The effect of a parallel free surface upon a submerged shallow synthetic jet

The interaction of a submerged shallow synthetic jet with a parallel free surface has gathered substantial interest, owing to its relevance to the operation of marine vehicles viz. ships that move close to the water surface. However, despite exhaustive research on the perturbation on a free surface, very few studies have experimentally investigated the effect of unconfined water surface height on the evolution and propagation of a submerged synthetic jet. This study experimentally investigates a synthetic jet submerged in a quiescent flow at shallow depths ejecting parallel to the free surface, through qualitative analysis and quantitative measurements. The qualitative study includes the visualization of the flow using Plane Laser Induced Fluorescence (PLIF) technique, whereas the velocity measurements are carried out by a five-beam Laser Doppler Velocimetry (LDV) probe. The primary objective of these analysis and measurements is to gain a physical insight into the characteristics of vortex ring in a synthetic jet ejected from a fixed orifice at different water depths and at varying Reynolds number. Our studies indicate that the behavior of the vortex rings drastically changes as the depth of the jet crosses a certain threshold. Although no significant change in the path of synthetic jet is observed beyond a threshold depth in our experiments, the jet trajectory shows an interesting dependence on the Reynolds number based on circulation for shallow water depths. It has been found that in the shallow depths, the vortex ring drifts upwards and interacts with the free surface at lower Reynolds number, whereas for larger Reynolds number, the vortex ring rebounds near the free surface and moves downward. Based on our observations, it can be concluded that the phenomenon of upward/downward flection of vortex rings depends both upon its circulation and water depth.


