3.1.1. Longitudinal plane visualisation

Figure 3 shows the instantaneous smoke visualisation images of the periodic vortex as a function of $ f_{b}$ and $ \alpha$ in the longitudinal plane ( $xy$ ). Figure 3(a) demonstrates the effect of $ f_{b}$ at a constant duty cycle of 50 %. The periodic vortex structure is shown for five different values of $ f_{b}$ ranging from 10 Hz to 50 Hz at the same phase, i.e. 20 ms after the start of the burst cycle. The vortex originates at the exposed electrode (see annotated visualisation at 10 Hz in figure 3 a) and convects along the wall at some angle $ \theta$ measured with respect to the wall surface. The vortex core location is marked with a white dot and denoted as $ x_{c}$ and $ y_{c}$ for horizontal and vertical coordinates, respectively. A clear difference in the periodic vortex behaviour can be observed at higher values of $ f_{b}$ . The periodic vortex structure is similar for lower and moderate values of burst frequency, i.e. for $ f_{b}=10$ Hz, 20 Hz and 25 Hz. However, the vortex topology changes at greater values of $ f_{b}=40$ Hz and 50 Hz. The downstream part of the periodic vortex rolls up in a clockwise direction, forming a dipole-like structure. The dipole formation takes place due to the boundary layer roll-up. The periodic vortex size is gradually reduced with increasing burst frequency. The vortex core is also located farther downstream for higher values of $ f_{b}$ . It indicates that the core velocity is a function of $ f_{b}$ . The upstream shear layer roll-up (spiral) behaviour is the same at all the values of $ f_{b}$ . The vortex topology is modified in the near-wall downstream region due to the interaction of the periodic vortex with the preceding vortex. These observations suggest that the burst frequency can control the periodic vortex size, speed, core trajectory and morphology.

Figure 3(b) demonstrates the effect of $\alpha$ at a constant burst frequency of 10 Hz. The periodic vortex is shown at the same phase, i.e. after 40 ms from the start of the burst cycle, for five different values of $ \alpha$ ranging from 50 % to 90 %. It may be noted that the axis limits are not the same in figures 3(a) and 3(b). Figure 3(b) indicates that the duty cycle does not influence the overall structure of the periodic vortex. The vortex size grows with an increase in $ \alpha$ in contrast to an increase of $ f_{b}$ . This can be verified by comparing the vortex sizes at $ \alpha =50\,\%$ and 90 %. The vortex core is located farther downstream and closer to the wall at higher duty cycle (see $ \alpha =80\,\%$ and 90 %). This means the vortex core travels at a small angle $ \theta$ relative to the wall for large duty cycles. It also indicates that the vortex core velocity and trajectory are both influenced by the duty cycle. The fluid present downstream of the periodic vortex is due to the remnant vortex. More remnant fluid is present at the higher duty cycle. This is because a higher duty cycle corresponds to a shorter idle time. A similar effect can be observed for higher values of $ f_{b} =$ 40 Hz and 50 Hz (see figure 3 a) where the absolute value of idle time is small because $ t_{b}$ is small. An identical observation was made in the previous work from the Schlieren measurements (Mishra & Panigrahi Reference Mishra and Panigrahi2017a ). Similar results confirm the accuracy of both measurement techniques and justify the negligible influence of seeding on the induced flow under the action of the electric field. Figure 3 confirms that the nature of periodic vortex structures is a strong function of $ f_{b}$ . Therefore, the role of $ f_{b}$ is investigated using the spanwise measurements in transverse and horizontal planes (see table 2), which have been presented in the following sections.

Figure 4.

Instantaneous smoke visualisation images of the periodic flow in transverse $ (zy)$ plane fixed at a distance of $x=10$ mm as a function of $ f_{b}$ , with $ \alpha =50\,\%$ . The results are shown at two different time instants, $t=$ $t_{0}-\Delta t$ and $t_{0}$ , where $t_{0}$ is the time instant at which the measurement plane intersects the vortex core centre and $\Delta t$ is equal to 3.2 ms.

Figure 5.

Instantaneous smoke visualisation images of the periodic flow in the $zy$ plane at a distance of $x=20$ mm as a function of $ f_{b}$ , with $ \alpha =50\,\%$ . The results are shown for two different time instants, $t=$ $t_{0}-\Delta t$ and $t_{0}$ , where $t_{0}$ is the time instant at which the vortex core intersects the selected measurement plane and $\Delta t$ is equal to 6.2 ms.

3.1.2. Transverse plane visualisation

Figures 4 and 5 show the instantaneous smoke visualisation images as a function of $ f_{b}$ in the transverse plane ( $zy$ ) fixed at $ x=10$ mm and $ x=20$ mm, respectively. The visualisation images of the moving vortex at two different instants, $ t=t_{0}-\Delta t$ and $ t=t_{0}$ , are shown, where $ t_{0}$ refers to the time instant when the vortex core intersects the selected measurement plane. At $ t=t_{0}-\Delta t$ , the vortex enters the measurement plane, where $ \Delta t$ is equal to 3.2 ms. Cell structures are formed due to spanwise instability present in the form of waves on the vortex periphery. The wave-like cell structures are initiated at $ f_{b} = 10$  Hz. At the moderate range, 10 Hz $\lt f_{b}\lt$ 40 Hz, the structures start separating from each other. This may be attributed to the increase in wave amplitude with the increase in burst frequency. The series of cell structures remain well organised over the entire span. The cell structures are separated at similar distances over the span (see the annotation in the visualisation images shown for 25 Hz at $ t=t_{0}-\Delta t$ ). A typical distance of 5 mm is observed containing one pair of cell-like structures for 10 Hz $\lt f_{b}\lt$ 40 Hz. It indicates that the spanwise wavelength is almost independent of $ f_{b}$ . At $ t=t_{0}$ , the formation of small streamwise vortices is observed at the top edge of the cell-like structures. The middle region (marked with a dotted line) corresponds to the vortex core, which remains nearly straight for $ f_{b}\lt 25$ Hz. At $ f_{b} =$ 40 Hz and 50 Hz, an instability wave appears in the vortex core region.

Figure 5 shows the spanwise structure of the periodic vortex in the downstream region i.e. transverse plane at $ x=20$ mm. At $ t=t_{0}-\Delta t$ , the cell structures observed at $ x=10$ mm are no longer present. However, several streamwise vortices are present which were in the nascent stage at $ x=10$ mm (see figure 4). They grow downstream and appear in pairs of counter-rotating vortices (vortex dipole) akin to mushroom-like structures. The streamwise vortex pairs are connected to each other like a string. The spanwise structure of a larger periodic vortex (corresponding to low $ f_{b}\lt 25$ Hz) is observed to be similar to the spanwise structure of the junction vortex reported by Allen & Naitoh (Reference Allen and Naitoh2007). The vortex core becomes irregular in comparison to the vortex core observed at $ x=10$ mm because of the instability growth. At $ f_{b} =40$ Hz, the vortex core is nearly diffused. The extent of vortex cross-section and spanwise structure in the wall-normal distance indicates the periodic vortex size reduces with an increase in burst frequency similar to the observation in the longitudinal plane (see figure 3).

