3.1 Fluid ejection and entrainment

Our vortex generator operates by compressing and relaxing its deformable bulb. Compression ejects fluid into the wake and relaxation entrains fluid by allowing the bulb to return to its original form. The temporal evolution of the average velocity of the fluid that is ejected or entrained by the propulsor is obtained by integrating the vertical velocity profile directly behind the nozzle exit of the propulsor. This average exit velocity across the nozzle diameter $D$ is defined in cylindrical coordinates as

(3.1)\begin{equation} U_{exit}(t) = \frac{4}{{\rm \pi} D^{2}} \int_{0}^{2{\rm \pi}}\int_{0}^{D/2} u(r,x = 0,t)r\,{{\rm d}}r\,{{\rm d}}\theta, \end{equation}

where $u(r,x=0,t)$ is the streamwise velocity component directly behind the nozzle exit at $x/D = 0$. Schematic representations of the velocity profiles at the nozzle exit and the temporal evolution of the averaged exit velocity are presented in figure 2 over the duration of one compression–relaxation cycle $T$. Positive and negative values of the exit velocity $U_{exit}$ correspond, respectively, to fluid ejection and fluid entrainment.

Figure 2. (a–d) Reference cylindrical coordinate system and schematics of the velocity profiles at the nozzle exit at different times during the compression–relaxation cycle. Fluid leaving the nozzle moves in the positive $x$-direction which is referred to as the streamwise direction. (e) Temporal evolution of the measured diameter averaged exit velocity $U_{exit}$ normalised by the characteristic velocity $U_{0}$ over the duration of one compression–relaxation cycle $T$. ( f) Non-dimensional stroke ratio obtained from integrating $U_{exit}$ in time.

The exit velocity $U_{exit}$ in figure 2 is normalised by $U_{0}$, the characteristic velocity of the system, which is defined as

(3.2)\begin{equation} U_{0} = \frac{8V_{0}}{{\rm \pi} D^{2} T}. \end{equation}

Here, $V_{0}$ is the total volume of fluid ejected from the propulsor during the duration of bulb compression, $T/2$. The velocity $U_{0}$ is the velocity of the equivalent constant uniform flow that yields the same ejected volume of fluid over the time of bulb compression. The velocity $U_{0}$ is 415 mm s$^{-1}$ for the data presented here, and the corresponding Reynolds number ${\textit {Re}}_{D}$ based on the nozzle exit diameter and $U_{0}$ is 5820.

Due to the particular design of our vortex generator, the exit velocity increases slowly when the compression starts, but rapidly catches up and reaches a maximum value of $U_{exit}/U_{0}\approx 1.9$ at $t/T=0.13$. At $t/T = 0.5$, the kinematics of the propulsor cause a transition from bulb compression to bulb relaxation. The fluid does not does respond directly to the transition of the kinematics and continues to flow out of the bulb for ${\rm \Delta} t/T\approx 0.07$ after bulb relaxation has begun. A shorter lag in the response of the flow with respect to the forcing kinematics is present at the end of cycle. The flow continues to be entrained for ${\rm \Delta} t/T\approx 0.04$ after the driving kinematics have stopped. The slight asymmetry in the fluid ejection–entrainment response to the symmetric compression–relaxation motion is characteristic of this particular driving kinematics and could be compensated for or enhanced by using more complex kinematics.

The length of an equivalent cylindrical column of fluid ejected by the vortex propulsor during compression and relaxation is called the stroke length. It is defined here as

(3.3)\begin{equation} L(t) = \int_{0}^{t} U_{exit} (\tau)\,{{\rm d}}\tau. \end{equation}

The stroke length is analogous to the distance travelled by the piston in a piston cylinder apparatus to eject the same volume of fluid. The temporal evolution of the stroke to diameter ratio $L/D$ is presented in figure 2( f). After the initial slow response of the flow to the compression kinematic, the stroke ratio increases nonlinearly during the compression phase for $0< t/T<0.57$. The stroke ratio attains a maximum value of 7 around $t/T=0.57$ and decreases approximately linearly with a rate ${{\rm d}}L/{{\rm d}}t= -5.7 D/T$ during the relaxation phase for $0.57< t/T<1.04$.

