3.1. Changes in flow characteristics due to nozzle flexibility

First, we examine how the development of the vortical structures in the flow fields is influenced by the fluid structure interaction as the characteristic stiffness parameter ( $Eh$ ) is varied for the given input flow parameters. Figure 5(a) illustrates the flow development from the $Eh = \infty\,\rm N\,m^-{^1}$ (rigid) nozzle for ${L}/{D}=2$ from (i) $t/t_{cycle}=0.41$ to (iv) $t/t_{cycle}=3$ , where $t_{cycle}$ is the time when the pump stops for each ${L}/{D}$ . The nozzle position relative to the flow field is shown as the black trapezoid at the top of each frame. As the fluid is expelled from the nozzle, the shear layer rolls up, forming a single primary vortex (PV) ring as it enters the quiescent tank. The plots show a cross-section cut across the middle of the ring, which is displayed as a single counter rotating vortex pair of opposite sign vorticity. This is consistent with Gharib et al. (Reference Gharib, Rambod and Shariff1998), wherein the ejected fluid from impulsive piston motion will form a single vortex ring for ${L}/{D} \lt 4$ . However, this behaviour changes when $Eh$ is sufficiently low enough (flexible enough) to perturb the input flow conditions.

Figure 5(b–e; i – iv) shows the flexible nozzle ( $Eh$ = 76, 54, 29, 19 N m^–1) flow fields, under the same input kinematic flow conditions from the pump ( ${L}/{D} \lt 2$ ). As $Eh$ decreases (becomes more flexible) the flow structures begin to deviate from the rigid case. For $Eh = 76\,\rm N\,m^-{^1}$ the vortex cores are slightly elongated, but become approximately circular after $t/t_{cycle}$ = 1. It is understood that the alteration is due to small-scale oscillations ( $\approx {0.02D}$ ) of the flexible nozzle. As $Eh$ is lowered further, the deformation of the nozzles becomes an order of magnitude more pronounced (up to $\approx {0.19D}$ ), multiple distinct vortex structures form, and the PV begins to significantly differ from the rigid case. For $Eh = 54\,\rm N\,m^-{^1}$ in figure 5(c), two cores form, where the secondary vorticity eventually leap frogs through the PV core, but the two cores do not separate from one another. The formation of this second core of vorticity corresponds to the nozzle oscillations and rapid collapse at $t=t_{cycle}$ , which is examined further below. For $Eh = 29\,\rm N\,m^-{^1}$ in figure 5(d), the PV separates from the nozzle prior to $t/t_{cycle} = 1$ . A weak secondary vortex (SV) structure forms, and separates from the nozzle at $t/t_{cycle}$ = 2. Notably the PV travels significantly further than the rigid case reaching $2.8D$ compared with $1.6D$ at $t/t_{cycle} = 2$ , defined by the averaged location of peak vorticity between the positive and negative cores. For the most flexible nozzle, $Eh = 19\,\rm N\,m^-{^1}$ , two separate vortex structures also form, but the SV is slightly larger than that produced by the $Eh = 29\,\rm N\,m^-{^1}$ nozzle. Interestingly, in this case, the PV travels a shorter distance of $2.4D$ below the nozzle at $t/t_{cycle} = 2$ compared with the slightly stiffer $Eh = 29\,\rm N\,m^-{^1}$ nozzle. Similar trends are observed at the other ejected volumes ( ${L}/{D} = 1, 4$ ), see Supplementary Material 1.

Figure 5. Vorticity and vector fields measured for (i) $t/t_{cycle}=0.41$ , (ii) $t/t_{cycle}=1$ , (iii) $t/t_{cycle}=2$ and (iv) $t/t_{cycle}=3$ for each nozzle given the same kinematic input from the pump for ${L}/{D} = 2$ . (a) Rigid nozzle ( $Eh = \infty\,\rm N\,m^-{^1}$ ); (b) $Eh=76\,\rm N\,m^-{^1}$ ; (c) $Eh=54\,\rm N\,m^-{^1}$ ; (d) $Eh= 29\,\rm N\,m^-{^1}$ ; (e) $Eh=19\,\rm N\,m^-{^1}$ .

To compare between the vortex rings produced by the different nozzles, we analyse cross-sectional cuts of the vortex rings generated by the $Eh = \infty\,\rm N\,m^-{^1}$ (rigid) nozzle and the $Eh = 29\,\rm N\,m^-{^1}$ nozzle, at $t/t_{cycle}=2$ for ${L}/{D}=1$ . This simplifies the comparison by focusing on single PV structures as shown in the vorticity plots in figures 6(a) and 6(b). Figures 6(c) and 6(d) display the spatial distribution of vorticity and velocity components measured along the cut lines. These cut lines are defined by the vertical ( $Y$ ) location of peak vorticity for each trial. As inferred by the distance travelled by the PV produced from the $Eh = 29\,\rm N\,m^-{^1}$ nozzle, the peak velocity is 2.5 times higher, and the peak vorticity is 2.8 times higher than the rigid nozzle case. Additionally, the wider spatial distribution velocity and vorticity suggests that the $Eh = 29\,\rm N\,m^-{^1}$ nozzle not only produces a stronger PV, but a larger PV compared with the rigid nozzle.

Figure 6. (a) Rigid nozzle PIV vector plot and vorticity at $t/t_{cycle}=2$ for ${L}/{D}=1$ . The solid line represents a cut section of the vortex ring and corresponds to the centred plots of vorticity and velocity; (b) $Eh = 29\,\rm N\,m^-{^1}$ nozzle PIV vector plot and vorticity, with cut line selected by the point of highest vorticity. (c) Vorticity across the vortex ring cut plane for the rigid and $Eh = 29\,\rm N\,m^-{^1}$ nozzles. (d) Vertical velocity across the vortex ring cut plane for the rigid and $Eh = 29\,\rm N\,m^-{^1}$ nozzles. Legend is the same for (c) and (d).