3.1.3. Horizontal plane visualisation

Figures 6 and 7 show the instantaneous spanwise structure of the periodic vortex in the horizontal plane $(zx)$ fixed at $ y=3$ mm and 6 mm, respectively. Three different time instants are selected to demonstrate the instability development i.e. $ t/t_{v} =0.25, 0.5$ and 0.75, where $ t_{v}$ is the vortex residence time in the respective plane. The vortex residence time $ t_{v}$ is calculated by observing the sequential flow visualisation images. The time $ t/t_{v}=0$ is the time when the smoke sheet corresponding to the front edge of the vortex approaches the measurement plane and $ t/t_{v}=1$ is the time when the smoke trace corresponding to the rear edge of the vortex leaves the respective measurement plane. Figure 6 demonstrates a wavy pattern in the downstream edge with mushroom-shaped vortices. The waviness corresponds to the outer turn of the periodic vortex and may be related to the instability wave development due to which the mushroom vortices are generated. The wider and nearly straight strip of finite width corresponds to the vortex core region. The visualisation images reveal that the vortex core remains nearly straight in the spanwise direction at all burst frequencies, except for $ f_{b} =50$ Hz, where minor oscillation can be observed in the core region. The vortex core resides in the plane for a shorter time interval at lower burst frequency values than the higher burst frequency. This is because of the difference in vortex size and the core travelling angle, $ \theta$ , which is higher for lower values of $ f_{b}$ . The downstream wavy structures start decaying at a lower burst frequency and diffuse for $ f_{b} =40$ Hz and 50 Hz. This may be attributed to the roll-up of the boundary layer, resulting in the formation of dipoles. The vortex core also resides in the plane for a longer time for $ f_{b} =40$ Hz and 50 Hz due to the smaller core travelling angle. Figure 7 shows the instantaneous visualisation images in the horizontal ( $ xz$ ) plane at $y=6$ mm. The vortex core width is greater at $y=6$ mm compared with that at $y=3$ mm. This is due to the spatial growth of the vortices away from the wall. The oscillation present in the vortex core for $ f_{b} =50$ Hz at $ t/t_{v}=0.25$ is temporally damped at $ t/t_{v}=0.5$ and 0.75 indicating that the vortex tends to stabilise with time. However, some minor fluctuations can still be observed in the vortex for $ f_{b} \geqslant 20$ Hz. A comparison of figure 7 with figure 6 reveals that waves appear on the travelling vortices both in front and back regions during their development in space and time. The waves in the front part of the vortex demonstrate greater growth in the wall-normal direction compared with that of the vortex in the rear part. The comparison of figures 4 and 5 with figures 6 and 7 indicates that the mushroom-like vortex structures seen in the outer regime of the developing vortex in the $ zy$ and $ zx$ planes are correlated with each other.

Figure 6.

Instantaneous smoke visualisation images of the periodic vortex in the horizontal $ (zx)$ plane fixed at $y=3$ mm as a function of $ f_{b}$ at $ \alpha =50\,\%$ . The values of residence time, $t_{v}$ (ms), are 12, 12, 10, 8 and 16 for an $ f_{b}$ (Hz) of 10, 20, 25, 40 and 50, respectively. The residence time $t_{v}$ is the duration of the flow structure in the respective plane. The time $t=0$ is set at the instant when the vortex approaches the selected plane.

3.2.1. Starting flow and vortex merging

Figure 8.

Development of the starting flow in burst mode for $ f_{b}=50$ Hz and $ \alpha =50\,\%$ . Instantaneous velocity vectors are superimposed on vorticity, $ \Omega _{z}$ (s⁻¹), with a common contour scale at all the time instants. The actuation signal is shown at the top of the contour plot for reference and the end of each burst cycle is marked with a filled circle (red in colour) where $ t_{b}$ is the burst cycle period.

The starting flow induced by a burst-modulated DBD-PA at higher burst frequencies is composed of several vortices. The vortices merge with each other and the total number of vortices involved in the merging process is a function of the burst cycle parameters. Figure 8 shows the development of starting flow at the end of each burst cycle for a burst modulated DBD-PA operating at $ f_{b}=50$ Hz and $ \alpha =50\,\%$ with the help of velocity vectors superimposed on isocontours of axial vorticity, $ \Omega _{z}$ (s⁻¹). The flow field results are presented for nine subsequent burst cycles from $ t/t_{b}=1$ to 9, where $ t/t_{b}=0$ corresponds to the start of actuation. At $ t/t_{b}=1$ , we can observe the presence of a single vortex generated during the first burst cycle which is known as the starting vortex. A vortex of greater size and higher strength is observed at $ t/t_{b}=2$ . This is due to the starting vortex merging with the second vortex generated during the second burst cycle. This happens quickly as the second vortex travels faster than the first vortex (Mishra & Panigrahi Reference Mishra and Panigrahi2017a ). Similarly, the third vortex is also merged at the end of the third burst cycle. Consequently, the vorticity magnitude and size of the resulting vortex are further increased. The vorticity distribution inside the vortex keeps changing because of the vortex merging. It may also be observed that the orientation of the velocity vectors at $ t/t_{b}=2$ and 3 is not the same as that at $ t/t_{b}=1$ .

As time progresses, the vortex moves downstream, and the distance between the remnant vortex and the exposed electrode increases. The remnant vortex keeps moving due to self-induced velocity even when no plasma actuation exists. Consequently, the subsequently generated vortices could not complete the merging process because they must travel more to reach the remnant vortex core. For example, at $ t/t_{b}=4-6$ , one can observe two patches of peak vorticity. At later times ( $ t/t_{b}=7-9$ ), the resultant vortical structure ingests the incipient vortices. Overall, figure 8 demonstrates that the vortices generated by the actuator in each burst cycle during the initial period merge at the end of the burst cycle. Finally, a large vortical structure is developed, leading to the quasi-periodic flow.

Figure 9.

Merging process of the second vortex with the starting vortex. The instantaneous flow field is shown using the velocity vectors superimposed on isocontours of negative values of $\lambda _{2}$ (s $ ^{-2}$ ). The burst mode operating parameters are $ f_{b} =25$ Hz and $ \alpha =50\,\%$ . The $\lambda _{2}$ contours labelled as 1 and 2 indicate vortex 1 (starting vortex) and vortex 2 (second vortex), respectively.

We now select the first two consecutive burst cycles $ (t/t_{b} \leqslant 2.05)$ to describe the vortex merging phenomenon. Figure 9 demonstrates the vortex merging process of the starting vortex and the second vortex when the actuator is working at $f_{b}=25$ Hz with a duty cycle of 50 %. The core of the vortex or coherent structure can be identified using the properties of the velocity gradient tensor. The $ Q$ -criterion (Hunt et al. Reference Hunt, Wray and Moin1988) and $ \lambda _{2}$ -criterion (Jeong & Hussain Reference Jeong and Hussain1995) are widely used techniques for identifying the vortical structures. Jeong & Hussain (Reference Jeong and Hussain1995) demonstrated that the $\lambda _{2}$ -criterion outperforms the other vortex identification technique. In the planar flows, the regions defined by $ -\lambda _{2}$ and $ +Q$ are equivalent (Jeong & Hussain Reference Jeong and Hussain1995). The vortex merging process is described using the $ \lambda _{2}$ -criterion evolution. Figure 9 shows the instantaneous flow field using the velocity vectors superimposed on isocontours of $\lambda _{2}$ -criterion for six sequential time instants $ t/t_{b}=1.4-2.05$ . At $ t/t_{b} =1.4$ , the contours of $ \lambda _{2}$ explicitly present two co-rotating (same sign of vorticity) vortices. The starting vortex is labelled as ‘1’, and the second vortex is marked as ‘2’. As the second vortex convects downstream, the distance between vortex 1 and vortex 2 decreases. At $ t/t_{b} =1.65$ , vortex 2 comes closer to vortex 1. Vortex 1 and vortex 2 co-rotate at $ t/t_{b} =1.75$ and mutual interaction occurs between the two vortices. While co-rotating, vortex 2 becomes stronger and vortex 1 becomes weaker, as apparent from the magnitude of $ \lambda _{2}$ at $ t/t_{b} =1.75$ and 1.85. It may be noted that plasma forcing is stopped at $ t/t_{b} =1.5$ . Finally, vortex 1 and vortex 2 completely merge at $ t/t_{b} =2.05$ .