The temporal velocity profile of the fluid ejected by the vortex generator is not constant and leads to a nonlinear variation of the stroke ratio. This allows us to characterise the formation process of a vortex ring generated by a nonlinear evolution of the stroke ratio, which has not yet received much attention. In addition to the nonlinear fluid ejection profile, the fluid entrainment process is unique to the design of our propulsor, and enables the periodic ejection of multiple vortex rings. The effect of the time-varying exit velocity profile and the fluid entrainment process on the temporal evolution of the propulsive force will be the subject of future investigations. Here, we focus solely on the flow field created by our propulsor. Our goal is to characterise the formation process of a vortex ring generated by the arbitrary fluid ejection profile and identify observable quantities that can aid a future optimisation of similar robotic devices that utilise pulsatile jet propulsion.

3.2 Vortex formation process

When fluid is ejected by our vortex propulsor a coherent vortex ring is formed. The growth of the vortex ring is presented by instantaneous snapshots of the flow field in figure 3. The arrows represent the two-dimensional velocity field $\boldsymbol {u}=(u,v)$ in the measurement plane. The out-of-plane vorticity component, $\omega$, is computed from the in-plane velocity field and is shown in the top half of the snapshots in figure 3. The colours in the bottom half of the snapshots indicate the swirling strength, $\lambda _{ci}$, which is the imaginary part of the complex eigenvalues of the velocity gradient tensor (Reference Zhou, Adrian, Balachandar and KendallZhou, Adrian, Balachandar, & Kendall, 1999). The swirling strength is a robust indicator of vortices in shear layers where high concentrations of vorticity exist and obfuscate the presence or absence of vortices.

Figure 3. Snapshots of experimentally observed vortex ring development during the compression phase of the vortex generator at (a) $L/D = 3.4$, (b,c) $L/D = 5.0$, (d) $L/D = 5.5$ and (e) $L/D = 5.9$. The colours in the top half of the snapshots indicate values of the experimentally measured out-of-plane vorticity component $\omega$. The colours in the bottom half of the snapshots indicate the experimentally obtained swirling strength $\lambda _{ci}$. (c) Zoomed-in view on a secondary vortex in the trailing shear layer with velocity vectors indicating the relative velocity with respect to the velocity measured in the centre of the secondary vortex. The vorticity associated with the secondary vortex in (d) is indicated by the small box.

The shear layer that emerges at the boundary between the ejected fluid and the surrounding quiescent flow immediately rolls up into a vortex ring. The coherent core is indicated by a localised region of non-zero values of the swirling strength (figure 3a). The vortex ring rapidly convects away from the nozzle exit while flow is still being ejected and a trailing shear layer appears between the vortex ring and the nozzle exit (figure 3b). At $L/D=5$, a secondary or trailing vortex is present in the trailing shear layer. Based on the vorticity concentration alone, it is difficult to distinguish between a strong shear flow and rotation, but the swirling strength distribution is conclusive  (figure 3b,c). The velocity vectors in the zoomed-in view of the secondary vortex in figure 3(c) represent the velocity relative to the velocity vector measured in the centre of the secondary vortex ($\boldsymbol {u}-\boldsymbol {u}_{vortex}$). The centre of the vortex is identified as the location of maximum swirling strength. This secondary vortex core is not persistent and is no longer visible in the swirling strength field for $L/D>5.5$ when the primary vortex travels further downstream and moves away from the trailing shear layer (figure 3d,e). The vorticity associated with the secondary vortex is still present and has moved closer to the primary vortex (indicated by the box in figure 3d) suggesting that the primary and secondary vortices begin merging.

