The interaction of a submerged shallow synthetic jet with a parallel free surface has gathered substantial interest, owing to its re levance 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 surfac e 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 quantitat ive 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 obje ctive 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 depend ence 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 vo rtex 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. Keywords: Synthetic jet; Submerged jet; vortex rings; interaction between a free surface and submerged jet; LDV; PLIF 2 1. 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 transfe rs 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 b oundary 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 and Wang 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 3 vehicles. However, the manoeuvring (particularly at reduced speeds) and docking strategies of autonomous or remotely operated underwater vehicles are more challenging and a compact zero - mass 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 t he distant field behaviour of a synthetic jet is very much identical to that of a continuous jet, it has many other a dvantages 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 vort ex 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 and Moin 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 surfa ce occurs in a plethora of environmental and geophysical flows, as well as in many technological applications. For instance, a typical example 4 of a shallow jet – free 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 presen ce 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 f rom 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 obl iquely, whereas some studies have limited their 5 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 wit h 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 jet s 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 propagat ion 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 6 quantitative measurements. The pri mary 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 time - averaged velocity measurements with a focus on vari ation with regard to Froude number and circulation based Reynolds number. Finally, the conclusions are presented in Section 4. 2. Experimental Setup and Methodology 7 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. Experiments are conducted to study the i nteraction 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 8 (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 s uch 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 19 D, 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 a n 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 rubbe r 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 obtai ned from the velocity signal in the vicinity of the diaphragm and is compared with the frequency of the diaphragm measured through the oscilloscope. The PLIF technique is employed for the qualitative examination of the flow is done. Figure 2b illustrates t he 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 t o 30 fps using a Basler® A501 -KC camera. Fluorescein and Rhodamine (6G) dyes are employed in our experiments. Velocity measurements in the cross 9 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 o f ± 3.7 % around the mean is observed in the measured velocity components considering uneven sampling. The five - beam probe is stationed on a 2D traverse that moves in transverse plane (YZ -plane) with a resolution of 6.25 µm and an a ccuracy of ± 300 µm. The physical parameters involved in the present study are frequency ( 𝑓) 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 𝑉d= 𝜋∆8(𝐷𝑐𝑎2+𝐷𝑐𝑦2) ⁄ . The average jet exit velocity can be calculated as 𝑈𝑜=𝑉𝑑𝐴𝑜𝑇𝑒 ⁄ where 𝐴𝑜 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 betw een 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 𝛤=(𝜋2128⁄ )(Δ2𝑓)(𝐷𝑐𝑎𝐷⁄)4, similar to Jabbal, et al. (2006). Next, it may be noted that the ge nerated 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 𝑅𝑒𝛤=𝛤𝜈⁄, similar to other prior studies (For instance, Melan der and 10 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 𝐹𝑟=𝑈𝑜√𝑔𝐻⁄ ; , where 𝑣 and 𝑔 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 non - dimensionalized with respect to D, and are represented as ℎ̃, 𝑥̃,𝑦 ̃and 𝑧 ̃respectively. Table 1 below presents all the values of different non -dimensional parameters at which experiments are conducted. Table 1 : Values of the relevant non -dimensional parameters for all experimental conditions at different frequencies and free s urface height. Frequency of actuation, 𝑓𝑎𝑐𝑡 (Hz) Diaphragm displacement (Δ in mm) L/D Reynolds number, 𝑅𝑒𝛤 Froude Number 𝐹𝑟 Actual Rounded ℎ̃ = 1.5 ℎ̃ = 3.0 ℎ̃ = 5.0 1.02 1 3.58 1.67 370 0.06 0.04 0.033 1.60 2 4.18 2.33 845 0.13 0.09 0.073 3.65 4 5.93 3.03 2663 0.42 0.30 0.23 5.88 6 8,35 4.55 5904 0.93 0.66 0.51 3. 