It is well known that the co-rotating vortices eventually merge to form a single vortex. Cerretelli & Williamson (Reference Cerretelli and Williamson2003) explained the physical mechanism behind the vortex merging using the vorticity field decomposition (described later in § 3.3.2) into symmetric and anti-symmetric components. The anti-symmetric vorticity field causes the co-rotating vortices to be pushed towards each other during the merging process. The anti-symmetric vorticity comprises two counter-rotating vortex pairs and their induced velocity readily pushes the two vortex centres to bring them together, leading to vortex merging. The critical ratio of core size, $ a_{c}$ , and separation distance, $ l_{c}$ , at which merging starts, is experimentally observed as $ a_{c}/l_{c} = 0.29\pm 0.01$ (Meunier & Leweke Reference Meunier and Leweke2001) and $ a_{c}/l_{c} = 0.24\pm 0.01$ (Meunier et al. Reference Meunier, Ehrenstein, Leweke and Rossi2002) for the merging of a pair of equal two-dimensional co-rotating vortices generated by impulsively rotated flat plates. We calculated the critical ratio at $ t/t_{b} = 1.4$ , 1.5 and 1.65, which gives $ a_{c}/l_{c} = 0.22\pm 0.01$ . A similar critical ratio of vortex core size and separation distance confirms the similarity of the vortex merging process in the DBD-PA to that of impulsively rotating flat plates. The minor difference in the value of the critical ratio may be attributed to the unequal strength of the vortex 1 and vortex 2 because they are generated at different instants.

Figure 10.

Formation of the periodic vortex from starting vortex for two different values of $ \alpha =50\,\%$ and 80 % at $ f_{b}=10$ Hz. The instantaneous isocontours of $ \Omega _{z}$ are shown for the first four consecutive burst cycles at the same phase, $ t/t_{b}=0.4$ . The contour levels are the same for each plot. The vortex core location is marked using a plus sign and mentioned inside the respective plot.

3.2.2. Formation of periodic vortices

The starting vortex is generated in a quiescent environment and the subsequent vortices generated during the burst cycle are exposed to the remnant flow from previous cycles. Figure 10 shows the comparison of vortices generated during four consecutive burst cycles from the start of the actuator for two different values of $ \alpha =50\,\%$ and 80 %, at a constant value of $ f_{b} =10$ Hz. The results are shown using the instantaneous flow field at the same phase, i.e. $ t/t_{b}=0.4$ , with the help of isocontours of vorticity, $ \Omega _{z}$ . For both values of $ \alpha$ , the starting vortex core is located nearly at the same location. The near-wall negative vorticity wraps the outer part of vortices in the upstream region, and the highest shear occurs at the interface of positive and negative vorticity. The vortices generated after the starting vortex are located farther downstream with respect to the starting vortex core location. There are minor differences between the structure and core location of the third vortex and fourth vortex for both duty cycles. The second vortex core is located farther downstream than the rest of the vortices. This is due to the mutual interaction between the starting vortex and the second vortex, which is similar to the leapfrogging of vortex rings. The leapfrogging phenomenon occurs when two identical coaxial vortex rings move along a common axis. Due to their mutual interaction, the leading vortex ring causes the rearward ring to shrink and accelerate. Consequently, the rearward ring moves faster. Concurrently, the induced velocity of the rearward ring causes the leading ring to grow and decelerate (Yamada & Matsui Reference Yamada and Matsui1978; Cheng et al. Reference Cheng, Lou and Lim2015) .

The second vortex travels farther downstream for $\alpha =80\,\%$ compared with that for $\alpha =50\,\%$ . The core of the third and fourth vortex is also located farther downstream for $\alpha =80\,\%$ than that for $\alpha =50\,\%$ . It indicates that a higher duty cycle increases the vortex core velocity. The geometric shape of the periodic vortex for $\alpha =80\,\%$ is also different. The vorticity distribution shows a circular pattern for the periodic vortex generated at $ \alpha =50\,\%$ and elliptic for $ \alpha =80\,\%$ (see the black dashed closed curve of the fourth vortex in figure 10). For a given $ f_{b}$ , the value of $\alpha$ determines the net energy addition per burst cycle and the time delay between the generation of the two consecutive vortices. A greater $ \alpha$ for a given $ f_{b}$ allows more time for the actuator to remain ON and a short idle time, increasing the net momentum exchange with the ambient fluid. This results in an increased convection speed of the vortex and greater vortex interaction (Mishra & Panigrahi Reference Mishra and Panigrahi2018).

Figure 11.

The $ u$ -velocity profile along (a) the streamwise $x$ -direction at $ y=5.2$ mm and (b) wall-normal $y$ -direction at $ x=6.2$ mm showing the differences in remnant flow from the start of the second vortex onward till the eighth vortex for $ \alpha =50\,\%$ and $ f_{b} =10$ Hz.

The differences between subsequent vortex structures are attributed to the initial conditions at the generation of the subsequent vortices. The differences in initial flow conditions at the start of each burst cycle can be better understood with the help of the velocity profiles extracted near the exposed electrode. Figure 11 shows the $ u$ -velocity profile at seven subsequent time instants just before the start of actuation in each burst cycle for $ \alpha =50\,\%$ and $ f_{b} =10$ Hz. The initial conditions are shown for the second burst to the eighth burst, i.e. at $ t_{0}+t_{b}-\Delta t$ to $ t_{0}+7t_{b}-\Delta t$ , where $ t_{0}$ refers to the start of the actuator and $ \Delta t=2$ ms is a constant. The profile is extracted along the streamwise distance at a fixed wall-normal distance of $ y=5.2$ mm and at a fixed streamwise distance of $ x=6.2$ mm.

It can be noticed that the velocity profiles during $5^{th}$ to $8^{th}$ vortex cycle overlap with each other in both streamwise and wall-normal directions. This data collapse indicates no change in initial remnant flow conditions for these burst cycles. The differences occur between $2^{nd}$ to $5^{th}$ vortex profile. A large deviation in velocity is observed in the downstream region and away from the wall. The velocity profile in the streamwise direction indicates that the second vortex velocity is higher in the region of $ 14\lt x\lt 24$ mm. The velocity profile in the wall-normal distance also indicates that the second vortex velocity has the greatest magnitude near the wall. This higher magnitude of velocity gradually reduces until the fifth profile. For the fifth to eighth vortex profile, there is a negligible difference in the near-wall region (up to $ y=12$ mm). Further away from the wall, $ 12\lt y\lt 25$ mm, the differences are still more significant in the case of the second vortex profile. This difference indicates that flow conditions are quite different near the exposed electrode before the start of the second burst cycle. This means that the second vortex is created in the fastest-moving remnant flow, which travels faster than the rest of the subsequent vortices. The small difference at later time instants suggests that vortices generated after the third vortex can be treated as repeating or periodic (or quasi-periodic) vortices at low and moderate burst frequencies.

Figure 12.

Periodic flow vortex core trajectory $ (x_{c},y_{c})$ over six consecutive burst cycles in the $xy$ plane, for (a) $ f_{b} =10$ Hz and (b) 20 Hz at a constant value of $ \alpha =50\,\%$ . The average value of the vortex core travelling angle $ \theta$ is mentioned inside the respective plot at the top left corner.