To confirm the merging of the primary and secondary vortices, we analyse the temporal evolution of the Lagrangian coherent structures in the positive and negative FTLE fields, which mark the boundaries of the vortex. Figure 4 shows the positive and negative FTLE ridges and fields atop the vorticity and swirling strength fields. The positive ridge is the upstream boundary of the vortex ring along which particle trajectories are attracted. The negative ridge is the downstream boundary of the vortex ring from which particle trajectories are repelled. The two ridges can be identified once the core of the primary vortex ring has moved one nozzle diameter away from the exit. The evolution of the streamwise location of the vortex core and of the intersections of the positive and negative FTLE ridges with the centreline are presented as a function of the stroke ratio $L/D$ in figure 4(e). As the vortex convects away from the nozzle exit, the positive FTLE ridge lags behind the vortex core (figure 4b,e). The positive FTLE ridge encloses the secondary vortex core indicated by the box in figure 4(b). For $L/D>5$ the positive FTLE ridge accelerates and catches up with the vortex core, pushing the vorticity associated with the secondary vortex core to merge with the vorticity of the primary core. The distance between the positive FTLE ridge and the core reaches a steady state at $L/D \approx 6$ such that the Lagrangian boundaries symmetrically enclose the vortex core (figure 4d).

Figure 4. Positive and negative FTLE ridges, computed from experimental data, indicating the boundaries of the vortex ring presented atop the vorticity and $\lambda _{ci}$ fields for (a) $L/D = 3.4$, (b) $L/D = 5.0$, (c) $L/D = 5.5$ and (d) $L/D = 5.9$. (e) Evolution of the streamwise location of the vortex core and of the crossing of the positive and negative FTLE ridges with the centreline as a function of the stroke ratio.

The evolution of the positive ridge during vortex merging enables the mixing of two dynamically different regions of fluid: fluid inside the primary vortex ring and fluid that is in the trailing shear layer including the secondary vortex ring (Reference O'Farrell and DabiriO'Farrell & Dabiri, 2010). At the end of merging, the positive and negative FTLE ridges are symmetric with respect to the vortex core and the vortex ring has separated from the trailing shear layer. A symmetric definition of the vortex ring by the FTLE ridges matches the intuitive idea of a pinched-off vortex ring, with the ridges acting as physical barriers that prevent additional vorticity from entering into the ring.

3.3 Evolution of vortex topology during vortex merging

The distance between the intersections of the positive and negative FTLE ridges with the centreline is used to define the streamwise length, $L_{o}$, of the vortex. The largest distance in the radial direction between the topmost and bottommost points on the positive FTLE ridge is used here to define the outer diameter, $D_{o}$, or height of the vortex. The definition of these vortex shape characteristics are indicated in figure 5(a) and their temporal evolution for $L/D>3$ is presented in figure 5(b). For lower stroke ratios, we cannot yet identify the downstream FTLE ridge to determine the vortex length.

Figure 5. Evolution of the shape and asymmetry of the experimentally observed vortex ring. (a) Vortex boundaries indicated by the FTLE ridges including the definition for the vortex streamwise length and outer diameter. Temporal evolution of the (b) vortex length and outer diameter and (c) the asymmetry parameter $a$ defined by (3.4) as a measure for the degree of asymmetry of the FTLE boundaries with respect to the vortex core. The shaded area indicates when vortex merging occurs.

Around $L/D=3$, the vortex outer diameter is approximately twice the nozzle diameter and larger than the streamwise length. The outer diameter increases approximately linearly during the rest of the compression phase of the bulb to $D_{o} = 2.4$ D at $L/D=6.5$. The streamwise length increases faster than the outer diameter as the positive FTLE ridge lags behind and reaches a maximum value of $L_{o}=2.65$ D at $L/D \approx 4.6$, yielding a minimum aspect ratio of $L_{o}/D_{o}=0.8$. For $L/D>4.6$, the positive FTLE ridge starts catching up with the vortex core and negative ridge (figure 4c) and the streamwise length rapidly decreases and converges to a value of $L_{o}=1.76$ D for $L/D>6$. This corresponds to an aspect ratio of $L_{o}/D_{o}=1.4$. Based on the velocity field (figure 3) and FTLE snapshots (figure 4), the merging of the secondary vortex with the primary vortex ring occurs for $5< L/D<6$. During this time interval, indicated by the shaded region in figure 5(b), the FTLE boundaries contract and push vorticity from the tail of the FTLE bound region towards the primary core line to merge.