Introduction
A synthetic jet can be defined as a train of vortex ring that originates completely from the working fluid, carries no net mass flux and yet transfers linear momentum to the fluid flow (Mohseni and Mittal 2015).Figure 1 shows the process of a synthetic jet actuation.As the vortex ring assisted by its self-effected velocity, moves downstream and interacts with the fluid, it enlarges till a point the energy of the circulation is unable to hold its growing size, eventually leading to its breakup into smaller pieces.As shown in the figure, a synthetic jet actuation system is usually constituted of an oscillating diaphragm, a cavity opening and a fixed boundary that encloses to form a cavity.The oscillations of the diaphragm are created at one end of the cavity, while the other end consists of an orifice.The diaphragm oscillations regulate the fluid suction and ejection rates through the orifice.
Fig. 1.Schematic of a synthetic jet actuation using orifice-diaphragm apparatus Synthetic jets are widely used in the industrial applications such as flow separation control over bluff bodies (Amitay, et al. 1997, Feng andWang 2012), stalled airfoils (Amitay, et al., 2001)and in duct flows (Amitay and Pitt, 2002;Glezer, et al., 2005).Fung and Amitay (2002) and Jagadeesh et al. (2009) have used synthetic jet with a view to control and maneuver the unmanned aerial vehicles.However, the manoeuvring (particularly at reduced speeds) and docking strategies of autonomous or remotely operated underwater vehicles are more challenging and a compact zeromass pulsatile jet also known as synthetic jet actuators can be used (Linden and Turner 2004, De Luca et al. 2014, Gil and Strzelczyk 2016).These synthetic jets have attracted much attention because although the distant field behaviour of a synthetic jet is very much identical to that of a continuous jet, it has many other advantages over a continuous jet.The most notable benefit of a synthetic jet as compared to a plain jet is that there is a supply of momentum at no net mass flux.
Since the suction and ejection mass flow rates induced by the diaphragm are exactly identical, no net mass flow of fluid occurs.This absence of net mass flux obviates the need for any additional pumping fluid, pump, moving components such as valves, piston etc.In addition, experiments carried out on synthetic jets have demonstrated that the vortex pairs predominate in the vicinity of the synthetic jet (Kumar et al. 2019).These vortex pairs entrain more fluid from surroundings and along with a larger volume flux, a synthetic jet also widens more swiftly in comparison to a continuous jet (Swift, 2003).Other applications of synthetic jets have also been reported, such as jet-vectoring (Smith & Glezer, 2002), electronic cooling (Anna. and Amitay, 2006, Travnicek et al. 2011, McGuinn et al. 2013) and in mixing enhancement (Freund andMoin 2000, Al-Atabi 2011).The applications of synthetic jets in cross-flow have also been reported in boundary layer control and drag reduction (Lardeau et al. 2011, Laouedj et al. 2015, Buren and Leong 2016, Kumar and Karn 2018).
Among the different applications of synthetic jets, there are many potential situations where the synthetic jet comes near to the unconfined surface, i.e. in the shallow water depths.It is well known that the interaction of a jet and an unrestrained surface occurs in a plethora of environmental and geophysical flows, as well as in many technological applications.For instance, a typical example of a shallow jetfree surface association can be observed in the wake of a cruising ship.The cruising ships sailing in the ocean generate a hydrodynamic signature in its wake and the detection of this wake is crucial in tracing the trajectory of the ship over time.The synthetic jet in such shallow water depths can alter the wake signature of these ships, and thus may prove beneficial for naval defense applications.
In general, a shallow jet is bound by an unrestrained surface above and a solid wall at the bottom and spans to an unlimited horizontal extent in the transverse direction.A lot of research has already been reported in the area of shallow jets.For instance, the work of Rodi (1982) has established that the shallow water conditions have a significant impact on mixing of a jet and its entrainment due to restricted extent of the presence of ambient fluid above and below the jet.On a similar note, Shinneeb et al. (2011) has substantiated that the variation in average velocity and turbulence statistics depends on the depth of jet from the free surface and confinement in lateral direction of jet expansion for an isothermal round jet in shallow waters.The implications of shallowness on the averaged flow field are mainly determined by the reduction in the velocity dissimilarity over the mixing layer due to the friction on the bottom surface (Prooijen, 2002), where the vertical confinement largely affects the large vortical structures and rapidly decreases the number and spectrum of vortices in downstream direction (Shinneeb, et al., 2011).