Next we analyse the temporal variation of vortex spacing ( $b$ ) and core diameter ( $a$ ) for each nozzle, as plotted in figures 7(a) and 7(b) for ${L}/{D}=2$ . The PV ring is tracked for this analysis, with $b$ defined as the distance between the peak vorticity values of each core, and $a$ is defined by the distance between the two points bounding 10 % of the maximum vorticity, averaged between the positive and negative cores. As shown in figure 7(a), the flexible nozzles generate vortex rings with wider spacing, with each ring converging to an approximately constant spacing of ${b}/{D} = 1.41 \pm 0.15$ , compared with ${b}/{D} = 1.19 \pm 0.13$ for the $Eh = \infty\,\rm N\,m^-{^1}$ (rigid) nozzle at $t=3t_{cycle}$ . The vortex ring core diameter ( $a$ ) varies slightly with $Eh$ , with the flexible nozzles all creating larger cores, as shown in figure 7(b). The $Eh = 54$ , $29$ and $19\,\rm N\,m^-{^1}$ nozzles converge to a core diameter of ${a}/{D} = 1.13 \pm 0.09$ , and the $Eh = 76\,\rm N\,m^-{^1}$ nozzle converges to ${a}/{D} = 1.01 \pm 0.09$ , compared with ${a}/{D} = 0.82 \pm 0.06$ for $Eh = \infty\,\rm N\,m^-{^1}$ at $t=3t_{cycle}$ . Here, the bounds are defined by the 95 % confidence interval for ${L}/{D} = 2$ . The other ejected volumes follow the same trends, but with smaller core diameter and spacing for ${L}/{D} = 1$ , and larger for ${L}/{D} = 4$ . Given the larger ( $b, a$ ), stronger ( $\omega _z$ ), and faster moving vortex rings generated by the flexible nozzles, the velocity produced by these nozzles should scale accordingly.

Figure 7. Primary vortex ring spatial parameters measured for ${L}/{D}=2$ . Points are plotted every 0.083 sec for clarity. (a) Vortex spacing ( $b$ ), normalised by the nozzle diameter ( $D$ ) over dimensionless time normalised by $t_{cycle}$ . (b) Vortex core diameter ( $a$ ), normalised with the nozzle diameter ( $D$ ), measured over time normalised by $t_{cycle}$ . Legend is the same for (a) and (b).

Figure 8. Temporal variation in vertical velocity spatially averaged across the nozzle exit ( $U_e$ ) measured $0.19D$ below each nozzle: (a) $U_e$ for ${L}/{D}=1$ across all $Eh$ values for ( $t/t_{cycle} = 0-4.5$ ); (b) $U_e$ for $Eh\approx \infty\,\rm N\,m^-{^1}$ (rigid), for all ${L}/{D}$ values. Here, $t_{acc}$ is defined as the time needed for the rigid nozzle to accelerate to 90 % of its maximum velocity.

The vertical velocity spatially averaged across the nozzle exit ( $U_e$ ), measured $0.19D$ below the nozzles, is plotted for ${L}/{D}=1$ , in figure 8(a). As expected by the distance travelled by the PV in the vorticity plots in figures 5(a)–5(e), $U_e$ increases as $Eh$ is reduced to a maximum at $Eh = 29\,\rm N\,m^-{^1}$ , and declines for $Eh = 19$ N/m. Furthermore, the time to reach the maximum $U_e$ becomes delayed with decreased $Eh$ . The $Eh = 19\,\rm N\,m^-{^1}$ nozzle does not accelerate to its peak velocity until $t/t_{cycle}=1.0$ , whereas the $Eh = 76\,\rm N\,m^-{^1}$ nozzle reaches its peak much quicker at $t/t_{cycle}=0.36$ .

To further examine the input flow conditions perturbed by the flexible nozzles, the output velocity from the rigid nozzle ( $Eh = \infty\,\rm N\,m^-{^1}$ ) is plotted for all ${L}/{D}$ values in figure 8(b). To compare the time scales of nozzle deformation with the input fluid flow, we define the time for the rigid nozzle to accelerate to 90 % of $U_{e}$ as $t_{acc}\approx 0.71$ sec, as shown in figure 8(b). Next, we examine how the nozzle deformations vary with $Eh$ to understand the kinematics affecting the flow fields.

Generally, each of the less stiff (more flexible) nozzles, ( $Eh = 54, 29, 19\,\rm N\,m^-{^1}$ ), exhibited similar nozzle deformation kinematics. The deformation is characterised by 3 important time phases: (i) initial tip deflection, (ii) periodic oscillations, (iii) nozzle collapse and return to original state. These three stages are plotted with overlaid time-coloured plot images of the $Eh = 54\,\rm N\,m^-{^1}$ nozzle for ${L}/{D}=4$ in figures 9(a)–9(e). See Supplementary Material 2 for similar plots for each of the flexible nozzles. During phase (i), the initiation of deformation, a travelling wave, or series of travelling waves, propagates down the nozzle. In turn, the first oscillation inward generates the largest deflection at the nozzle tip while the pump is running, defined as $t_{max\;def}$ . This motion is shown for the $Eh = 54\,\rm N\,m^-{^1}$ nozzle in figures 9(a) and 9(d). In phase (ii), $t_{max\;def}\lt t\lt t_{cycle}$ , the nozzles return to their initial position but with an over correction, expanding into the opposite plane of the initial tip deflection. These oscillations continue at varying frequencies depending on $Eh$ as shown in figures 9(b) and 9(e), until the pump turns off at $t/t_{cycle}=1$ marking the end of phase (ii). In phase (iii), $t\gt t_{cycle}$ , each less stiff nozzle ( $Eh = 54, 29, 19\,\rm N\,m^-{^1}$ ) collapses asymmetrically inward due to the decreased pressure from the quick decrease in velocity as the pump shuts off as shown from $t_{cycle}\lt t\lt 2t_{cycle}$ in figures 9(c) and 9(f). The $Eh = 54\,\rm N\,m^-{^1}$ nozzle tip immediately reopens after partially collapsing. In contrast, the $Eh = 29$ and $19\,\rm N\,m^-{^1}$ nozzles fully collapse and remain in the collapsed position for 9 and 25 s, respectively, before reopening and oscillating back to their original position. Each nozzle consistently collapsed on one preferential axis for every trial.