3.2.3. Core trajectory of periodic vortices

Figure 12 shows the trajectory of the periodic vortex core in the $ xy$ plane over six consecutive burst cycles, for $ f_{b} =10$ Hz and 20 Hz at a constant value of $ \alpha =50\,\%$ . The centre of the peak vorticity region is considered as the location of the vortex core. The core travelling angle, $ \theta$ (also refer to figure 3) is calculated as $ \theta =\tan ^{-1}(y_{c}/x_{c})$ . The vortex core travels at a smaller angle for the higher value of $ f_{b}$ . The periodic vortex for $ f_{b} =10$ Hz travels at an angle, $ \theta$ , of 29.8°. However, it travels at an angle of 22.7° for $ f_{b} =20$ Hz. For a greater value of burst frequency, $ \theta$ is reduced further to a lower value. The decrease in values of $ \theta$ is due to faster movement of the vortex in the streamwise direction and slower movement in the vertical direction for greater values of $ f_{b}$ . The starting vortex in the continuous mode travels at an angle of 31° (Whalley & Choi Reference Whalley and Choi2012). The periodic vortex travels at an angle smaller than that of the starting vortex (Mishra & Panigrahi Reference Mishra and Panigrahi2017a ). Xue et al. (Reference Xue, Gao, Xi and Liu2020) reported that vortex core travelling angle decreases from 23° to 17° as $ f_{b}$ increases from 5 to 50 Hz. In the present study, we also observe a similar behaviour and nearly the same value of $ \theta = 17$ ° at $ f_{b}=50$ Hz and $ \alpha =50\,\%$ .

At higher values of $ f_{b}$ , the vortex size is small and these small vortices march parallel to the wall due to less momentum transfer because of the shorter duration of plasma-generated forcing. This is the reason behind a long residence time of the spanwise core structure at higher values of $ f_{b}$ , as observed in figures 6 and 7. The scatter in the core trajectory is attributed to the nature of the periodic vortex, which is different in each burst cycle due to the nonlinear interaction of the periodic vortex with the remnant flow. The core location is more scattered in the downstream region due to greater unsteadiness attributed to the diffusion of the periodic vortex with the remnant flow.

3.2.4. Self-similarity of periodic vortices

The characteristics of the vortex generated due to impulsive action are self-similar over a time interval if scaled by proper scaling parameters. Cantwell (Reference Cantwell1986) reported a detailed mathematical treatment of the transient motion when a viscous incompressible fluid is forced from an initial state of rest. The self-similarity parameters derived by Cantwell (Reference Cantwell1986) have been used to test the self-similar behaviour of impulsively generated vortices such as junction vortex (Allen & Lopez Reference Allen and Lopez2007; Allen & Naitoh Reference Allen and Naitoh2007), starting vortex (Whalley & Choi Reference Whalley and Choi2012; Mishra & Panigrahi Reference Mishra and Panigrahi2017a ) and periodic vortices (Mishra & Panigrahi Reference Mishra and Panigrahi2017a ). The self-similarity parameters for space and velocity are given as follows:

(3.1) \begin{equation} \xi = \frac {x}{M^{1/3}t^{q/3}}\,, \quad \eta = \frac {y}{M^{1/3}t^{q/3}}\,, \quad \zeta = \frac {u}{M^{1/3}t^{(q/3)-1}}, \end{equation}

where $ M$ is the amplitude parameter for the force acting on the fluid, $ q$ is the forcing parameter, and $ q = 1,2$ and 3 for impulse, constant (step) and ramp forcing (Cantwell Reference Cantwell1986). We assume $M\sim \nu U_{a}$ and $q=2$ for (3.1), where $ U_{a}$ is the average velocity of the time-averaged mean flow field. The average velocity is calculated as $ U_{a} = U_{max}/2$ , where $ U_{max}$ is the absolute maximum of $ u$ -velocity in the time-averaged mean flow field. It may be noted that instead of $ U_{a}$ , vortex core velocity $ u_{c}$ was used in our earlier investigation (Mishra & Panigrahi Reference Mishra and Panigrahi2017a ). However, it is observed that $ U_{a}$ is nearly the same as that of $ u_{c}$ . The time can be made non-dimensional using the definition, $ t^{*} = tU_{a}^{2}/\nu$ , where $ \nu$ is the kinematic viscosity of the fluid. The non-dimensional time is an equivalent of Reynolds number (Allen & Naitoh Reference Allen and Naitoh2007).

Figure 13.

Phase-averaged flow field in similarity coordinates $ (\xi ,\eta )$ with self-similar velocity parameter $ (\zeta )$ at some selected non-dimensional time interval, $ t^{*}$ , for different values of burst frequency: (a) 10 Hz; (b) 25 Hz; and (c) 40 Hz. The duty cycle $ \alpha =50\,\%$ is constant. The periodic vortex core location, $ (\xi _{c},\eta _{c})$ , is marked using a black dot with the respective value mentioned inside the brackets.

Figure 13 shows the periodic vortex evolution in similarity coordinates $ (\xi ,\eta )$ for three different values of $ f_{b}=10$ Hz, 25 Hz and 40 Hz using the phase-averaged data. The velocity vectors are superimposed on the isocontours of $ \zeta$ and the results are presented for nearly the same values of $ t^{*}$ . The vortex core location in the similarity coordinates ( $ \xi _{c},\eta _{c}$ ) corresponding to ( $x_{c},y_{c}$ ) in the absolute coordinates is identified by the visual inspection of velocity vectors on similarity coordinates and marked with a black dot. The respective coordinate values are also mentioned inside the plots. It should be noted that the flow field presented in figure 13 for the vortex characterisation contains more than one vortex. However, the self-similar parameters are valid for the incipient vortex present in the upstream region. For $ f_{b}=10$ Hz, the core location occurs at $ (0.89,0.51)$ for $ t^{*} = 208$ and at $ (0.89,0.47)$ for $ t^{*} = 286$ . It can be noticed from the vortex core trajectory presented in figure 12 that the core location in the $ y$ -direction is undulating in nature. The lower value of $ \eta _{c} = 0.47$ at the later time can be due to these undulations. The undulations in the core trajectory are possible if the vortex core oscillates along its rotation axis. The starting vortex core location remains constant at $ (0.7,0.52)$ (Whalley & Choi Reference Whalley and Choi2012). For $ f_{b}=25$ Hz, the value of $ (\xi _{c},\eta _{c})$ is $ (1.1,0.45)$ at $ t^{*} = 209$ . For $ f_{b}=40$ Hz, we have marked two core locations $ (0.72,0.3)$ at $ t^{*} = 52$ and $ (1.3,0.48)$ at $ t^{*} = 210$ corresponding to incipient vortex and remnant vortex, respectively. It can be noticed that the core location of vortex for $ f_{b}=40$ Hz at $ t^{*} = 210$ is obscure. It may be noted that the values of $ \xi _{c}$ and $\eta _{c}$ from the phase-averaged data match well with those from the instantaneous data.

The differences between the core locations for nearly the same $ t^{*}$ at different burst frequencies can be attributed to the change in their velocity magnitude. This is also evident from the reduction in peak values of $ \zeta$ for 25 Hz and 40 Hz burst frequency. For $ f_{b}=10$ Hz, the value of $ t$ is large and convection velocity, $ U_{a}$ , is small, and for a greater value, $ f_{b}=25$ Hz and 40 Hz, the value of $ t$ is small and $ U_{a}$ is greater in magnitude. The vortex core location in the streamwise direction, $ \xi _{c}$ , shifts to the higher side for a greater vortex convection velocity. Thus, it indicates that the core location of the impulsively created periodic vortex on the similarity coordinates changes with the operating conditions. For a given duty cycle, $ \xi _{c}$ increases and $ \eta _{c}$ decreases with increasing burst frequency. A greater value of $ \xi _{c}$ and lower value of $ \eta _{c}$ at higher $ f_{b}$ is attributed to greater vortex convection velocity and smaller size of the vortex, respectively.