Apart from jet depth from the free surface, there are other factors that affect the association of the jet vorticity to the unrestrained surface in shallow water jets as reported in the literature such as, jet angle with the free surface, surface contamination, jet velocity and shape of the jet (Gharib and Weigand, 1996;Archer, et al., 2007;Anthony, et al., 1991;Bernal, et al., 1989;Bernal and Kwon, 1998).Gharib and Weigand (1996) studied the connection and disconnection of a vortex ring advancing towards a free surface obliquely, whereas some studies have limited their investigations to the normal interaction (Archer, et al., 2007).Other studies have focused on the vortex ring moving parallel to the free surface (Bernal and Kwon, 1998).Bernal and Kwon (1998) has also reported experimental results with both clean and contaminated surface conditions and demonstrated that surface contamination has a profound influence on the behaviour of a single vortex ring.Along similar lines, Bernal, et al.(1989) investigated the effect of the surface contamination has on a vortex pair moving parallel to the free surface and reported that even a very small amount of surface active agent in bulk water significantly modifies the dynamics of vortex ring interaction with the free surface because the vorticity diffuses away from the surface and rolls up into secondary vortices, altering the global evolution of the vortex ring.In addition, investigations into the interaction of a laminar two-dimensional vortex pair with clean and contaminated free surfaces shows that rebounding of the primary vortex near the free surface is also possible due to generated secondary vorticity beneath the contaminated surface, but rebounding is absent for clean surface (Bernal, et al., 1989).In conclusion, it is worth pointing out here that the interaction of a vortex ring with a free surface is an intriguing problem in fluid dynamics owing to the complicated flow physics surrounding the vortex ring growth, instability and breakdown.Although a lot of research has been done on the different aspects of the interaction of vortex rings with the free surface, a systematic study on the behavior of synthetic jets in shallow water depths and the interaction of the formed vortex rings with the free surface, has not received considerable attention.
Thus, in the current paper, we focus our attention on exploring the effects of Froude number and Reynolds number variation on the evolution and propagation of a submerged synthetic jet.
This study experimentally investigates a synthetic jet that is submerged in a quiescent fluid medium and that ejects parallel to the unrestrained surface, through qualitative analysis and quantitative measurements.The primary objective of these analysis and measurements is to gain a physical insight into the characteristics of vortex ring in synthetic jet ejected from a fixed orifice at different values of Froude number and Reynolds number based on circulation.The current paper is structured as follows: Section 2 furnishes the details on the experimental setup and methodology.
Thereafter, in Section 3, we deal with the results and discussion on flow visualization and timeaveraged velocity measurements with a focus on variation with regard to Froude number and circulation based Reynolds number.Finally, the conclusions are presented in Section 4. Experiments are conducted to study the interaction of the vortex rings with the free surface, at different unconfined surface heights (H) and actuation frequencies (fact) of the jet.The experiments are carried out in a water tank of dimensions 1000 mm (Length) × 500 mm (Width) x 500 mm (Height).The water tank experimental setup is made with 12 mm acrylic sheets, allowing for complete optical view from all sides, with a free water surface that can be treated to be contaminated.The actuator is stationed at the middle of 500 mm × 500 mm wall in such a way that the spacing between the walls from all sides and the synthetic jet is 250 mm.The orifice diameter (D) in the current experiments is 13 mm and the distance from the center of the orifice to bottom and side walls is greater than 19D, ensuring that there are negligible blockage effects in the present study.The synthetic jet generation unit comprises a circular cavity of inner diameter (Dca) equal to 70 mm and a height of 60 mm.At one extreme of the cavity boundary, an arrangement to install an orifice plate of 3mm thickness is made and on the opposite end of the cavity a circular diaphragm is attached, which is further interlinked to the electromagnetic actuator through a stainless steel bar.The diaphragm is made of nitrile coated nylon rubber sheet with 1.5 mm thick having a circular steel disc (Dcy) of 40 mm diameter at center.A frequency controller unit drives the oscillations of the electromagnetic actuator through a sinusoidal voltage signal.The frequency in the flow field is also obtained from the velocity signal in the vicinity of the diaphragm and is compared with the frequency of the diaphragm measured through the oscilloscope.