To quantify the asymmetry of the FTLE bound area with respect to the vortex core location, we introduce the following asymmetry parameter:

(3.4)\begin{equation} a = \frac{L_{p}-L_{n}}{D_{o}}, \end{equation}

with $L_{n}$ the distance from the streamwise location of the vortex core to the leading nFTLE intersection, and $L_{p}$ the distance from the streamwise location of the vortex core to the lagging pFTLE intersection with the centreline (figure 5a). Values of $a$ close to zero indicate symmetric FTLE boundaries with respect to the vortex core, higher positive values indicate an asymmetric tail-heavy FTLE-enclosed area. For $3< L/D<4.5$, the asymmetry increases similarly to the vortex length due to the lagging of the positive FTLE ridge. For $L/D>4.5$, the positive FTLE ridge catches up with the core line and the asymmetry parameter decreases and reaches zero at $L/D=6.2$. The FTLE boundaries become fully symmetric with respect to the vortex core after vortex merging.

3.4 Evolution of integral quantities during vortex merging

The variation in the vortex ring topology during merging affects the amount of vorticity and its distribution inside the vortex core, which influences the vortex circulation, hydrodynamic impulse and energy (Reference Gharib, Rambod and ShariffGharib et al., 1998; Reference de Guyon and Mullenersde Guyon & Mulleners, 2021c). By analysing the FTLE ridges and the local extrema in the pressure field of vortex rings created by synthetic jets, Reference Lawson and DawsonLawson and Dawson (2013) demonstrated that vortex rings can continue to grow after pinch-off due to interaction with their environment. To quantify the evolution of the vortex ring development during the entire formation process, we analyse here the temporal evolution of the integral quantities of the vortex, including its circulation, energy, and resulting translational velocity.

The circulation of the vortex ring is computed as the surface integral of the average between the positive and negative out-of-plane vorticity

(3.5)\begin{equation} \varGamma = \iint_{A} \frac{|\omega|}{2}{{\rm d}}A. \end{equation}

We have calculated the circulation within the area bound by the FTLE ridges and within a rectangular area centred around the vortex centre such that it only contains the primary vortex ring (figure 6a). The extent of the rectangular box is defined based on the trailing pressure maximum following the procedure presented by Reference Lawson and DawsonLawson and Dawson (2013). The temporal evolution of the non-dimensional circulation in both integration areas is presented in figure 6(b,c). The circulation is non-dimensionalised by the characteristic velocity $U_{0}$ and the nozzle outlet diameter $D$ in figure 6(b) and by $U_{0}$ and the vortex diameter $D_{v}$ in figure 6(c). The vortex diameter normalised by the nozzle diameter is presented in figure 6(d).

Figure 6. Temporal evolution of the vortex circulation and vortex diameter. Circulation values obtained by integration of the experimentally obtained out-of-plane vorticity within the area bound by the positive and negative FTLE ridges are presented in violet, values obtained by integration within a rectangular box centred around the primary vortex centres are presented in turquoise. The light blue shaded region represents the uncertainty based on variations in the box height. (a) Sketch of the two integration areas. (b) Circulation normalised by $U_{0}$ and the nozzle diameter $D$. (c) Circulation normalised by $U_{0}$ and the vortex diameter $D_{v}$. (d) Evolution of the vortex diameter normalised by the nozzle diameter.

