Amitay (Reference Van Buren, Whalen and Amitay2014b) (figure 12 where $St=0.046$ and $0.028$ are equivalent to $L_o/D_{eq}=4.54$ and $7.46$, respectively).

4.3. Streamwise corner vortices

Later in the jet phase and upstream of the primary vortex ring, four streamwise vortices – specifically two counter-rotating sets – are apparent in the coloured isosurfaces of $\lambda _{ci}$ marked ‘SCV’ in figure 25. These streamwise vortices stretch from the ends of the rectangular orifice up to mid-section of the newest vortex ring. Similar coherent structures have been detected in continuous jets from elliptic and square nozzles (Husain & Hussain Reference Husain and Hussain1993; Grinstein & DeVore Reference Grinstein and DeVore1996), in isolated vortex rings from elliptic nozzles (Adhikari Reference Adhikari2009) and in synthetic jets from rectangular orifices (Wang et al. Reference Wang, Feng, Wang and Li2018a). They are the result of tilting and stretching induced by the deformed primary vortex ring. To understand their formation, it is helpful to think of the cylindrical shear layer that precedes the vortex ring as a train of coaxial vortex lines that encircle the jet. As the shear layer rolls up starting from its downstream end, the vorticity is concentrated into the nascent vortex ring core. A small amount of residual vorticity remains in the upstream section of the shear layer which has classically been referred to as the braid region of continuous jets (this is sketched as a dashed line in figure 3a). These unconsolidated braid vortex lines are much weaker than the vortex ring; therefore, their celerity and rate of self-induced deformation is much slower. Instead, the much stronger nearby vortex ring has a dominant influence on the changing shape of the vortex lines. As the highly curved ends of the elongated vortex ring move ahead of its middle, the greatest velocity induction on the braid vortex lines occurs in the region where their separation distance is at a minimum, i.e. the mid-section. Thus, they deform in a manner which is the opposite of the primary ring, i.e. the middle moves downstream faster than the ends (see dashed line marked ‘SCV’ in figures 3b–3d). The uneven induction from the primary vortex ring has the effect of tilting the originally spanwise oriented vorticity ($\omega _z$) into the streamwise direction ($\omega _x$). Furthermore, the tilting of these vortex lines tends to consolidate this sparse vorticity while stretching intensifies it into the streamwise corner vortices.

Figure 25.

Isosurface of $\lambda _{ci}$ at 4 % of $\lambda _{ci,max}$ for case Re3/St3 at $\phi =210^{\circ }$: $(a)$ axonometric projection, $(b)$ side view and $(c)$ top view with arrows indicating rotational direction (blue, streamwise corner vortices with $\omega _x<0$; red, streamwise corner vortices with $\omega _x>0$; grey, primary vortex ring).

The streamwise corner vortices exist in the four quadrants of square, rectangular and elliptic jets and therefore do not cross the centreline planes where experimental data are frequently obtained. Thus, they are often undetected in experimental studies of synthetic jets, as was true for the cases in this study that did not include a 3-D flow field reconstruction. Despite this difficulty, their presence may be significant. For one, as the vortices are tilted and intensified by stretching they induce flow normal to the jet axis. This tends to enhance entrainment (Liepmann & Gharib Reference Liepmann and Gharib1992; Husain & Hussain Reference Husain and Hussain1993) and can cause the ejection of flow out of the jet between each counter-rotating pair (Gutmark & Grinstein Reference Gutmark and Grinstein1999). In addition, the interaction of the streamwise vortices with the primary vortex ring can also inhibit axis switching (Grinstein & DeVore Reference Grinstein and DeVore1996) and influence the structure of the vortex ring in such a way as to accelerate instability, transition and breakdown (Liepmann & Gharib Reference Liepmann and Gharib1992; Gutmark & Grinstein Reference Gutmark and Grinstein1999; Adhikari Reference Adhikari2009).