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, ℎ̃= 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 th e synthetic jet centerline at five different downstream locations ( 𝑥̃= 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 u sed in the current paper. 11 3.1. 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 𝑥̃= 10 to 𝑥̃= 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 ℎ̃= 1.5. At the same shallow water depth, as 𝑅𝑒𝛤 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 𝑥̃= 6 to 8. In both the cases, vortex ring drifts upwards toward the free surfa ce 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 tha t 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. ℎ̃= 1.5 ℎ̃= 3.0 𝑅𝑒𝛤=370 12 𝑅𝑒𝛤=845 𝑅𝑒𝛤=2663 𝑅𝑒𝛤=5904 Fig. 3. Visualization of flow patterns in XZ plane at ℎ̃= 1.5 and ℎ̃ = 3.0, Reynolds number varying from 𝑅𝑒𝛤=370 𝑡𝑜 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. When the momentum of synthetic jet is incre ased further by increasing the 𝑅𝑒𝛤 to 2663 and 5904, the vortex ring moves along its centerline beyond 𝑥̃= 5 and 𝑥̃= 8 for 𝑅𝑒𝛤 = 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 r ecirculation zone in the vicinity of the jet flow. In general, the synthetic jet creates a larger recirculation zone around it extended to 13 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 𝑅𝑒𝛤 = 370 and 845, whereas the occurrence of rebounding at 𝑅𝑒𝛤 = 2663 and 5904. In Figures 3 (e -h), the non -dimensional water depth, ℎ̃ has been increased from 1.5 to 3. At 𝑅𝑒𝛤 = 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 𝑅𝑒𝛤 = 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 surf ace, somewhere between 𝑥̃= 8 to 10. As shown in Figure 4, at ℎ̃= 5 the vortex rings moves close to the synthetic jet centerline and no significant variation is observed in the PLIF images at 𝑅𝑒𝛤 = 370 and 845. At 𝑅𝑒𝛤 = 2663, the trajectory of vortex ring oscillate along the synthetic jet centerline and some undulations are observed, whereas at 𝑅𝑒𝛤 = 5904, in far field region after 𝑥̃=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 ℎ̃ = 1.5 case at for the attainable range of 𝑅𝑒𝛤 in our experiments. It is indeed possible that even at larger ℎ̃ , the deflection of these vortex rings can be observed at higher 𝑅𝑒𝛤. Thus, it can be suggested that the deflection of the vortex rings from the free surface depends not only on 𝑅𝑒𝛤, but possibly also on Fr. It is also possible that the vortex deflection may be a function of 𝑅𝑒𝛤𝐹𝑟 and a critical 𝑅𝑒𝛤𝐹𝑟 may exist that determine the transition of the vortex ring deflection at each ℎ̃. For instance, in our experiments at ℎ̃=1.5, an upward flection of the vortex ring trajectory is observed for 𝑅𝑒𝛤𝐹𝑟 in the range of 22 – 110, whereas a downward flection is noticed at much higher 𝑅𝑒𝛤𝐹𝑟 aroun d 1100 – 5500. However, to substantiate this hypothesis, further systematic experiments will 14 be required at a much larger range of variation of both 𝑅𝑒𝛤 and 𝐹𝑟. Moreover, since all the images shown here are instantaneous snapshots and there are many r andom 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 i t 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. Fig. 4. Visualization of flow patterns for a synthetic jet in XZ plane for ℎ̃ = 5.0 at 𝑅𝑒𝛤 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. (a) (b) (c) (d) 15 3.2. Velocity measurements In the above section, the shown PLIF images revea l 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. 