The core location $ (\xi _{c},\eta _{c})$ is a function of impulse magnitude. In the present study, the plasma forcing magnitude is the same at all test conditions because $ \phi _{pp}$ and $ f_{ac}$ are constant. The changes in burst parameters are responsible for altering the impulse behaviour in the present study. A larger value of $ f_{b}$ for a given duty cycle has smaller $ t_{on}$ . Considering that the total impulse is the product of applied force and the duration of application, it can be concluded that the total impulse is smaller at a higher burst frequency. The core location further indicates that the vortex growth rate is different in the streamwise and wall-normal directions. The vortex development is slower in the wall-normal direction for all the test conditions. The complete collapse of data in similarity coordinates depends upon the scaling parameters, and the core location ( $ \xi _{c}, \eta _{c}$ ) changes accordingly (Allen & Chong Reference Allen and Chong2000). The amplitude parameter and forcing parameter in (3.1) are two important quantities responsible for obtaining the perfect similarity coordinate. Here, the forcing parameter is the same for all the actuation conditions as the applied force is constant. However, the amplitude parameter depends on the average velocity, which is a function of burst signal parameters. As a result, the values of $ \xi _{c}$ and $\eta _{c}$ for a given set of measurements are affected.

3.2.5. Vorticity distribution of periodic vortex

Figure 14.

Non-dimensional axial vorticity distribution of the periodic vortex along (a) horizontal and (b) vertical section lines at seven sequential time instants. The experimental data are compared with the Lamb–Oseen vortex model. The horizontal $ (xx)$ and vertical $ (yy)$ section lines pass through the vortex core with the origin fixed at the vortex centre, as shown in inset 1. Inset 2 and inset 3 highlight the region where the experimental data deviates from the Lamb–Oseen curve. The actuator is operating at $ f_{b}=10$ Hz and $ \alpha =50\,\%$ .

There are several models for vortex characterisation in the literature, such as the Rankine vortex, Lamb–Oseen vortex and so forth. The vorticity distribution of the Lamb–Oseen vortex is expressed as

(3.2) \begin{equation} \Omega _{z} = \frac {\Gamma }{\pi a^{2}} \exp (-r^{2}/a^{2}), \end{equation}

where $ r$ is the radial distance in the cylindrical coordinate system $ (r,\varphi )$ , $ a$ is the vortex core radius and $ \Gamma$ is the circulation. The circulation per unit area, $ \Gamma /(\pi a^{2})$ , represents the peak magnitude of vorticity. The Lamb–Oseen model has good agreement with the laboratory results of vortex behaviour (Leweke et al. Reference Leweke, Dizès and Williamson2016). We have used the peak vorticity from the experiments. The core diameter was found by fitting the best match between experimental data and the Lamb–Oseen vortex model (3.2). Figure 14 shows the non-dimensional vorticity distribution and comparison with the Lamb–Oseen vortex model. The origin of the coordinate system is fixed at the centre of the vortex. The vorticity distribution across the vortex core is plotted for both horizontal $ (xx)$ and vertical $ (yy)$ directions. The vorticity distribution is normalised by the peak vorticity magnitude (i.e. the vorticity magnitude at the vortex centre) at the respective time instant. The vorticity profile of a developing periodic vortex is extracted at seven sequential time instants in the range of $ t/t_{b}=0.16-0.64$ with unequal time steps, where $ t/t_{b} =0$ corresponds to the start of the burst cycle. The distances are normalised by the instantaneous vortex core diameter $ (2a)$ . It can be observed that at all time instants ( $ t_{1} \text { to } t_{7}$ ), the experimental data collapse well onto a single curve within the vortex core, indicating the self-similarity of vorticity distribution.

The vorticity profile of the Lamb–Oseen model matches well with the experimental values inside the vortex core. However, the experimental values deviate outside the core region i.e. in the downstream region $ (x/a\gt 1$ , see the top plot of figure 14) and in the near-wall region ( $ y/a\lt -1$ , see the bottom plot of figure 14). The deviation in the front region of the travelling vortex is attributed to the straining effect due to the interaction between the vortex and the remnant flow field present in the downstream region. The profile in the near-wall region departs from the theoretical vortex model due to the viscous effects in the wall boundary layer and the acceleration of fluid in that region due to the plasma-generated forcing. Furthermore, it may be noticed that the experimental values in these two regions also do not collapse on each other. This may be attributed to the effect of the wall boundary layer and the transient evolution of the shear flow, respectively. The experimental data for later times at $ t/t_{b} =0.54$ and 0.64 in the near wall and the downstream regions shows a similar trend. This may be because the plasma forcing stops at $ t/t_{b} =0.5$ and the remnant flow is more diffused as time advances. As a result, the vortex is subjected to a very weak strain and gradually decays for $ t/t_{b} \gt 0.5$ .

3.3.1. Spanwise vorticity distribution

Figures 15 and 16 show the temporal evolution of vorticity, $ \Omega _{y}$ , in the $ zx$ plane for continuous and low frequency ( $f_b$ = 10 Hz and 20 Hz) burst mode actuation. It compares $ \Omega _{y}$ for the starting vortex of continuous mode and periodic vortices generated in burst mode. The instantaneous flow field is used for the starting vortex and phase-averaged results are used for the periodic vortices. Here, it may be noted that the differences between the instantaneous and phase-averaged results are insignificant (refer to (2.3)). Figure 15 shows the cross-stream vorticity, $ \Omega _{y}$ , for the measurement plane fixed at the wall-normal distance of $ y= 3$ mm. It can be observed that the starting vortex and periodic vortex at $ f_{b} =10$ Hz have similar temporal evolution of the vorticity distribution. The spatial distribution of $ \Omega _{y}$ at the last time step ( $ t=t_{0}+9.8\Delta t$ for the starting vortex and $ t=t_{0}+12.4\Delta t$ for the periodic vortex) also indicates that both have similar features. The $ \Omega _{y}$ distribution of the periodic vortex at $ f_{b}=20$ Hz is more regular compared with the starting vortex and the periodic vortex at 10 Hz. The strength or peak value of $ \Omega _{y}$ is highest for the starting vortex and reduces with an increase in burst frequency. The orientation of vorticity distribution changes with time for the starting vortex and periodic vortex at $ f_{b} = 10$ Hz. Our observation of other burst frequency cases indicates that $ \Omega _{y}$ has lower randomness at higher burst frequency. The $ \Omega _{y}$ is distributed in three rows and arranged in a fixed pattern with alternating signs of vorticity (see figure 15). This well-arranged pattern of $ \Omega _{y}$ grows, stretches and convects downstream with time. The stretching particularly dominates in the middle row containing the vortex core.

Figure 15.

Temporal evolution of cross-stream vorticity, $ \Omega _{y}$ , in the $ zx$ plane at $ y= 3$ mm for three different test conditions. The instantaneous flow field and phase-averaged results are presented for the starting vortex and periodic vortex, respectively. The initial phase, $ t_{0}$ , is arbitrarily chosen and the value of $ \Delta t =2$ ms is the same for all the plots.