Experimental Setup and Methodology
The PLIF technique is employed for the qualitative examination of the flow is done.Figure 2b illustrates the PLIF imaging plane (or, XZ plane) at the orifice center.XZ plane is chosen as the imaging plane in our experiments to capture the vortex roll-up in the side view.Figure 2c further provides details on the PLIF setup.A COHERENT ® INNOVA-90, 4W continuous wave Argon-Ion Laser is used in our experiments with a plano-concave cylindrical lens of focal length (15 mm) to generate a laser sheet of 1.5 mm thickness.The LIF images are captured with a resolution of 1280×1280 and a frame rate between 10 to 30 fps using a Basler ® A501-KC camera.Fluorescein and Rhodamine (6G) dyes are employed in our experiments.Velocity measurements in the cross streamwise and streamwise direction are carried out using five-beam LDV probe as presented in Figure 2d.Hollow glass spheres coated with silver in the diameter range of 8 µm -12µm are used as external flow seeding particles.At each point, the velocity data is sampled over a minimum 60 operational cycles, and an arithmetic mean is obtained.A maximum fluctuation of ± 3.7 % around the mean is observed in the measured velocity components considering uneven sampling.The fivebeam probe is stationed on a 2D traverse that moves in transverse plane (YZ-plane) with a resolution of 6.25 µm and an accuracy of ± 300 µm.
The physical parameters involved in the present study are frequency () and amplitude of oscillation of the actuator (∆), averaged jet exit velocity (Uo), ejection time in a cycle (Te), water free surface height (H), orifice diameter (D) and the geometrical parameters of the diaphragm such as inner cavity diameter (Dca) and diameter of the disk (Dcy).As suggested by Mohseni(2006), the ejected volume (Vd) for the diaphragm can be calculated as . The average jet exit velocity can be calculated as   =       ⁄ where   denotes the cross-sectional area of the orifice.In the current setup, since the actuation frequency and the diaphragm displacement are coupled with each other (Δ varied between 1.2 mm at 1 Hz to 2.2 mm at 6 Hz.), a similar inter-dependence is observed in the Reynolds number and Strouhal number, defined conventionally based on the jet ejection velocity, Uo.Therefore, an alternative dimensionless parameter has to be introduced that assimilates the effect of variation in both ejection velocity and actuation frequency.Hence, circulation of the synthetic jet at orifice exit is defined as  = ( 2 128 ⁄ )(Δ 2 )(   ⁄ ) 4 , similar to Jabbal, et al. (2006).Next, it may be noted that the generated vortex rings in the current study gain an induced velocity that is dependent on the circulation of the vortex ring.Thus, Reynolds number defined based on circulation is introduced as   =   ⁄ , similar to other prior studies (For instance, Melander and Hussain, 1989).Since the flow involves a free water surface and the experiments are conducted at different water depths, Froude number is defined conventionally as  =   √ ⁄ ; , where  and  represent the kinematic viscosity of water and acceleration due to gravity, respectively.
Further, the free surface height and the distances from the center of the orifice are nondimensionalized with respect to D, and are represented as ℎ ̃,  ̃,  ̃and  ̃respectively.