The $Eh = 76\,\rm N\,m^-{^1}$ nozzle was observed to have different kinematics than the less stiff nozzles. While the pump was moving, small-scale ( $\approx {0.02D}$ ) axisymmetric oscillations moved the nozzle slightly inward and outward until the pump stopped. Similar to the other nozzles, the pressure change at the nozzle exit created a small peak in deflection ( $\approx {0.04D}$ ) but the $Eh = 76\,\rm N\,m^-{^1}$ nozzle did not exhibit any collapse. Afterward, it oscillated back and forth until the viscous forces from the surrounding water damped the nozzle motion to a halt.

Figure 9. Imaging of the $Eh = 54\,\rm N\,m^-{^1}$ nozzle for ${L}/{D} = 4$ . The colour plot scale bar applies to each image, with the start and finish defined as follows in the panel descriptions. (a, d) Nozzle deformation for $t = $ $0 - t_{max\;def}$ . (b, e) Nozzle deformation for $t = t_{max\;def} - t_{cycle}$ . (c, f) Nozzle deformation for $t = t_{cycle} - 2t_{cycle}$ .

The bottom millimetre of the flexible nozzles was also quantitatively tracked. As expected, reducing the nozzle stiffness resulted in greater maximum deformation magnitudes at $t=t_{max\;def}$ , ranging from $0.02D$ to $0.19D$ for the $Eh = 76$ and $29\,\rm N\,m^-{^1}$ nozzles, respectively. Although the $Eh=19\,\rm N\,m^-{^1}$ nozzle is the least stiff, the peak deformation was lower at only $0.15D$ . This will be discussed further in § 3.2. The $Eh = 54\,\rm N\,m^-{^1}$ tip deformation is shown in figure 10(a), illustrating the previously described deformation of the nozzle’s bottom millimetre. It is noted that prior to its inward deflection, the nozzle undergoes a small radial expansion due to jet acceleration and the positive internal pressure gradient. However, the magnitude of this deformation is significantly smaller than the large inward deflection and therefore does not appear prominently in figure 10(a). For reference, a plot illustrating the tip deformation of all the flexible nozzles is shown in figure 11(a). In figure 10(a), phases (i), (ii) and (iii) are delineated by orange, blue and green shading. In phase (i), orange, the nozzle tip shape becomes a slightly non-axisymmetric elongated oval at $t=t_{maxdef}$ . In phase (ii), green, the nozzle quickly returns to an approximately axisymmetric shape with smaller periodic oscillations. In phase (iii), blue, the nozzle collapses from the negative pressure gradient within the nozzle becoming highly non-axisymmetric. The nozzle tracking shown is from the XY plane to clearly show the large collapse. Once the nozzle reopens at $t=t_{free response}$ , the oscillation behaviour can be modelled as an underdamped free oscillation since no external forces are acting on the nozzle, other than the initial condition of starting in a collapsed state. This is highlighted with yellow shading, in figures 10(a) and 10(b). The natural frequency of the flexible nozzles is found by transforming the positional periodic motion of the free oscillations into the frequency domain using FFTs. The data used for this frequency analysis begin when the nozzle returns to its original position ( $X/D=0$ ) after collapsing due to the change in pressure. The data used for the FFT analysis were obtained from views from both the XY and XZ planes.

For each nozzle, the frequency spectrum reveals a clear peak, indicating resonance at these specific frequencies under free oscillation, with an example frequency spectrum shown in figure 10(c). This resonant peak corresponds to the damped natural frequency ( $\omega _{d}$ ) for each nozzle. A summary of the variation of $\omega _d$ with $Eh$ is shown in figure 10(d), which is intuitive in that the stiffer nozzles have a higher $\omega _d$ values as stiffness scales with natural frequency. These $\omega _{d}$ values can be incredibly useful in that they can be used to predict the timing of $t_{max\;def}$ . The maximum variation in measured $\omega _{d}$ , was for the $Eh = 54\,\rm N\,m^-{^1}$ nozzle where $\omega _{d} = 0.737 \pm 0.013$ Hz based on the 95 % confidence interval.

Figure 10. (a) Temporal variation of position ( $X/D$ ) for the $Eh=54\,\rm N\,m^-{^1}$ nozzle tip. The $t_{free reponse}$ line indicates the start of the data used to find the frequency spectrum. (b) Tracked nozzle deflections used for FFT to quantify the frequency spectrum. The start is initiated by when the nozzle tip passes its original position ( $X/D=0$ ), after undergoing collapse due to the negative pressure gradient. (c) Frequency spectrum obtained from the positional data. Here, $\omega _{d}$ is identified as the most prevalent frequency in the frequency domain. (d) Value of $\omega _{d}$ for the different $Eh$ nozzles found using the same process as outlined here.