The spatial development of $ \Omega _{y}$ can be noticed by shifting the $ zx$ plane to a larger wall normal distance. Figure 16 shows the temporal evolution of $ \Omega _{y}$ for the measurement plane fixed at a distance of $ y= 6$ mm. The vorticity distribution at $ y=6$ mm is different from the distribution of $ \Omega _{y}$ at $ y=3$ mm. The peak magnitude of $ \Omega _{y}$ is reduced for all the test conditions in comparison to that at $ y=3$ mm. A decrease in the peak magnitude of $ \Omega _{y}$ indicates the spatial decay of 3-D structure, which means that $ \Omega _{y}$ quickly evolves in space, and its effect is significant during the initial development stage of starting and periodic vortex. It may be noted that the top and bottom slices of the vortices represent the early and later stages of the vortex, respectively. The vorticity distribution looks organised at a later time, $ t= t_{0}+ 8.4\Delta t$ , for the starting vortex. This is in contrast to the periodic vortex, which shows an organised distribution at an early stage i.e. $ t=t_{0}+0.8\Delta t$ for $ f_{b}=10$ Hz and $ t=t_{0}$ for $ f_{b}=20$ Hz. It is to be noted that the three-row structure present at $ y=3$ mm is absent at $ y=6$ mm for $ f_{b} =20$ Hz. The behaviour of $ \Omega _{y}$ represents the characteristics of 3-D flow structures. Therefore, figures 15 and 16 indicate that the 3-D effects are spatially and temporally evolving during the development of both starting and periodic vortex.

Figure 16.

Temporal evolution of cross-stream vorticity, $ \Omega _{y}$ , in the $ zx$ plane at $ y= 6$ mm for three different test conditions. Instantaneous flow field and phase-averaged results are presented for the starting vortex and periodic vortex, respectively. The initial phase, $ t_{0}$ , is arbitrarily chosen and the value of $ \Delta t =2$ ms is the same for all the plots.

Figures 17 and 18 demonstrate the temporal and spatial evolution of streamwise vorticity, $ \Omega _{x}$ , in the $ zy$ plane fixed downstream of the exposed electrode at $ x=10$ mm and 20 mm, respectively. Figure 17 reveals that the orientation of $ \Omega _{x}$ changes with time and the variation of the starting vortex is more than the periodic vortices. The strength of $ \Omega _{x}$ of the starting vortex in continuous mode is also more than the periodic vortices in burst mode. It indicates that the strength of the 3-D structure is related to the nature of vortex development. The vorticity distribution of the periodic vortex at $ t=t_{0}$ is arranged in a regular pattern with alternating signs of vorticity. A comparison of $ \Omega _{x}$ distribution at time instant $ t=t_{0}$ with the respective flow visualisation images (refer to figure 4) suggests that the vorticity distribution corresponds to the circular-cell like flow structure. It may be noted that the circular cell size and spacing between them correlate well with the size of vorticity patches and their spacing. The alternating sign of vorticity indicates that the circular cells have rotational motion in the opposite sense with respect to each other.

Figure 17.

Temporal evolution of streamwise vorticity, $ \Omega _{x}$ , in the $ zy$ plane at $ x= 10$ mm for three different test conditions. Instantaneous flow field and phase-averaged results are presented for the starting vortex and periodic vortex, respectively. The initial phase, $ t_{0}$ , is arbitrarily chosen and the value of $ \Delta t =2$ ms is the same for all the plots.

At a farther downstream distance of $ x=20$ mm (see figure 18), the vorticity distribution is similar to the flow structures of flow visualisation images presented in figure 5. The strength of $ \Omega _{x}$ increases when the vortex moves downstream from $ x=10$ mm to 20 mm, indicating growth of the streamwise vorticity. The circular cells near the wall surface at $ x=10$ mm are absent in the downstream location. The $ \Omega _{x}$ distribution also indicates that the starting vortex creates more intense streamwise vortices than the periodic vortices. The starting vortex and periodic vortices at low values of $ f_{b}$ are comparatively larger. Hence, it can be concluded that a larger vortex of greater strength generates higher 3-D effects. It may be noted that the magnitude of $ \Omega _{x}$ and $ \Omega _{y}$ are quite comparable. The strength of $ \Omega _{x}$ and $ \Omega _{y}$ can also be compared with the strength of $ \Omega _{z}$ (for example, compare with the magnitude in figure 10). It suggests that the 3-D structures are strong and can play an active role in the flow control process.

Figure 18.

Temporal evolution of streamwise vorticity, $ \Omega _{x}$ , in the $ zy$ plane at $ x= 20$ mm for three different test conditions. Instantaneous flow field and phase-averaged results are presented for the starting vortex and periodic vortex, respectively. The initial phase, $ t_{0}$ , is arbitrarily chosen and the value of $ \Delta t =2$ ms is the same for all the plots.

The comparison of $ \Omega _{x}$ with the respective flow visualisation images (see figure 5) indicates that the distribution of $ \Omega _{x}$ corresponds to the streamwise vortices (mushroom-shape structures) that are generated due to the instability modes present in the outer part of the vortex. Allen & Auvity (Reference Allen and Auvity2002) observed that unstable waves appear on the vortex forming in front of the moving piston and concluded that the centrifugal instability is responsible for the formation of waves. Allen & Naitoh (Reference Allen and Naitoh2007) also observed these waves on the outer turn of the junction vortex and observed that the instability occurs on the outer turn of the vortex due to centrifugal forces leading to the formation of streamwise vortices. The instantaneous flow field shows a strong shear in that region, which acts as the source of instability (refer to figure 10). The high-shear zone is the region between positive and negative vorticity. The spanwise measurements presented in figures 15– 18 reveal that the burst frequency not only governs the size of periodic vortices, but also influences the interaction between the vortices which is responsible for altering the 3-D flow field induced by the DBD-PA.

3.3.2. Signatures of elliptic instability

The periodic vortices generated by the actuator are examples of convecting co-rotating vortex systems of unequal strengths. The co-rotating vortices are subjected to strain because of their mutual interaction. The strained vortices are influenced by two types of instabilities known as elliptic instability (Waleffe Reference Waleffe1990) and Crow instability (Crow Reference Crow1970). The elliptic instability reduces the time to dissipate the vortex by eventually breaking into turbulence. In this section, we try to relate the oscillations observed in the periodic vortex core to the elliptic instability phenomenon. In support of our argument, we have compared the theoretical predictions of streamlines and vorticity distribution for an elliptical flow. An elliptical flow is the superposition of pure rotation (about the $ z$ -axis) and a stagnation point flow (potential flow) with stretching at an angle $\beta = -45$ ° (Landman & Saffman Reference Landman and Saffman1987; Waleffe Reference Waleffe1989, Reference Waleffe1990; Leweke & Williamson Reference Leweke and Williamson1998). It is an exact solution of the two-dimensional Navier–Stokes equation. The base flow, $ U$ , for the elliptical flow, is given as

(3.3) \begin{equation} U = \begin{bmatrix} 0 &-\gamma - \epsilon &0 \\[1ex] \gamma - \epsilon &0 &0\\[1ex] 0 &0 &0 \end{bmatrix} \begin{bmatrix} x\\[1ex] y\\[1ex] z \end{bmatrix}. \end{equation}

Here, $\gamma = \Omega /2$ is the rotation and $\epsilon$ is the strain rate. For an elliptical flow, $|\epsilon |\lt |\gamma |$ . The stream function, $\psi$ , and the aspect ratio of elliptic streamlines, $E$ , are then written as

(3.4) \begin{equation} \psi = -0.5[(\gamma -\epsilon )x^{2} + (\gamma +\epsilon )y^{2}], \qquad E = \sqrt {(\gamma +\epsilon )/(\gamma -\epsilon )}. \end{equation}