Due to the initial tail-heavy asymmetry of the FTLE boundaries with respect to the vortex core, the circulation inside the FTLE contour is higher than the circulation in the box until $L/D\approx 5.5$. The width of the FTLE boundary converges to the width of the rectangular contour post vortex merging and the two circulation curves in figure 6(b,c) converge to $\varGamma /(U_{0}D)\approx 4.3$ and $\varGamma /(U_{0}D_{v})\approx 2.5$. The final value of the vortex circulation when normalised by the vortex diameter instead of the nozzle diameter is close to the non-dimensional values reported for propulsive vortex rings generated with a piston cylinder (Reference Gharib, Rambod and ShariffGharib et al., 1998) and drag vortices behind a translating cone (Reference de Guyon and Mullenersde Guyon & Mulleners, 2021c).

The difference between the circulation in the FTLE contour and in the rectangular contour for $L/D>5.5$ is attributed to vorticity outside the FTLE contour in the radial direction. For $L/D<5.5$, the difference is attributed to circulation in the trailing shear layer that will eventually be fed into the primary vortex ring through merging. The maximum circulation in the FTLE boundary in measured when the streamwise length of the vortex is also maximal (figure 5b). The circulation in the rectangular contour gradually increases and does not attain a local maximum prior to converging at $L/D>5.5$. When the circulation inside the vortex ring is normalised by the vortex diameter instead of the nozzle diameter, the non-dimensional circulation already reaches 90 % of its final value after $L/D=3.6$, which is well before vortex merging takes place. The additional volume and associated vorticity that is added to the main vortex ring due to merging does not significantly alter the non-dimensional circulation based on the vortex size, but primarily leads to an increase in the vortex core diameter (figure 6d). The increase in the vortex diameter during the merging process is confirmed for vortex rings generated in different configurations, such as orifice-generated vortex rings (Reference Limbourg and NedićLimbourg & Nedić, 2021b). After merging, the vortex core diameter converges to $D_{v}/D=1.73$ at the same time as the non-dimensional circulation converges.

Based on the evolution of the vortex circulation, we can calculate a vortex formation number. The formation number of a vortex ring is typically obtained as the stroke ratio at which the circulation in the entire domain equals the steady-state circulation value inside the vortex ring. Following this convention, we obtain a vortex formation number of 3.3 which is within the range of formation numbers observed for propulsive vortex rings generated for different stroke ratios (Reference DabiriDabiri, 2009). Note that the evolution of the total circulation in the domain is not included in the figures as it quickly exceeds the axis range selected for display.

The non-dimensional energy of the vortex ring serves as a measure for the distribution of the vorticity inside the vortex ring (Reference de Guyon and Mullenersde Guyon & Mulleners, 2021c). According to Reference Gharib, Rambod and ShariffGharib et al. (1998), the non-dimensional energy $E^{*}$ is defined as

(3.6)\begin{equation} E^{ * } = \frac{E}{\sqrt{\rho I \varGamma^{3}}}, \end{equation}

with $E$ the kinetic energy, $\rho$ the fluid density, $I$ the impulse and $\varGamma$ the circulation of the vortex ring. The kinetic energy is computed as

(3.7)\begin{equation} E = {\rm \pi}\rho \iint_{A} \omega \psi\,{{\rm d}}A, \end{equation}

where $\psi$ is the streamfunction, obtained from integrating the Cauchy Riemann equations for the axisymmetric vortex ring. The streamfunction is computed here in the entire domain and integrated within the integration area $A$. Similar to the procedure applied to compute the circulation (figure 6a), we consider again two integration areas. One area is bound by the FTLE ridges, the other one is a rectangular area centred around the vortex centres. The impulse of the vortex ring is computed as

(3.8)\begin{equation} I = {\rm \pi}\rho \iint_{A} \omega |r|^{2} \,{{\rm d}}A, \end{equation}

where $|r|$ is the distance away from the vortex centreline.