4.4. Interstitial streamwise vortices

The streamwise corner vortices were not the only streamwise structures identified upstream of the vortex ring. Analysis of the three-component vorticity field revealed additional sets of counter-rotating structures along the jet's span. We refer to these as interstitial streamwise vortices due to their position between the corner vortices. Vorticity associated with the nascent interstitial streamwise structures is present at phase $\phi =210^{\circ }$, but the structures are too immature to be detected via vortex identification techniques like $\mathcal {Q}$-criterion or $\lambda _{ci}$. By $\phi =330^{\circ }$ some of the interstitial streamwise vortices have become coherent enough to appear in $\lambda _{ci}$ isosurfaces; however, a more complete view of the structures can be obtained by plotting the vorticity field. Figure 26 depicts isosurfaces of streamwise vorticity, $\omega _x$, coloured by rotational sense (blue, $\omega _x<0$; red, $\omega _x>0$) upstream of the primary vortex ring while other structures in the vicinity are plotted using isosurfaces of $\lambda _{ci}$ coloured grey. Six to eight sets of counter-rotating streamwise vortices can be identified in the figure. The outermost vortex sets are the streamwise corner vortices (SCV) identified in the prior phase, while the inner sets constitute multiple pairs of the interstitial streamwise vortices (ISV). Some of the interstitial streamwise vortices are also visible in the isosurfaces of $\lambda _{ci}$ in figure 27.

Figure 26.

Isosurface of $\omega _x$ at 1.5 % $\omega _{x,min}$ (blue) and 1.5 % $\omega _{x,max}$ (red) with isosurface of $\lambda _{ci}$ at 3 % of $\lambda _{ci,max}$ (grey) for case Re3/St3 at $\phi =210^{\circ }$.

Figure 27.

Isosurface of $\lambda _{ci}$ at 3 % of $\lambda _{ci,max}$ for case Re3/St3 at $\phi =330^{\circ }$: $(a)$ axonometric projection, $(b)$ side view and $(c)$ top view with arrows indicating rotational direction (green, arc-shaped vortices; grey, primary vortex ring and streamwise secondary structures).

Sets of streamwise vortices distributed along the span of synthetic jets have also been reported in other studies. For example, Amitay & Cannelle (Reference Amitay and Cannelle2006) speculated that streamwise structures were present inside their synthetic jets based on traces left on the centreline plane cross-stream vorticity fields. They hypothesized that the streamwise structures were related to azimuthal waves in the primary vortex ring during transition. However, we are not aware of any study which definitively identifies the formation mechanism.

The formation process may be related to rib vortex generation in the braid regions of shear layers (Bernal & Roshko Reference Bernal and Roshko1986) and continuous jets (Liepmann & Gharib Reference Liepmann and Gharib1992; Husain & Hussain Reference Husain and Hussain1993). However, unlike shear layers and continuous jets, there is no upstream primary or roll vortex in the synthetic jet at this phase to stretch the streamwise structure in the opposite direction as the downstream vortex ring. The timing of the appearance and development of the interstitial vortices may provide a hint as to how they form. At $\phi =210^{\circ }$ the streamwise vorticity associated with the vortices had just started to appear while by $\phi =330^{\circ }$ the interstitial vortices were well established. These phases correspond to the point in the actuation cycle shortly after the start of the suction phase and shortly before the end of the suction phase, respectively. Therefore, the suction cycle in synthetic jets may have an equivalent role in streamwise vortex formation as induction from an upstream vortex in continuous jets. However, if the interstitial vortices form from the vorticity sheet in the braid region, like streamwise corner vortices or rib vortices, it would imply that the sketch of vortex lines in figure 3 is oversimplified. The picture of how these structures come to be is obviously incomplete; as a result, additional work is required to ascertain the details of interstitial streamwise vortex formation in synthetic jets, and to determine if they are equivalent to the rib vortices found in other flows.

4.5. Arc-shaped structures

The self-induced deformations of the non-circular vortex ring also lead to the generation of secondary structures downstream of the ring. Slightly after their formation, these appear as two sets of counter-rotating vortices wrapped around the primary vortex ring in the green $\lambda _{ci}$ isosurfaces marked ‘ASV’ in figure 27. These structures are much weaker than the vortex ring, and so the isosurfaces containing their cores appear broken and irregular in the figure.