Waleffe (Reference Waleffe1990) showed that an Ince equation governs the stability of an elliptical flow. He obtained the analytical expressions for the localised solution to small perturbations of the velocity $ (\boldsymbol {u} )$ and vorticity $ (\boldsymbol {\omega })$ . The corresponding solutions for $ \boldsymbol {u}$ and $ \boldsymbol {\omega }$ in cylindrical coordinates $ (r,\varphi ,z)$ are given as follows:

for the velocity,

(3.5) \begin{align} u_{r} = & \,\frac {\sqrt {3}}{2} e^{\sigma t} \left [ J_{0} \left (\sqrt {3}\mu r\right ) +\frac {1}{3} J_{2}\left (\sqrt {3}\mu r\right )\right ]\cos (\mu z)\sin \left (\varphi + \frac {\pi }{4}\right ), \end{align}

(3.6) \begin{align} u_{\varphi } = & \,\frac {\sqrt {3}}{2} e^{\sigma t} \left [ J_{0} \left (\sqrt {3}\mu r\right ) -\frac {1}{3} J_{2}\left (\sqrt {3}\mu r\right )\right ]\cos (\mu z)\cos \left (\varphi + \frac {\pi }{4}\right ), \end{align}

(3.7) \begin{align} u_{z} = & \,e^{\sigma t} J_{1} \left (\sqrt {3}\mu r\right ) \sin (\mu z)\sin \left (\varphi + \frac {\pi }{4}\right ); \end{align}

for the vorticity,

(3.8) \begin{align} \omega _{r} = & \,\sqrt {3} \mu e^{\sigma t} \left [ J_{0} \left (\sqrt {3}\mu r\right ) +\frac {1}{3} J_{2}\left (\sqrt {3}\mu r\right )\right ]\sin (\mu z)\cos \left (\varphi + \frac {\pi }{4}\right ), \end{align}

(3.9) \begin{align} \omega _{\varphi } = & \,\sqrt {3} \mu e^{\sigma t} \left [ J_{0} \left (\sqrt {3}\mu r\right ) -\frac {1}{3} J_{2}\left (\sqrt {3}\mu r\right )\right ]\sin (\mu z)\sin \left (\varphi + \frac {\pi }{4}\right ), \end{align}

(3.10) \begin{align} \omega _{z} = & \,2\mu e^{\sigma t} J_{1} \left (\sqrt {3}\mu r\right ) \cos (\mu z)\cos \left (\varphi + \frac {\pi }{4}\right ). \end{align}

Figure 19.

Qualitative description of elliptic flow: (a) streamlines and (b) corresponding vorticity field in the 3-D domain along with sample experimental images for comparison. The total flow is the sum of the base flow and perturbation field (Waleffe Reference Waleffe1990). The visualisation image shows oscillations in the vortex core.

Here, $ J_{i}, (i=0,1,2)$ are Bessel functions of the first kind, $ \sigma = ({9}/{16})({\epsilon }/{\gamma })$ is the maximum growth rate, $ t$ is the time and $ \mu = \lambda \sqrt {(E^{2}+1)/2E^{2}}$ is the axial wavenumber of the perturbation.

Figure 19 demonstrates the qualitative nature of an elliptical flow with the help of streamlines and vorticity distribution. The elliptical flow streamlines are given as $ \psi = \psi _{b} + \psi _{p}$ , where $ \psi _{b}$ and $ \psi _{p}$ denote the streamlines of base flow and perturbation, respectively. Here, $ \psi _{b}$ corresponds to streamlines of circular flow, i.e. $ \psi$ with zero strain $ (\epsilon = 0)$ in (3.4). The perturbation streamlines, $ \psi _{p}$ , are obtained from the perturbation velocity components (see (3.5)–(3.7)) using the stream function definition in axisymmetric cylindrical coordinate. The origin of the coordinate system is fixed at the centre of the unperturbed vortex. The unperturbed Lamb–Oseen vortex has circular streamlines (refer to figure 19 a). The perturbation has a two-lobe structure near the core. When the perturbation is added to the base flow, the vortex centre is displaced from its origin and the streamlines of total flow become elliptic.

Figure 19(b) shows the vorticity distribution of Lamb–Oseen vortex (see (3.2)) with and without vertical vorticity perturbation, $ \omega _{z}$ (see (3.10)) using the following numerical values of parameters, $ \gamma = -1 , \epsilon =0.8, a=1.75, t=1$ and $ \lambda =1$ . The base flow vorticity is organised in a circular form similar to the base flow streamlines. The vorticity distribution is constant over the span and the vortex core is straight. The perturbation vorticity $ \omega _{z}$ is arranged as patches of alternating sign and distributed in the form of Crow footprints. When the perturbation vorticity is added to the base flow, it displaces the core from its origin and introduces the sinusoidal oscillation into the core. The experimental visualisation image included in figure 19(b) for comparison with stability analysis results demonstrates the presence of oscillation of vortex core for $ \alpha =50\,\%$ and $ f_{b} =50$ Hz. Similar oscillations in the vortex core of small amplitude can be observed in the flow visualisation images presented in figures 4, 5 and 7. The successful comparison between the visualisation images and stability analysis confirms the role of elliptic instability in the generation of 3-D vortex structures by the DBD-PA.

Figure 20 shows the distribution of $ \Omega _{y}$ for $ f_{b}=40$ Hz and 50 Hz. It can be noticed that the spatial organisation of $ \Omega _{y}$ for 40 Hz and 50 Hz is similar to the spatial organisation of $ \Omega _{y}$ observed for 20 Hz (refer to figure 15). The stretching of the middle row at a higher burst frequency is less than that at 20 Hz. The characteristic pattern of $ \Omega _{y}$ is similar to the spanwise distribution of theoretical perturbation vorticity, $ \omega _{z}$ , as shown in figure 19. Similar spatial organisation of vorticity was also observed for the elliptical instability of counter-rotating vortex pairs in the underwater experiment by two rotating plates (Leweke & Williamson Reference Leweke and Williamson1998) and direct numerical simulation (Laporte & Corjon Reference Laporte and Corjon2000) studies. The elliptic instability also occurs in a co-rotating vortex pair (Meunier & Leweke Reference Meunier and Leweke2005) with a similar spatial pattern of vorticity.

Figure 20.

Phase-averaged vorticity, $ \Omega _{y}$ , from the $ zx$ plane at $ y= 3$ mm for $ f_{b}=40$ Hz and 50 Hz, at constant $ \alpha =50\,\%$ .

Figure 21.

Streamlines (as seen by an observer moving with the phase velocity, $u_{p}=f_{b}\lambda _{x}$ ) superimposed on the contours of $\lambda _{2}$ -criterion for the periodic vortex generated at $ f_{b}=20$ Hz and $ \alpha =50\,\%$ .

3.3.3. Galilean streamlines

To better demonstrate the signature of elliptical instability present in the periodic vortices, we next present a qualitative comparison of the vortex streamline. Figure 21 shows the streamlines superimposed on isocontours of $\lambda _{2}$ -criterion (described in § 3.2.1) corresponding to the phase-averaged vorticity field. The streamlines are constructed by subtracting the vortex convection velocity, $ u_{c}$ , from the instantaneous $ u$ -velocity field. The vortex core velocity is approximated as $u_{c}=f_{b}\lambda _{x}$ , where $\lambda _{x}$ is the distance between two consecutive matured vortices. The vortex core velocity $u_{c}$ can be termed as phase velocity or celerity, and the streamlines plotted in this manner mimic the flow patterns as observed by a moving observer. This is equivalent to the Galilean transformation, which changes the reference frame by subtracting a constant velocity from the entire velocity field. It helps to resolve all the vortices from an instantaneous flow field (Agrawal & Prasad Reference Agrawal and Prasad2002).