The non-dimensional energy for the two integration areas is presented in figure 7(a). The evolution of $E^{*}$ within the FTLE boundary is only shown as a reference. The asymmetric shape of the FTLE ridges for $L/D<6$ indicate that the primary vortex ring is not isolated and we cannot directly interpret the value of the non-dimensional energy bound by the FTLE ridges as a measure of the vorticity distribution within the vortex ring. The non-dimensional energy inside the rectangular contour that contains only the primary vortex ring drops to a steady-state value of 0.26 after only three stroke ratios. This would point again towards a vortex formation number around 3. The formation number can be interpreted here as the non-dimensional time required for the vorticity to accumulate into a stable distribution but it does not mean that the vortex will not accept additional vorticity or impulse. After the initial decrease, the non-dimensional energy of 0.26 is maintained for the remainder of the vortex merging process while other quantities, such as the circulation and diameter of the vortex ring (figure 6b–d), continue to evolve. The continued evolution of the circulation and size of vortex rings after their pinch-off or for formation times beyond the formation number has been observed by Reference Lawson and DawsonLawson and Dawson (2013) and Reference Limbourg and NedićLimbourg and Nedić (2021b).

Figure 7. Temporal evolution of (a) the non-dimensional energy of the vortex and (b) the translational velocity of the experimentally observed vortex according to (3.9) and based on the tracking of the vortex centre locations. Values obtained by integration in the area bound by the FTLE ridges are presented in violet, values obtained by integration within a rectangular box centred around the vortex centres are presented in turquoise.

The limit value of the non-dimensional energy observed in this study, 0.26, is lower than the value of 0.33 which is typically observed in the literature for vortex rings generated with a constant piston or jet velocity (Reference NitscheNitsche, 2001; Reference Shusser, Gharib and MohseniShusser, Gharib, & Mohseni, 1999). This lower value is attributed to the kinematics of the vortex propulsor, which influences the velocity and acceleration profile of the jet that feeds the vortex ring (Reference de Guyon and Mullenersde Guyon & Mulleners, 2021a; Reference Krieg and MohseniKrieg & Mohseni, 2013a; Reference Limbourg and NedićLimbourg & Nedić, 2021a). The nozzle exit velocity decelerates for $t/T>0.2$ up to the end of the compression phase of the vortex generator (figure 2e), which resembles the negative sloping velocity programs implemented by Reference Danaila, Luddens, Kaplanski, Papoutsakis and SazhinDanaila, Luddens, Kaplanski, Papoutsakis, and Sazhin (2018) and Reference Krueger and GharibKrueger and Gharib (2003). The non-dimensional energy of the vortex ring can also be manipulated to values below 0.33 by varying the nozzle geometry, such as a converging nozzle or a nozzle with a temporally varying exit diameter (Reference Krieg and MohseniKrieg & Mohseni, 2021).

The additional volume and circulation that will merge with the primary vortex ring for $L/D>3$ does not alter its non-dimensional energy. This is a great example of the Kelvin–Benjamin variational principle, which states that a vortex will only accept additional vorticity if this does not disturb the vorticity distribution, and thus the non-dimensional energy of the new configuration (Reference BenjaminBenjamin, 1976). The vorticity level in the trailing jet is high enough to allow it to penetrate into the primary vortex ring. The primary vortex is able to accept the additional vorticity from the secondary vortex by increasing its radius (figure 6d) and redistributing the additional vortical fluid such that its non-dimensional energy remains constant (figure 6a).

The reason the secondary vortex can merge with the primary vortex in the first place is due to their relative translation velocities (Reference MaxworthyMaxworthy, 1972). The vortex translation velocity is calculated from the non-dimensional energy, circulation, and diameter of the vortex ring (Reference de Guyon and Mullenersde Guyon & Mulleners, 2021c; Reference SaffmanSaffman, 1970):

(3.9)\begin{equation} U_{v} = \frac{\varGamma}{{\rm \pi} D_{v}} \left(E^{ * } \sqrt{\rm \pi} + \frac{3}{4}\right). \end{equation}

Equation (3.9) is valid for steady vortex rings with a thin core (Reference SaffmanSaffman, 1970). An error of ${\approx } 3\,\%$ is obtained for the vortex translational velocity by taking into account the effect of the core thickness, based on the second-order correction proposed by Reference FraenkelFraenkel (1972) and Reference Shusser, Gharib and MohseniShusser et al. (1999) for Norbury vortices (Reference NorburyNorbury, 1972).