Results and Discussion
In the present work, the effect of the shallow water on the flow characteristic of synthetic jet is presented at three distinct values of water depths, ℎ ̃= 1.5, 3.0 and 5.0 and at four different frequencies.As shown in Figure 2b, the PLIF images are captured in a vertical plane aligned with synthetic jet axis and perpendicular to the free surface of the water.The velocity distributions are measured along the synthetic jet centerline at five different downstream locations ( ̃= 0.5, 2, 5, 8 and 12) in vertical transverse planes (referred hereafter as XZ planes).The terms free surface, unrestrained surface and unconfined surface have been interchangeably used in the current paper.

Flow visualization
In this section, the qualitative analysis of the vortex ring and the free surface interaction is studied through flow visualization.First, the effect of water depths from the unconfined surface (variation in Froude number) and second, the implications of variations in Reynolds number are discussed subsequently.The vortex interaction with the free surface is observed between  ̃= 10 to  ̃= 12 as shown in Figure 3a.This causes perturbations on the free surface and it has been observed that these perturbations are most dominant for ℎ ̃= 1.5.At the same shallow water depth, as   is increased from 370 to 875, the vortex ring interacts with the free surface earlier than before.As shown in Figure 3 (b), the vortex interacts with the free surface between  ̃= 6 to 8. In both the cases, vortex ring drifts upwards toward the free surface and the upper counter rotating vortices of the vortex ring lose its vorticity and disappear earlier than the lower counter rotating vortices.The underlying physics behind this has been well explained by Bernal and Kwon (1988), where it was reported that the association of the vortex rings and the unrestrained surface may result in the curtailment of vortex lines, diminishing the vortex ring upper half that revolves in counter clockwise direction as seen from the cross sectional view.When the momentum of synthetic jet is increased further by increasing the   to 2663 and 5904, the vortex ring moves along its centerline beyond  ̃= 5 and  ̃= 8 for   = 2663 and 5904, respectively.Further, a striking and interesting observation is that the leading vortex ring rather than interacting with the free surface, takes a curvilinear trajectory, bends and moves away from the free surface.However, the low strength vortices (trailing jet) moves upwards just like the above case.The reason could be the formation of a non-symmetric recirculation zone in the vicinity of the jet flow.In general, the synthetic jet creates a larger recirculation zone around it extended to several diameters downstream.When this recirculation zone is symmetric, the synthetic jet follows the normal path i.e. along its centerline.But when this recirculation zone is non-symmetric as in shallow waters, the synthetic jet changes its trajectory depending on the momentum.This may be one of the reasons of vortex core interaction at low   = 370 and 845, whereas the occurrence of rebounding at   = 2663 and 5904.
In Figures 3 (e-h), the non-dimensional water depth, ℎ ̃ has been increased from 1.5 to 3. At   = 370 and 845, the vortex rings either moves along or close to the synthetic jet centerline or with without interacting with the free surface.However, at   = 2663 and 5904, the leading vortex rings moves upward and shows a curvilinear deflection, although the trailing ring continues to move upward and interact with free surface, somewhere between  ̃= 8 to 10.As shown in Figure 4, at ℎ ̃= 5 the vortex rings moves close to the synthetic jet centerline and no significant variation is observed in the PLIF images at   = 370 and 845.At   = 2663, the trajectory of vortex ring oscillate along the synthetic jet centerline and some undulations are observed, whereas at   = 5904, in far field region after  ̃= 8, the path of leading and trailing vortex rings diverges.Most importantly, this deflection is however observed to be not so significant as compared to ℎ ̃ = 1.5 case at for the attainable range of   in our experiments.It is indeed possible that even at larger ℎ ̃ , the deflection of these vortex rings can be observed at higher   .Thus, it can be suggested that the deflection of the vortex rings from the free surface depends not only on   , but possibly also on Fr.It is also possible that the vortex deflection may be a function of    and a critical    may exist that determine the transition of the vortex ring deflection at each ℎ ̃.For instance, in our experiments at ℎ ̃= 1.5, an upward flection of the vortex ring trajectory is observed for    in the range of 22 -110, whereas a downward flection is noticed at much higher    around 1100 -5500.However, to substantiate this hypothesis, further systematic experiments will be required at a much larger range of variation of both   and .Moreover, since all the images shown here are instantaneous snapshots and there are many random effects that disturb the jet flow, these images cannot qualitatively characterize and completely capture the mechanism behind jet curvature.As mentioned before, the PLIF images are captured after few cycles of synthetic jet actuation, and it is indeed possible that such different trajectories may only happen during the initial stages of the synthetic jet actuation.Therefore, to gain a better insight into the flow physics, the time averaged LDV measurements are also discussed in the next section.

Velocity measurements
In the above section, the shown PLIF images reveal only the qualitative nature of the vortex ring.The PLIF images are indicative of the instantaneous flow field, and may be affected by free from attenuation of laser sheet intensity across flow the field.Therefore, velocity measurements are also carried out in the XZ plane using LDV to analyze the flow behavior quantitatively.

Time averaged velocity in the XZ plane
Similar to the velocity distribution along the centerline, time average velocity distribution for ℎ ̃= 1.5 (Figure 5a) in XZ plane shows the shifting of streamwise velocity towards the free surface.
In the near vicinity, or  ̃= 0.5, the unrestrained surface has no substantial effect.At  ̃= 2, the velocity distribution is relatively symmetrical, although the axis of symmetry is shifted upwards.
In far field, velocity distribution is unsymmetrical about the centerline of synthetic jet and drift upwards.The cross stream velocity distribution in Figure 5(b) shows a strong upward movement of vortex ring at  ̃= 0.5 which is insignificant between  ̃= 2 to 5. Between  ̃= 8 to 12, a strong upward movement of fluid is observed due to vortex interaction and breakdown at free surface as seen in Figure 3(a).For ℎ ̃= 3 and ℎ ̃=5, flow is akin to a deeply submerged jet and the unconfined surface has a very minimal impact on the velocity distribution.A backflow zone with negative velocity is realized in the near field at  ̃= 0.5 around the orifice boundary due to the generation of slight suction, in all cases.Again, a symmetry to both sides can be observed as far as this backflow region is concerned and no notable influence of the free surface is evident.In contrary to the  = 845, in Figure 5