Choi & Park (Reference Choi and Park2022, Reference Choi and Park2024) proposed that synchronising the maximum deformation with the input fluid acceleration, i.e. timing the release of elastic energy to the fluid, creates the optimal conditions for thrust from a starting jet. Based on this, we hypothesise that the optimal nozzle stiffness for increasing thrust can be predicted using the damped natural frequency of the nozzle, such that the optimal condition will occur when $t_{acc}=t_{max\;def}$ , where $t_{acc}$ is the time to reach peak fluid acceleration from the rigid nozzle, and $t_{max\;def}$ is the time of maximum nozzle deformation while the pump is running. Using the damped natural frequency, $\omega _d$ , of each nozzle we predict $t_{max\;def}$ to occur at ${1}/{4}$ of an oscillatory cycle, i.e. $t_{max\;def} = {1}/{4\omega _{d}}$ . If $t_{acc}=t_{max\;def}= {1}/{4\omega _{d}}$ , we can estimate the optimal nozzle damped frequency, $\omega _{d, optimal}$ , for the present flow conditions (measured $t_{acc}$ ). Under the current flow conditions, $\omega _{d, optimal}=0.36\,\rm Hz$ . For the tested nozzles, the $Eh = 29\,\rm N\,m^-{^1}$ nozzle with $\omega _{d} = 0.38 \pm 0.008\,\rm Hz$ Hz is closest to $\omega _{d, optimal}$ , shown in figure 10(d). In the following section, we will confirm this prediction of the optimal nozzle stiffness (nozzle stiffness that will lead to greatest increase in thrust) by further examining the fluid output for each nozzle stiffness and $L/D$ .

Figure 11. (a) Temporal variation in nozzle tip deflection for ${L}/{D}=4$ . (b-c) Spatially averaged velocity measured $0.19D$ beneath the nozzle exit ( $U_e$ ) for $Eh = 54, 29, 19\,\rm N\,m^-{^1}$ for all ${L}/{D}$ normalied by the steadystate rigid nozzle velocity ( $U_0$ ) measured for each respective ${L}/{D}$ ; (b) $U_e/U_0$ for $Eh = 54\,\rm N\,m^-{^1}$ ; (c) $U_e/U_0$ for $Eh = 29\,\rm N\,m^-{^1}$ ; (d) $U_e/U_0$ for $Eh = 19\,\rm N\,m^-{^1}$ .

3.2. Optimal conditions, fluid impulse and circulation

To further investigate the relationship between $t_{max\;def}$ and $t_{acc}$ , we examine the measured output velocities from the $Eh = 54, 29$ and $19\,\rm N\,m^-{^1}$ nozzles, all of which have different $t_{max\;def}$ based on their respective $\omega _d$ . Three different timing scenarios can occur ( $t_{acc}\gt t_{max\;def}$ , $t_{acc}\lt t_{max\;def}$ , $t_{acc}=t_{max\;def}$ ), which is summarised by the nozzle tip deflection plot in figure 11(a) and the output velocity profiles from the $Eh = 54, 29$ and $19\,\rm N\,m^-{^1}$ nozzles in figures 11(b)–11(d). The $Eh = 76\,\rm N\,m^-{^1}$ nozzle velocity is not plotted as it matches the same timing scenario as the $Eh = 54$ N/m nozzle, but with much lower magnitude. For reference, $t_{max\;def}$ can be interpreted from figure 11(a) as the time to reach the first big peak in inward deflection.

In the first scenario, where $t_{acc}\gt t_{max\;def}$ , the nozzle tip reaches maximum inward deformation prior to the end of input fluid acceleration of the unperturbed flow. This condition is met by the $Eh = 54$ N m^–1 nozzle, where there is a slight enhancement in peak output velocity shown in figure 11(b). Here the nozzle contraction occurs at $t_{max\;def} = 0.65t_{acc}$ , after which the velocity decreases, as elastic energy is no longer being supplied to the flow after $t_{max\;def}$ . In contrast, after $t_{max\;def}$ the nozzle extracts energy back from the fluid to re-expand to its original position. In the re-expansion process a suction velocity is generated that counters the input flow direction and causes a decline in output velocity as the input fluid flow continues to accelerate until $t_{acc}$ . This mismatch in timing results in the output velocity profile reaching the maximum velocity at time $t_{max vel}$ , where $t_{max\;def}\lt t_{max vel} \lt t_{acc}$ , as the nozzle re-expansion temporarily impedes the output velocity growth after $t_{max\;def}$ .

The second scenario ( $t_{acc}\lt t_{max\;def}$ ) is met by the $Eh=19\,\rm N\,m^-{^1}$ nozzle where the nozzle contraction occurs at $t_{max\;def} = 1.4t_{acc}$ shown in figure 11(d). In this case, elastic energy supplied by the nozzle is transferred to the fluid throughout all of $t_{acc}$ , which supplements the velocity growth for a much longer period resulting in higher peak velocities compared with $Eh = 54\,\rm N\,m^-{^1}$ . However, the response of the nozzle lags behind the input acceleration of the fluid, so a portion of the elastic energy stored within the nozzle has not been fully transferred to the flow before $t_{acc}$ . As the input acceleration ends at $t_{acc}$ , there is a decrease in the growth rate of velocity. However, as the nozzle is still supplying elastic energy to the fluid, the velocity continues to grow until time $t_{max vel}$ where $t_{acc}\lt t_{max vel} \lt t_{max\;def}$ . Due to the delay in $t_{max\;def}$ , the acceleration is slower, and the peak velocity is lower compared with the optimal case.