3.2.1. Time averaged velocity in the XZ plane Similar to the velocity distribution along the centerline, time average velocity distribution for ℎ̃= 1.5 (Figure 5a) in XZ plane shows the shifting of streamwise velocity towards the free surface. In the near vicinity, or 𝑥̃= 0.5, the unrestrained surface has no substantial effect. At 𝑥̃= 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 𝑥̃= 0.5 which is insignificant between 𝑥̃= 2 to 5. Between 𝑥̃= 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 ℎ̃= 3 and ℎ̃=5, flow is akin to a deeply submerged jet an d 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 𝑥̃= 0.5 around the orifice boundary due to the generation of slight suction, in all cases. Again, a sym metry 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 𝑅𝑒𝛤= 845, in Figure 5(c) and (d) at 𝑅𝑒𝛤=2663 , the velocity distribution is skewed downwards as s een by downward movement of synthetic jet in Figure 5(d). In Figure 5(b) at 𝑅𝑒𝛤=845, the distribution of cross stream velocity shows only the upward movement of fluid at 𝑥̃= 0.5 between -1 ≤ 𝑧̃≤ 0.4. At the same location 𝑥̃= 0.5 for 𝑅𝑒𝛤=2663 in 16 Figure 5(d), the downward as well as upward movement of fluid is observed between -0.3 ≤ 𝑧̃ ≤ 0.5 and -1 ≤ 𝑧̃≤ -0.3 respectively. Since this region is in suction zone of synthetic jet (less than 𝑥̃= 2), the suction stroke combined with shallow water depth affects the formation and propagation of vortex ring. The cross stream velocity distribution at 𝑅𝑒𝛤=845 and 𝑅𝑒𝛤=2663 shown in Figure 5(b) and (d) are opposite to each other at 𝑥̃= 8 and 12 supporting the observation in the image shown in Figure 3. Figure 6 and Figure 7 show the time averaged velocity distribution of synthetic jet at water depth equal to ℎ̃= 3 and 5. In Figure 6 (a) and (c), as compared to the above cases no significant change s in the stream wise velocity distribution of synthetic jet is observed at 𝑅𝑒𝛤=845 and 𝑅𝑒𝛤=2663 except the profile is more symmetric 𝑅𝑒𝛤=2663 in comparison to 𝑅𝑒𝛤=845. In Figure 6(b) and (d), the trend of cross stream velocity distribution is similar at 𝑥̃= 0.5 and at 𝑥̃= 2 for 𝑅𝑒𝛤=845 and having inversion point at 𝑧̃ = - 0.3 and -0.62 respectively at 𝑥̃= 0.5. The far field has a random upward and downward movement of fluids except at 𝑥̃= 5 for 𝑅𝑒𝛤=845 , where a significant downward motion is observed between 𝑧̃ = 0.6 and 1.8. 𝑢𝑎𝑣𝑔(𝑚/𝑠) 𝑣𝑎𝑣𝑔(𝑚/𝑠) (a) (b) 17 Fig. 5. Time averaged stream -wise velocity and their corresponding cross stream velocity distribution for a shallow water depth of ℎ̃= 1.5 at 𝑅𝑒𝛤 equal to (a, b) 845 and (c, d) 2663. (The upward (+ 𝑣𝑎𝑣𝑔) and downward ( -𝑣𝑎𝑣𝑔) movement of fluid are shown by dotted arrow. The red dotted lines are obtained by connecting the velocity maxima, to compare the vortex trajectory at different 𝑅𝑒𝛤. 𝑢𝑎𝑣𝑔(𝑚/𝑠) 𝑣𝑎𝑣𝑔(𝑚/𝑠) (a) (b) (c) (d) 18 Fig. 6. Time averaged stream -wise velocity and their corresponding cross stream velocity distribution for a shallow water depth of ℎ̃ = 3.0 at 𝑅𝑒𝛤 equal to (a, b) 845 and (c, d) 2663. (The upward (+ 𝑣𝑎𝑣𝑔) and downward ( -𝑣𝑎𝑣𝑔) movement of fluid are shown by the dotted arrow. 𝑢𝑎𝑣𝑔(𝑚/𝑠) 𝑣𝑎𝑣𝑔(𝑚/𝑠) (c) (d) (a) (b) (d) 19 Fig. 7. Time averaged stream -wise velocity and their corresponding cross stream velocity distribution for a shallow water depth of ℎ̃ = 5.0 at 𝑅𝑒𝛤 equal to (a, b) 845 and (c, d) 2663. (The upward (+ 𝑣𝑎𝑣𝑔) and downward ( -𝑣𝑎𝑣𝑔) movement of fluid are shown by the dotted arrow. In Figures 7(a) and (b), the streamwise velocity distribution is symmetric about the jet centerline at all downstream locations for both the frequencies. The magnitude of the cross stream velocity is also varies close to the synthetic jet centerline. There fore it can be concluded that at ℎ̃ = 5, the free surface have negligible effect on the flow pattern of the synthetic jet. In summary, The time averaged velocity flow field of synthetic jet in shallow water at different depths ℎ̃= 1.5, 3 and 5 are shown in F igures 5, 6 and 7 for 𝑅𝑒𝛤=845 and 𝑅𝑒𝛤=2663 . At ℎ̃= 1.5, vortex ring drifts upwards and an interaction of upper counter rotating vortices with unrestrained surface is noted at 𝑅𝑒𝛤=845. However, rebounding of vortex ring near the free surface in far -field is observed at 𝑅𝑒𝛤=2663 . 3.2.2. 