The temporal evolution of the translational velocity was computed for the FTLE boundary and the rectangular contour enclosing solely the primary vortex and is presented in figure 7(b). The translational velocity based on the tracking of the vortex centres is included in figure 7(b) for comparison. The translational velocity of just the primary vortex ring based on the integral quantities rapidly increases during the first three stroke ratios to a value of $U_{v}/U_{0}=0.8$. The velocity based on the tracking the core converges to $U_{v}/U_{0}=1$ after four stroke ratios. This rapid increase in the translational velocity causes the core to move away from the nozzle exit. At $L/D=3$, the core has already move one diameter away from the outlet (figure 4a,e) which makes it harder to directly feed additional vorticity into the primary vortex, but it does not mean that the vortex is not able to accept additional vorticity at a later stage. The early physical distancing of the primary vortex ring is a direct consequence of the specific time-varying outlet velocity profile of our vortex generator.

The translational velocity based on the integral values computed for the FTLE boundary increases beyond the velocity of the primary vortex for $4< L/D<5.5$. This difference is attributed to the higher translational velocity of the secondary vortex in the tail of the FTLE-bounded area. The secondary vortex catches up with the primary vortex and they merge. The vortex diameter, circulation, and translational velocity all converge to new post-merging values for $L/D>5.5$ while the non-dimensional energy remains at its limiting pre-merging value in agreement with the Kelvin–Benjamin variational principle.

3.5 Fluid entrainment and detrainment during vortex merging

The entrainment and detrainment of fluid into the merging of the primary and secondary vortex is visualised using a Lagrangian approach. Artificial seed particles are placed inside and outside the vortex boundaries marked by the FTLE ridges $L/D = 3.7$ and are convected with the flow. The initial time of particle seeding is selected such that the FTLE ridges demarcating the vortex have fully formed and form a closed contour. The locations of the seed particles at different time instants during the compression stage of the bulb are presented in figure 8. The top half of the snapshots in figure 8 present the results for particles that were initially inside the FTLE boundaries and show fluid detrainment. The bottom half of the snapshots in figure 8 present the results for particles that were initially outside the FTLE boundaries and show fluid entrainment.

Figure 8. Snapshots of artificial seed particles convected by the experimentally measured flow. Top half shows particles initially inside the FTLE boundaries, bottom half shows particles initially outside the FTLE boundaries. White symbols correspond to points along the FTLE ridges that are non-hyperbolic, filled coloured symbols correspond to points along the FTLE ridges that are hyperbolic. Initial position of the particles at (a) $L/D = 3.7$, and their location after convection by the flow at (b) $L/D = 4.4$, (c) $L/D = 5.0$, (d) $L/D = 5.5$, (e) $L/D = 5.9$ and ( f) $L/D = 6.4$.

By definition, fluid particles move around the negative ridge and are attracted by the positive FTLE ridge (Reference HallerHaller, 2002; Reference Shadden, Katija, Rosenfeld, Marsden and DabiriShadden, Katija, Rosenfeld, Marsden, & Dabiri, 2007). Occasionally, particles surpass the positive FTLE ridge to enter the vortex boundary (figure 8d–f). The particles leave through the formation of lobes in the negative FTLE ridge, in the process known as tail shedding (Reference de Guyon and Mullenersde Guyon & Mulleners, 2021b; Reference Shadden, Dabiri and MarsdenShadden et al., 2006).