For the optimal conditions, the $Eh=29\,\rm N\,m^-{^1}$ nozzle tip reaches peak inward deflection at the same exact time as the end of the input fluid acceleration, $t_{max\;def} = t_{acc}$ . This results in all of the elastic energy stored in the nozzle being imparted to the flow within $t_{acc}$ as shown in figure 11(c), as well as a greater tip deflection shown in figure 11(a). In this case, all of the elastic energy from the nozzle is supplied to the fluid within the same duration in which the input flow is accelerating, maximising the growth in the fluid velocity. Additionally, it is understood that the pressure change from acceleration to steady velocity input flow and the formation of a vortex ring creates a low pressure region at the nozzle tip at the same time as $t_{max\;def}$ , creating a positive feedback loop that magnifies the inward tip deflection and output (Gao et al. Reference Gao, Wang, Yu and Chyu2020). This behaviour is seen as a slight inflection point for all of the flexible nozzles in figure 11(a) at $t\approx t_{acc}$ , which results in the $Eh = 29\,\rm N\,m^-{^1}$ nozzle deflecting an additional $0.027{X}/{D}$ . Importantly, the nozzle deformation behaviour is consistent across all ${L}/{D}$ values during the fluid acceleration period, ensuring that this optimal timing is effective regardless of ejected volume.

For the $Eh = 54, 29$ and $19\,\rm N\,m^-{^1}$ nozzles with ${L}/{D} = 2$ and $4$ , there is a decline in output velocity after $t_{max\;def}$ , or $t_{acc}$ , depending on which occurs later, despite the input fluid velocity remaining constant. This decrease is attributed to the nozzle re-expanding back to its original position after $t_{max\;def}$ , creating a counter flow opposing the input flow conditions and extracting energy back from the fluid to reopen. As the nozzles return to their original position, the piston motion continues and then accelerates the fluid back to the input velocity until $t_{cycle}$ . In contrast, for ${L}/{D}=1$ , $t_{cycle}$ occurs before the nozzle can completely return to its original position, negating the need to overcome the entire counter flow generated by the nozzle reopening, maximising the relative flow output. The effect of ${L}/{D}$ is explored further in § 3.3.

To compare the vorticity generated as $Eh$ is varied, the total circulation ( $\Gamma$ ) is calculated using

(3.1) \begin{equation} \Gamma = 0.5 \int _{C} |\omega _z| \, {\rm d}A .\end{equation}

The 90 % vorticity contour is chosen as the integration area, $C$ , within the field of view defined by $-1.5\lt x/D\lt 1.5$ , and $-0.5\lt y/D\lt 5.5$ . Here, $|\omega _{z}|$ is the vorticity perpendicular to the measurement plane, as the flow is axisymmetric ( $\omega _x = \omega _y = 0$ ). Circulation is non-dimensionalised by dividing the temporal data by the circulation generated by the rigid nozzle ( $\Gamma _0$ ) at $t/t_{cycle}=1$ . Figure12(a) shows the temporal variation of total circulation ( ${\Gamma }/{\Gamma _{0}}|_{t/t_{cycle}=1}$ ) as $Eh$ is varied for ${L}/{D}=1$ . Similar to the trend observed to this point, the least stiff nozzles, $Eh=19\,\rm N\,m^-{^1}$ N/m and $Eh=29\,\rm N\,m^-{^1}$ , produced the largest relative circulation, ${\Gamma }/{\Gamma _{0}} = 2.87$ and $2.91$ , respectively. However, the $Eh = 19\,\rm N\,m^-{^1}$ , nozzle requires nearly twice as much time ( $t/t_{cycle}=2$ ) to reach peak circulation compared with the $Eh = 29\,\rm N\,m^-{^1}$ nozzle ( $t/t_{cycle}=1.1$ ). It is noted that $\Gamma$ is found to be equal in both the parallel (XY) and perpendicular (XZ) planes to the preferential axis of nozzle collapse for all experiments. See Appendix A for circulation plots comparing the two measurement planes.

To quantify the effect of nozzle stiffness on the output thrust, hydrodynamic impulse is calculated using (Saffman Reference Saffman1993)

(3.2) \begin{equation} I_h = \frac {1}{2}\rho _{f}\int _{C}x\times \omega \ {\rm d}V, \end{equation}

Figure 12. Temporal variation in total circulation and impulse for ${L}/{D}=1$ . (a) Total circulation ( $\Gamma$ ) normalised by the rigid nozzle ( $Eh=\infty$ ) circulation at $t/t_{cycle}=1$ . (b) Total impulse ( $I$ ), normalised by the rigid nozzle ( $Eh=\infty\,\rm N\,m^-{^1}$ ) impulse at $t/t_{cycle}=1$ . Legend is the same for (a) and (b).

where cylindrical coordinates are used with position vector $\boldsymbol{x} = (r, \theta, y)$ with the origin located at the centre of the nozzle exit, with positive $y$ in flow direction. The vorticity vector is simplified and given as $\omega _z$ for consistency. Considering ${\rm d}V = 2\pi r{\rm d}A$ , hydrodynamic impulse can be calculated as $I_h = \pi \rho _{f} \int _{C} r^2 |\omega _{z}|{\rm d}A$ . Here, $\rho _{f}$ is the fluid density at room temperature and $r$ is the radial distance to each integration element from the nozzle centre line. In figure 12(b), the temporal distribution of impulse ( ${I_h}/{I_0}|_{t/t_{cycle}=1}$ ) is plotted, where $I_0$ is the rigid nozzle impulse measured at $t/t_{cycle}=1$ . The measured hydrodynamic impulse increases as $Eh$ declines (becomes less stiff), up to $Eh=29\,\rm N\,m^-{^1}$ . Similar to circulation, the impulse generated by the least stiff nozzle ( $Eh=19\,\rm N\,m^-{^1}$ ) eventually achieves similar amounts of hydrodynamic impulse to the $Eh=29\,\rm N\,m^-{^1}$ nozzle, but not until $t/t_{cycle}=2.75$ , compared with $t/t_{cycle}=0.8$ , which is predicted to result in much lower force production which scales with impulse generated per time (Saffman Reference Saffman1993). As previously discussed in § 3.1, the flexible nozzles collapse under the negative pressure gradient formed upon fluid deceleration. Importantly, the hydrodynamic impulse generated from the flexible nozzles has no dip at $t=t_{cycle}$ , which would indicate a loss in the pressure impulse (Gao et al. Reference Gao, Wang, Yu and Chyu2020). In contrast, the unfavourable pressure gradient formed within the rigid nozzle results in a clear loss in hydrodynamic impulse at $t=t_{cycle}$ due to the generation of negative pressure impulse, as seen in figure 12(b). Through the nozzle collapse mechanism, the negative pressure gradient is instead suppressed as shown in figure 12(b), resulting in no additional losses in hydrodynamic impulse. A summary of the measured impulse at $t_{cycle}$ across all $Eh$ and ${L}/{D}$ is presented in figure 13(a). These results confirm that the $Eh = 29\,\rm N\,m^-{^1}$ nozzle produces the largest increase in thrust or hydrodynamic impulse as predicted from our nozzle deformation measurements (optimal nozzle condition), and any increase or decrease in $\omega _{d}$ results in a reduction in output thrust.