These structures form slightly later than the streamwise vortices at a phase when the ends of the vortex ring in the major axis plane are approaching the jet centreline. Similar structures have been reported by Adhikari (Reference Adhikari2009) and O'Farrell & Dabiri (Reference O'Farrell and Dabiri2012) for experiments with isolated elliptic vortex rings in water and by Wang et al. (Reference Wang, Feng, Wang and Li2018a) for a synthetic jet in water. Adhikari (Reference Adhikari2009) referred to them as ‘arc-like structures’. To the authors’ knowledge they have not been previously reported in synthetic jets formed in air, presumably due to the difficulty of identifying such weak structures at higher $Re$. At the conditions for the present experiment, the full topology of the arc vortices could not be extracted. Nor was sufficient temporal resolution obtained to determine the precise mechanism of their formation. Wang et al. (Reference Wang, Feng, Wang and Li2018a) suggested that the two counter-rotating, streamwise vortex sets on either side of the deforming vortex ring were part of a single loop-shaped structure based on their experiment with an $AR=3$ rectangular orifice synthetic jet in water. This possible topology is sketched out in figures 3(c) and 3(d) as a dash-dotted line marked ‘ASV’. However, unlike Wang et al. (Reference Wang, Feng, Wang and Li2018a), Adhikari (Reference Adhikari2009) and O'Farrell & Dabiri (Reference O'Farrell and Dabiri2012) who tested nozzle $AR$ values between 2 and 5, no discernible vortex pair was seen in the minor axis plane, which should bisect such a structure. Therefore, the arc-shaped vortices may have a different topology near the axis of moderate $AR$ synthetic jets.

It has been shown in other studies that arc-like vortices are formed from the same residual shear layer vorticity as the streamwise corner vortices discussed in § 4.3 (Adhikari Reference Adhikari2009). In the minor axis plane, the outward travelling sections of the vortex ring move downstream slower than the jet centreline velocity, and so a section of the shear layer passes through the centre of the vortex ring and overtakes its minor axis. The high extensional strain rate with spanwise elongation in the flow downstream of the vortex ring stretches the shear layer fluid, which causes the vorticity to be amplified and concentrated into the arc-shaped vortices. Details on the formation process of these structures are provided by O'Farrell & Dabiri (Reference O'Farrell and Dabiri2012). Continued induction from the comparatively stronger vortex ring eventually wraps the arc vortices around itself such that the ends of the secondary structure reside upstream of the ring.

The presence of these arc-shaped vortices has several effects on the jet flow field. First, the two arc-shaped vortices have a rotational sense which mutually induces higher momentum fluid inside the jet radially outward from the jet axis. Note that the formation bridges may also contribute to this jet outflow. The transported fluid appears as lobes of elevated momentum at the edge of the jet in the major axis plane around $x/h=8-20$ (labelled ‘L’ in figure 16a). These lobes occur at a streamwise position in the major axis plane where one would expect the jet to contract if only the axis switching of the primary vortex ring was considered. Therefore, we can infer that these secondary structures enhance mixing of the non-circular jet. Similar lobes of elevated momentum can be identified in figure 2(b) (ii) of Van Buren & Amitay (Reference Van Buren and Amitay2016) for an $AR=12$ rectangular orifice synthetic jet.

The arc-shaped vortices also leave a tell-tale sign in the major axis plane vorticity contours. The semi-circular ends connecting the two arc-shaped structures appear as two islands of elevated vorticity below the primary vortex ring at a height of $x/h = 8.5$ in the phase-locked data presented in figure 11. As the ends of the secondary structure are rolled around the travelling vortex ring, they sweep out two arcs that wrap back into the manifold of the jet. This appears as two ‘horn’ shaped projections of vorticity away from the jet axis in the time-averaged vorticity contours (labelled ‘H’ in figure 13). Similar vorticity ‘horns’ can again be found by careful inspection of the vorticity contour plot in figure 7(b) (ii) of Van Buren & Amitay (Reference Van Buren and Amitay2016) for an $AR=12$ rectangular orifice synthetic jet.

The ‘horn’ shaped vorticity contours were present for all conditions tested. They did not change appreciably with $Re$ but were dependent on $St$. Specifically, as $St$ increased, the angle the ‘horns’ made with the jet axis was reduced from roughly $45^{\circ }$ to almost $0^{\circ }$, indicating the arc-shaped vortices remained closer to the main jet. This decreasing protrusion of the arc-shaped vortices from the jet with increasing $St$ is reflected in a reduction of the major axis plane normalized entrainment rate in the upper jet.