The streamlines are presented for the flow field generated by the DBD-PA at $ f_{b}= 20$ Hz and three different phases, $t/t_{b}=0.25, 0.5$ and $0.75$ , where $t=0$ is the start of a burst cycle. The periodic vortex streamlines are temporally evolving and streamlines are elliptic in shape, similar to that in figure 19. The streamlined patterns show a similar pattern between the two co-rotating anticlockwise vortices. The critical points of flow patterns can also be identified from the streamlines pattern (Perry & Fairlie Reference Perry and Fairlie1975; Perry & Chong Reference Perry and Chong1987). The critical point between the two coexisting periodic vortices is identified as a moving saddle point, $S$ , which is displaced in the direction of flow as the vortices march downstream. The mutual interaction between the vortices generates the saddle point. The hyperbolic streamlines around the saddle point indicate the local dominance of strain and the elliptic streamlines correspond to the vorticity-dominated flow. It suggests that the local strain-dominated zones influence the vortex rotational motion.

3.3.4. Vorticity decomposition

The perturbation associated with the instantaneous flow field can be separated from the base flow using the vorticity decomposition technique (Cerretelli & Williamson Reference Cerretelli and Williamson2003; Leweke et al. Reference Leweke, Dizès and Williamson2016). It can be used to deduce the vorticity responsible for the vortex merging (Cerretelli & Williamson Reference Cerretelli and Williamson2003) and perturbations leading to 3-D instabilities of the vortex (Leweke & Williamson Reference Leweke and Williamson1998). The total vorticity, $ \Omega (x,y)$ (i.e. $ \Omega _{z}$ in the present study), can be decomposed as

(3.11) \begin{align} \Omega (x,y) &= \underbrace {\frac {1}{2}\left [\Omega (x,y)+\Omega (x,-y)\right ]}_{\text {symmetric part:}\,\Omega _{s} (x,y)} + \underbrace {\frac {1}{2}\left [\Omega (x,y)-\Omega (x,-y)\right ]}_{\text {antisymmetric part:}\,\Omega _{a} (x,y)}, \end{align}

(3.12) \begin{align} \Omega (x,y) &= \Omega _{s} (x,y)+ \Omega _{a} (x,y), \end{align}

where $ \Omega _{s} (x,y)$ and $ \Omega _{a} (x,y)$ are the symmetric and anti-symmetric parts of the total vorticity, respectively. The anti-symmetric part becomes zero when the vortex is symmetric (e.g. circular).

Figure 22 shows the results of vorticity decomposition performed at a randomly selected phase for the periodic vortex generated by DBD-PA at $ f_{b}=10$ Hz and 20 Hz. The decomposition is carried out using the instantaneous vorticity, $ \Omega _{z}$ , from the longitudinal-plane measurement, and the coordinates are fixed at the centre of the vortex. For $ f_{b}=10$ Hz (refer to figure 22 a), the total vorticity has a complicated distribution that is neither elliptic nor circular. The symmetric part corresponds to the base flow and the anti-symmetric part represents the perturbations. The contour values of the anti-symmetric part are of significant strength. The peak magnitude of perturbation is approximately 20 % of the total vorticity maxima. For $ f_{b}=20$ Hz (refer to figure 22 b), the total vorticity distribution is elliptic in nature. It can be observed that the symmetric part has a circular vorticity distribution at the vortex core. The anti-symmetric part about the origin (i.e. vortex core centre) is well organised. The maximum contour values of the anti-symmetric part are aligned at an angle, $ \beta$ , in the range of 40°–45°. The angle $ \beta$ is defined as the angle between the axis line (horizontal or vertical) and a line starting from the origin and passing through perturbation maxima. The elliptical flow theory shows that the vortex core is displaced due to the stretching at $ \beta =45$ °.

Figure 22.

Vorticity decomposition of instantaneous flow field for (a) $ f_{b} =10$ Hz and (b) $ f_{b} =20$ Hz at a constant value of $ \alpha =50\,\%$ . The contour values in panels (a) and (b) are normalised by the local maxima of total vorticity and have common contour levels.

Leweke & Williamson (Reference Leweke and Williamson1998) observed that the anti-symmetric part exhibits a characteristic feature of a two-lobe structure with their maxima aligned in the stretching direction for each sign of the vorticity. The resulting structure is similar to the theoretical perturbation vorticity for the elliptic instability, as shown in figure 19. Here, in the anti-symmetric part, we can observe four lobes labelled as 1–4; see figure 22(b). The appearance of the four lobes is attributed to the presence of both positive (anticlockwise) and negative (clockwise) vorticity. The counter vortices readily push towards each other to produce oscillation in the vortex core. It can be noticed that the structure of lobes 1 and 2 are quite similar to the perturbation vorticity $ \omega _{z}$ , indicating the presence of elliptical instability in periodic vortices. The peak magnitude of perturbation is approximately 30 % of the total vorticity maxima for $f_{b}=20$ Hz and greater than the peak values observed for $ f_{b} =10$ Hz. We believe that the external strain on the periodic vortex is due to the remnant flow, which depends on the duty cycle $ \alpha$ and $ f_{b}$ .

The large periodic vortex generated at low $ f_{b}$ and low $ \alpha$ is subjected to the minor strain due to greater separation distance between the subsequent vortices. In that situation, the vortex will be primarily influenced only by the action of centrifugal instability and does not have signatures of elliptic instability. When the distance between the vortices is small at higher burst frequency, the centrifugal action at the vortex outer turn weakens due to vortex mutual interaction. At higher values of $ f_{b}$ , even for a low-duty cycle (say approximately 50 %), the idle time scale is small, resulting in a shorter distance between the vortex pair, and interaction between the vortices becomes more prominent, leading to elliptic instability. The centrifugal instability is related to the wave propagation on the vortex outer portion generating the vortices in the transverse plane, and the elliptic instability is responsible for producing the oscillation in the vortex core. The periodic vortices experience both the instability mechanism and the significance of their relative effect depends on the actuation conditions. The periodic vortices at moderate and higher $ f_{b}$ are dominated by the elliptic instability during its development.

We have presented the phase-averaged results of wall-normal and streamwise vorticity at multiple phases and for different conditions, which indicates that the instability patterns are regular. The elliptic nature of the streamlines and the nature of associated perturbation from the vorticity decomposition are also shown to demonstrate the signatures of elliptic instability. Furthermore, we observed the elliptic instability phenomenon at moderate frequencies where the strain is significant. It is worth mentioning that such regular flow patterns cannot be attributed to natural randomness or plasma discharge fluctuations that occur at a much smaller time scale. Therefore, it is justified to conclude that the spanwise vortex characteristics reported in the present paper are due to the flow instability phenomenon and not because of the spanwise randomness in the plasma discharge.

Performing the detailed stability calculations from the experimental measurements is difficult for the DBD actuation case as the perturbation flow field is neither directly accessible nor can it be obtained precisely early in the evolution, particularly as a function of time. For the simultaneous measurements of growth rate and wavelength, we require simultaneous measurement in two different planes using two synchronised cameras (Meunier & Leweke Reference Meunier and Leweke2005). The light sheet must also be oriented at the travelling angle of the vortex core to always pass through the core (Allen & Naitoh Reference Allen and Naitoh2007). The vortex core angle in the present case also depends on the actuation conditions, leading to difficulty in the precise mounting of the camera. Meunier & Leweke (Reference Meunier and Leweke2005) also observed that the wave amplitude is difficult to measure experimentally if the amplitude is not large enough in between subsequent images. These reasons restrict us from exercising the detailed stability calculations associated with the stability diagram.