Interestingly, as a result of these differing measured output velocities with nozzle stiffness (despite the same kinematic input), the fluid slug length expelled also differs with flexibility. Specifically, the flexible nozzle deformations and collapse after $t/t_{cycle}=1$ expel additional fluid relative to the rigid case, increasing the total ejected volume that can positively contribute to the circulation of the vortex ring. However, after sufficient time the nozzle reopens, refilling itself returning the total ejected volume to equilibrium for mass conservation. The nozzle refilling process will be discussed further in § 3.4. As such, the effective expelled fluid slug length needs to be derived by integrating the $U_e$ profiles and dividing by the nozzle diameter to yield $({L}/{D})_{eff} ={1}/{D} \int U_e(t){\rm d}t$ . The circulation produced by the different nozzles should scale linearly with increasing $({L}/{D})_{eff}$ (Didden Reference Didden1979). This is confirmed in figure13(b), where the data show a linear trend as expected from Gharib et al. (Reference Gharib, Rambod and Shariff1998).

Figure 13. Summary of the total measured impulse and circulation from each nozzle over all ${L}/{D}$ values. (a) Normalised impulse plotted versus damped natural frequency $\omega _d$ , with the predicted optimal $\omega _{d, optimal}$ condition shown. (b) Measured total circulation versus effective ejected volume ( $({L}/{D})_{eff}$ ).

The lowest ejected volume, ${L}/{D}=1$ , resulted in the highest normalised impulse compared with the rigid nozzle data, as shown in figure 13(a). This is contributed to four aspects: initial increase of fluid velocity from the nozzle elastic energy, no loss in pressure impulse due to fluid deceleration, high relative increase in $({L}/{D})_{eff}$ and formation of a single coherent vortex structure. The first aspect is related to the flexible nozzles imparting elastic energy when $t\lt t_{max\;def}$ , which occurs early in the fluid expulsion process. As the deformation behaviour of the nozzles, and thus elastic energy supplied, is approximately constant across all ${L}/{D}$ , the relative gain of this energy is maximised for the lowest ejected volume. Additionally, after $t_{max\;def}$ the flexible nozzles extract energy back from the flow to re-expand, which creates a suction velocity that counters the input flow direction. For ${L}/{D}=1$ , the pump turns off when the nozzle re-expansion process is taking place, whereas the larger ejected volumes must completely overcome this suction velocity to continue to expel additional fluid.

The second aspect, the suppression of the negative pressure gradient from nozzle collapse, is amplified at lower ejected volumes. For lower ejected volumes, a slight loss of hydrodynamic impulse due to negative pressure impulse has a proportionally larger effect on the total produced. The flexible nozzles collapse inward when this unfavourable pressure region is formed, and in turn not only suppresses the negative impacts of the negative pressure region, but also expel additional fluid volume.

The third aspect, the relative increase in effective ejected volume, $({L}/{D})_{eff}$ , is maximised at lower ejected volumes. The added volume from the nozzle deformation is approximately constant across all ${L}/{D}$ for a given $Eh$ . Consequently, a small increase in ejected volume can significantly enhance the normalised impulse and circulation when a small ${L}/{D}$ is ejected from the pump’s perspective.

The fourth aspect is attributed to the formation of a single coherent vortex structure which maximises the pressure impulse created per unit volume (Krueger & Gharib Reference Krueger and Gharib2003). The impulse gain from pressure impulse can be a quite substantial contribution of the total impulse when forming an isolated vortex ring, thus can be significantly leveraged to maximise thrust efficiency per unit volume from this type of propulsor. This aspect is studied in § 3.3.

3.3. Vortex ring formation and relative vorticity contributions

Figure 14. Measured number of vortex structures for all measured $({L}/{D})_{eff}$ and $Eh$ .