To understand how particles can cross FTLE ridges we compute the strain rates normal to the ridges. The strain rates allow us to verify whether the ridges are indeed hyperbolic Lagrangian coherent structures (Reference Green, Rowley and SmitsGreen et al., 2010; Reference HallerHaller, 2002). The positive FTLE ridge, which repels particles, is a hyperbolic repelling material line if the strain rate normal to the ridge is positive. A negative FTLE ridge is an attracting material line if the strain rate normal to the ridge is negative (Reference HallerHaller, 2001). The sign of the strain rates on the positive and negative FTLE ridges are indicated by the markers in figure 8. The negative FTLE ridge continuously maintains negative strain rates on the ridge throughout the process, confirming that it is a hyperbolic attracting material line (Reference Green, Rowley and HallerGreen et al., 2007). The entrainment of fluid particles into the vortex ring across the positive FTLE ridges occurs where the ridges are locally non-hyperbolic (figure 8b–e). The positive FTLE ridge evolves into an entirely hyperbolic repelling ridge post vortex merging when the ridges symmetrically enclose the primary vortex core (figure 8f).

3.6 Evolution of pressure field during vortex merging

The Lagrangian analysis provides detailed and accurate insight into the formation and development of the vortex ring generated by our propulsor. However, the Lagrangian analysis is computationally expensive, requires time-resolved flow field data, and is not suitable for in situ optimisation and control of the time-dependent exit velocity profile. Local instantaneous pressure measurements would be preferred for this purpose. To evaluate the potential of pressure-based indicators of vortex formation we need to explore how pressure features in the flow field relate to the previously extracted Lagrangian boundaries.

The pressure field is computed from a direct integration of the velocity field (Reference Dabiri, Bose, Gemmell, Colin and CostelloDabiri, Bose, Gemmell, Colin, & Costello, 2014) and presented in figure 9 for selected snapshots. The FTLE ridges are added atop the pressure fields for comparison. The pressure minima reliably indicate the location of the vortex core in all the snapshots. The pressure minimum becomes stronger with increasing stroke ratio. A high pressure region called the leading pressure maximum forms ahead of the vortex core and has a local maximum where the negative FTLE ridge intersects with the centreline (figure 9a). Local high pressure regions or trailing pressure maxima, emerge aft of the primary vortex core for $L/D > 3.5$. These regions are scattered pre vortex merging (figure 9b,c) and combine to a single more coherent region centred around the intersection of the positive FTLE ridge with the centreline post merging (figure 9d).

Figure 9. Snapshots of the pressure field derived from the experimentally measured velocity field at (a) $L/D = 3.4$, (b) $L/D = 5.0$, (c) $L/D = 5.5$ and (d) $L/D = 5.9$. (e) Evolution of the streamwise location of the vortex core, of the crossing of the positive and negative FTLE ridges with the centreline and of location of the leading and trailing pressure maxima (l.p.m., t.p.m.) along the centreline as a function of the stroke ratio.

The evolution of the streamwise locations of the vortex core, of the intersection of the positive and negative FTLE ridges with the centreline, and of the location of the leading and trailing pressure maxima along the centreline are summarised in figure 9(e). The trajectory of the leading pressure maximum and the negative FTLE ridge coincide perfectly. The trailing pressure maximum is initially ahead of the positive FTLE ridge which lags behind the primary vortex core. For $4.4< L/D<6$, the trailing pressure maximum cannot be reliably identified in all snapshots. We use a linear interpolation to fill the gaps. The locations of the trailing pressure maximum and the positive FTLE ridge match closely post-merging.

The leading and trailing pressure maxima, like the positive and negative FTLE ridges, reliably indicate the upstream and downstream bounds of the vortex ring (Reference Lawson and DawsonLawson & Dawson, 2013). The ridges and the pressure maxima symmetrically enclose the primary vortex core and act as physical barriers that prevent additional fluid from entering the vortex ring. The pressure maxima can serve as reliable observables for vortex ring formation and shedding that can be used for future optimisation and control of the driving jet profile of bio-inspired vehicles and other vortex generating systems. A pressure-based methodology does not require time-resolved data and is computationally less expensive than calculating the FTLE field. Our results disclose new possibilities to incorporating pressure sensors and probes in the flow to measure vortex ring properties in situ.