Time averaged velocity along synthetic jet centerline
The time-averaged streamwise velocity (uavg) and cross-stream velocity (vavg) along the synthetic jet centerline at three different water depths of ℎ ̃= 1.5, 3 and 5 are measured for four different values of   = 370, 845, 2663 and 5904.The velocity field is studied by non- Finally, it is also worth mentioning the reasons behind some of the apparent differences behind the instantaneous PLIF and time-averaged LDV measurements.Although the PLIF clearly shows the different trajectories of the leading and trailing vortex rings, no such observation can be made from the time-averaged velocity measurements.The reason behind the different trajectories can be attributed to the accumulation of initiating vortex rings, which sometimes follow a slightly different path than the following train of vortex ring due partly to the variation in the local flow field.Also, some differences between the PLIF measurements and the LDV measurements is obvious since the former is instantaneous while the latter entails time-averaging.For instance, the trailing vortex ring, which is visible in the PLIF, is not evident in the time-averaged LDV measurements because of its extremely low velocities as compared to that of leading vortex rings.
Clearly, it is also possible that such different trajectories may only occur only during the initial stages of the synthetic jet actuation.Indeed, more controlled experiments (such as Time-resolved PIV measurements) are required to ascertain the phenomenon under question.In fact, the deflection of these vortex rings is a three-dimensional phenomenon and any systematic study into this necessitates that the entire three-dimensional flow physics be captured, to arrive at some substantial conclusions.To fully understand the mechanisms involved in the vortex interaction and rebounding at free surface, a time resolved velocity flow field analysis and PLIF is required in both the transverse planes, which could not be carried out in the current study because of experimental limitations and probably could be taken up by future studies.

Conclusions
The interaction of a submerged shallow synthetic jet with a parallel free surface is of considerable interest because of its relevance to the operation of marine vehicles.In the current study, we explore a submerged synthetic jet ejecting laterally to the free surface in motionless flow It is concluded that the vortex rings may interact or rebound from the free surface depending on its circulation strength and water depth.

Fig. 2 .
Fig. 2. Experimental setup (a) showing the cross-sectional view of the cavity, (b) the orientation of axes at the orifice center, (c) for flow visualization using PLIF technique, and (d) for velocity measurement using LDV.

Fig. 3 .
Fig. 3.Visualization of flow patterns in XZ plane at ℎ ̃= 1.5 and ℎ ̃ = 3.0, Reynolds number varying from   = 370  5904 as shown on the left.The horizontal white lines denote the free surface.The red dotted line depicts the average trajectory of trailing and leading vortex rings, obtained from a continuous sequence of images.

Fig. 4 .
Fig. 4.Visualization of flow patterns for a synthetic jet in XZ plane for ℎ ̃ = 5.0 at   equal to (a) 370 (b) 845 (c) 2663 and (d) 5904.The horizontal line on the top denotes the free surface.The red dotted line depicts the trajectory of trailing and leading vortex ring.
through PLIF and LDV measurements, to gain a physical insight into the characteristics of vortex ring in synthetic jet ejected from a fixed orifice at different water depths and various actuation frequencies of the jet.Experiments are carried out for three distinct non-dimensional free surface heights of 1.5, 3 and 5 from the centerline of the orifice, and at value of Reynolds number based on circulation in the range of 370-5904.It was found that at shallow water depths such as ℎ ̃ = 1.5, vortex rings gravitate to the unrestrained surface at lower value of Reynolds number of 370 and 845, but bounce back and move away from the free surface at greater values of Reynolds number 2663 and 5904.The results also show no significant impact of the free surface on the formation and propagation of synthetic jet at greater water depths of ℎ ̃ = 3 and ℎ ̃ = 5 in the range of Reynolds number attainable in our experiments.At high   , both the leading and trailing vortex rings are visible in the PLIF images, although the trailing vortex is not visible in the LDV measurements, because of time-averaging, that nullifies the low velocity as compared to the leading vortex ring.

Table 1 :
Table 1 below presents all the values of different non-dimensional parameters at which experiments are conducted.Values of the relevant non-dimensional parameters for all experimental conditions at different frequencies and free surface height.