As shown in § 3.1 and Supplementary Material 1, larger ejected volumes from the more flexible nozzles result in multiple vortex structures, even when $({L}/{D})_{eff}\lt \approx 4$ . This is summarised in figure 14, where the number of vortex structures and measured $({L}/{D})_{eff}$ is plotted. Krieg & Mohseni (Reference Krieg and Mohseni2021) showed that vortex ring pinch-off can be predicted using the output velocity ( $U_e$ ) and the characteristic velocity ( $U_c$ ) of the vortex ring. They showed that a vortex ring will pinch-off and stop gaining vorticity when the ring exceeds its feeding velocity, which is ( $2U_{e}$ ) for fully developed Poiseuille, or pipe, flow. For the current flow conditions, it is assumed that by the end of fluid acceleration the flow is fully developed into Poiseuille flow allowing for this subsequent pinch-off prediction to be used. The $U_c$ value of the ring is derived in (Krieg & Mohseni Reference Krieg and Mohseni2021), and is as shown here

(3.3) \begin{equation} U_c = \sqrt {\frac {\rho _f \pi \Gamma ^3}{4I}} \geqslant 2U_{\text{e}}. \end{equation}

Here, all variables are defined as described in § 3.2. Figure 15(a) shows $U_c$ compared with $2U_{e}$ . As shown in (a), the primary ring is expected to pinch-off at $t_{pinch} = 0.33 t/t_{cycle} = 1 sec$ , when $U_{c}$ surpasses $2U_{e}$ . We confirm this prediction by calculating the pLCS and nLCS ridges in the FTLE field in this time frame to show material boundaries indicative of pinch-off (Shadden et al. Reference Shadden, Dabiri and Marsden2006). In figures 16(a)–16(c), a closed LCS loop from overlaying the nFTLE and pFTLE fields is shown, indicating a pinched off vortex ring after the predicted $t_{pinch}$ . As vortex ring pinch-off isn’t an instantaneous process, the additional time steps shown at $t=1.5t_{pinch}$ and $t=3.75t_{pinch}$ in figures 16(b) and 16(c) provide additional insight that a single vortex structure is formed with a closed material boundary from the nLCS. As time progresses for ${L}/{D}=4$ , additional vorticity is produced and SV structures form in the wake of the PV. However, the primary closed loop of vorticity continues to travel downward without gaining additional fluid from the trailing jet formed in its wake, indicating an accurate prediction of $t_{pinch}$ .

Figure 15. Predicting PV pinch-off for $Eh=29\,\rm N\,m^-{^1}$ , ${L}/{D}=4$ . (a) Temporal evolution of $U_c$ and $2U_e$ to predict PV pinch-off when $U_c\gt 2U_e$ ( $t_{pinch}=0.33t/t_{cycle}\approx {1.0}$ sec). (b) Circulation contributions from PV and SV, calculated using the vorticity contours informed by the nLCS and pLCS.

Through examining FTLE fields for each case within the studied parameter space, we can confirm closed off structures within the flow field. With this information, the vorticity fields measured directly from PIV can be used to separate the total circulation produced by each nozzle into the PV ring and the trailing jet formed in its wake. Figures 16(d)–16(e) show vorticity contours at the same instants as the FTLE fields in figures 16(a)–16(c), which are used to delineate circulation into separate parts of the flow, as informed by closed LCS contours in the FTLE field.

Figure 16. The FTLE fields and vorticity plots from the $Eh=29\,\rm N\,m^-{^1}$ nozzle with ${L}/{D}=4$ . The red contours show attractive pLCS, and the blue contours show repelling nLCS: (a) nLCS and pLCS for $t=t_{pinch}$ ; (b) nLCS and pLCS for $t=1.5t_{pinch}$ ; (c) nLCS and pLCS for $t=3.75t_{pinch}$ ; (d–e) vorticity contours corresponding to the same time steps as the FTLE plots directly above each frame.

Figure 15(b) shows the circulation contributions from the PV and SV structures and their sum of total circulation for $Eh = 29\,\rm N\,m^-{^1}$ . By informing the vorticity calculation through the use of the FTLE fields, the circulation from separate parts of the flow were confidently calculated without question as to whether or not vorticity was included from the trailing jet when calculating circulation for the PV and vice versa. As expected, the PV does not gain additional vorticity after the predicted $t_{pinch}$ . Interestingly, the PV contains approximately 50 % of the total circulation produced, while the piston has only moved ${1}/{3}$ of its total prescribed motion for ${L}/{D} = 4$ . This highlights that the benefits of utilising a flexible nozzle are most pronounced at a low ejected volume as there is a relatively diminished return in total circulation for the latter ${2}/{3}$ of the piston motion, even though the piston travels a greater distance after the PV has pinched off. Additionally, for the $Eh=29\,\rm N\,m^-{^1}$ nozzle ${L}/{D}=1$ , $t_{pinch} \approx t_{cycle}$ , indicating that the pump turns off when the PV exceeds the feeding velocity. In turn, this optimises the ejected volume by entraining the maximum vorticity within the PV, while minimising the piston motion. In figure 17(a) we compare the PV circulation measured from each ejected volume for the $Eh = 29\,\rm N\,m^-{^1}$ nozzle and despite multiple vortex structures forming for ${L}/{D} = 2, 4$ , we confirm that the PV strength ( $\Gamma _{PV}$ ) does not grow with additional ejected volume beyond ${L}/{D} = 1$ . Given the exit (escape) velocity ( $U_e$ ) measured for each nozzle is approximately constant for $t/t_{pinch}\lt 1$ we expect to find similar pinch-off results from varied ${L}/{D}$ .

Figure 17. Primary vortex circulation. (a) Temporal development of only the PV circulation for all ${L}/{D}$ values generated with the $Eh=29\,\rm N\,m^-{^1}$ nozzle. (b) Temporal development of PV circulation from each $Eh$ nozzle for ${L}/{D}=2$ .