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 ℎ̃= 1.5, 3 and 5 are measured for four different values of 𝑅𝑒𝛤= 370, 845, 2663 and 5904 . The velocity field is studied by non - (c) 20 dimensionalizing the streamwise and cross -stream velocity by the corresponding slug velocity (Uo). The slug velocity is defined as the average ejection velocity of expelled fluid through orifice in forward stroke of the diaphragm. Figure 8 illustrates the variation of the non -dimensional streamwise velocity and the non -dimensional cross -stream velocity at different transverse dista nces. As shown in Figure 8, the overall value of uavg/Uo , is lowest for 𝑅𝑒𝛤=5904 and highest for 𝑅𝑒𝛤=370. This is because, increasing the frequency increases the slug velocity and also changes the velocity distribution across the orifice. The incre ment in the time averaged velocity along the centerline is slower than the slug velocity due to widening of the suction region from the periphery of the orifice towards the center at an increased frequency. Since the slug velocity remains constant at a par ticular frequency, the profile of the curves in Figure 8 is a function of uavg and vavg only. In Figure 8(a) for ℎ̃= 1.5, the streamwise velocity of synthetic jet at an 𝑅𝑒𝛤=370 decline sharply in magnitude due to the upward movement of vortex ring towar ds the free surface as evident in the distribution of cross stream velocity shown in Figure 8(b). At an 𝑅𝑒𝛤=845, the peak of cross stream velocity gets shifted to 𝑥̃= 3 with reduced magnitude in comparison to 𝑅𝑒𝛤= 370 which gives a reduced slope for s treamwise velocity distribution. However, at both the frequencies, the vortex ring moves upward and interacts with the free surface and finally disappears by losing its vorticity. At greater circulation 𝑅𝑒𝛤=2663 and 𝑅𝑒𝛤=5904 , the synthetic jet behavi or differs due to vortex rebounding from the free surface (see Figures 3c and 3d). This trend is in response to the negative value of cross stream velocity in Figure 8(b) between 𝑥̃= 0 to 6 which is just opposite to the trend observed at 𝑅𝑒𝛤=370 and 𝑅𝑒𝛤=854 . Similar observation is also found in Figure 8(c) for an increased water depth until ℎ̃= 3 with an oscillating value of uavg. In Figure 8(d), the absolute value of cross stream velocity at all four 𝑅𝑒𝛤 decreases 21 in comparison to the above case, indicating that the effect of the free surface is reduced at this depth. When the free surface height is increased to ℎ̃ = 5.0, the velocity distribution trend are similar at all the values of 𝑅𝑒𝛤, as shown in Figures 8(e) and 8(f). It has been observed from PLIF images and centerline velocity distributions that the synthetic jet shows similar trends at lower value of 𝑅𝑒𝛤 = 370 and 845, but the behavior gets altered at higher value of 𝑅𝑒𝛤 equal to 2663 and 5904 . Since the behav ior at the extreme frequencies has already been reported in a recent study (Kumar et al. 2017), we focus our attention only on 𝑅𝑒𝛤= 845 and 𝑅𝑒𝛤=2663 , while presenting the cross -stream velocity distribution in the following section. 22 Fig. 8. Non-dimen sional time averaged stream wise and cross -stream velocity with respect to transverse distance at (a, b) ℎ̃= 1.5, (c, d) ℎ̃= 3.0, and (e, f) ℎ̃= 5.0 at different frequencies. 23 Finally, it is also worth mentioning the reasons behind some of the apparent differe nces 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. T he re ason 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, som e 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, i t is also possible that such different trajectories may only occur only during the initial stages of the synthetic jet actuation. Indeed, m ore 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 tha t 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 beca use of experimental limitations and probably could be taken up by future studies. 4. 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 24 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. Exp eriments 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 a t shallow water depths su ch as ℎ̃ = 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 significan t impact of the free surface on the formation and propagation of synthetic jet at greater water depths of ℎ̃ = 3 and ℎ̃ = 5 in the range of Reynolds number attainable in our experiments. At high 𝑅𝑒𝛤, 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. It is concluded that the vortex rings may interact or rebound from t he free surface depending on its circulation strength and water depth. Acknowledgements The first author gratefully acknowledges the contribution of Prof. P. K. Panigrahi and Prof.A. K. Saha for their support and guidance in carrying out the experime nts at IIT Kanpur. We also thank Mr. Sri Ragunath V for his assistance in the preparation of this manuscript.