This process was repeated for each nozzle and the vorticity contained within the PV as $Eh$ is varied follows the same trend as total circulation as shown in figure 13(b). The temporal variation of circulation measured for each of the PV cores produced by each nozzle from ${L}/{D}=2$ is plotted in figure 17(b). One point to note is that the $Eh = 19\,\rm N\,m^-{^1}$ PV pinches off significantly later than $Eh = 29\,\rm N\,m^-{^1}$ , as shown by the prolonged increased in circulation; however, the circulation contained within the PV is lower due to the lack of momentum flux (exit velocity) compared with the $Eh =29\,\rm N\,m^-{^1}$ nozzle. The $Eh=29\,\rm N\,m^-{^1}$ nozzle creates the strongest PV, and further explains why the high non-dimensionalised impulse is reached for ${L}/{D}=1$ , as the pump turns off immediately following PV pinch-off maximising the gain from pressure impulse. However, there is a very weak secondary structure formed for $Eh = 29\,\rm N\,m^-{^1}$ , ${L}/{D}=1$ , when the nozzle collapses after $t_{cycle}$ , yet the strength of this structure is very small and almost the entirety of the ejected fluid is contained within the PV.

3.4. Stopping vortex formation and implications for bio-propulsors

To provide a complete picture of an entire cycle of the movement of the flexible nozzles, we investigate the flow structure produced by the nozzles reopening after collapse. Given that the nozzles were manufactured using clear silicone, we can perform PIV within the structure of the flexible nozzles (Choi & Park Reference Choi and Park2022). As described in § 3.1, the less stiff nozzles ( $Eh = 54, 29, 19\,\rm N\,m^-{^1}$ ) collapse due to a pressure change at the outlet after the pump stops moving at $t_{cycle}$ . This is in contrast to the low pressure region formed within rigid nozzles upon fluid deceleration causing a decrease in pressure impulse (Gao et al. Reference Gao, Wang, Yu and Chyu2020). The flexible nozzles compliantly deform when this negative pressure gradient is formed and as a result see no loss in impulse as would be characterised by a slight dip in hydrodynamic impulse (Krueger & Gharib Reference Krueger and Gharib2005). After collapse, the nozzles reopen at time $t_{open}$ causing a suction velocity, which creates a stopping vortex of varied strength within the nozzle itself, similar to stopping vortices found in rigid nozzles that positively contribute to pressure impulse (Gao et al. Reference Gao, Wang, Yu and Chyu2020). The time to reach $t_{open}$ is 0.95 s after $t_{cycle}$ for $Eh=54\,\rm N\,m^-{^1}$ , whereas $t_{open}$ does not occur until 9 and 25 s after $t_{cycle}$ for $Eh = 29$ and $19\,\rm N\,m^-{^1}$ , respectively. Additionally, the time for the nozzles to fully refill ( $t_{refill}$ ) (the time at which the circulation of the stopping vortex is the highest), is delayed with decreased nozzle stiffness. A summary of these time scales is shown in table 2 and figure 20(b) shows the nonlinear increase of $t_{refill}$ as $Eh$ is varied. Given that the $Eh = 76\,\rm N\,m^-{^1}$ nozzle does not collapse, it does not produce a strong stopping vortex within the nozzle interior, and is not included in this stopping vortex analysis. It should be noted that whilst both the $Eh = 76\,\rm N\,m^-{^1}$ and rigid nozzles do generate a beneficial stopping vortex below the nozzle exit after fluid deceleration is complete, it is very weak compared with the stopping vortex generated by the nozzles that collapsed and is outside the nozzle itself.

Table 2. Nozzle reopening time scales.

In terms of total momentum produced, forming stronger stopping vortices within the nozzle can increase the thrust of a bio-inspired propulsor, due to increased forward momentum within the nozzle itself (Xiaobo et al. Reference Xiaobo, Wei, Qiang and Jun2021). It has been demonstrated that for the swimming performance of siphonophores, animals that propel themselves with cyclical jets with refill cycles, the refill phase augments their distance travelled by 17 % over theoretical results which only generate thrust from the jet phase (Sutherland et al. Reference Sutherland, Gemmell, Colin and Costello2019). As the fluid accelerates back into the siphonophore, or in our case the nozzle, it generates a high pressure region that should augment thrust. Thus, quantifying this stopping vortex is imperative, in terms of summing the total thrust produced by our varied flexibility nozzles. Returning to jet propelled aquatic animals such as jellyfish and squid, they primarily produce forward momentum in the expelling cycle of their motion, but their swimming efficiency is linked to the generation of positive momentum in the refilling cycle known as passive energy recapture (Gemmell et al. Reference Gemmell, Colin and Costello2018). The locomotion of jellyfish particularly is significantly augmented by the presence of a stopping vortex which can account for up to 50 % of the jellyfish’s travel per cycle (Gemmell et al. Reference Gemmell, Costello, Colin, Stewart, Dabiri, Tafti and Priya2013).

Figure 18. Stopping vortices produced varied $Eh$ nozzles at $t=0.75(t_{refill}-t_{open})$ . (a) Shows $Eh = 54\,\rm N\,m^-{^1}$ . Note that the $Eh=54\,\rm N\,m^-{^1}$ nozzle immediately reopens, and thus there is secondary vorticity labelled in the frame, separate from the stopping vortex. Only vorticity contained within the stopping vortex is included for the circulation calculation. (b) Shows $Eh = 29\,\rm N\,m^-{^1}$ and (c) $Eh = 19\,\rm N\,m^-{^1}$ .

Figure 19. Circulation produced by the stopping vortices formed within the $Eh = 54, 29,$ and $19\,\rm N\,m^-{^1}$ nozzles after $t_{open}$ ; (a) ${\Gamma _{stop}}/{\Gamma _{0}}|_{t/t_{cycle}=1}$ , (b) ${\Gamma _{stop}}/{\Gamma _{x}}$ . Legend is the same for (a) and (b).

Figures 18(a)–18(c) show stopping vortices forming in the interior of the flexible $Eh=54, 29$ and $19\,\rm N\,m^-{^1}$ nozzles at $t=0.75(t_{refill}-t_{open})$ , or the time at which the stopping vortex has reached 75 % of the maximum $\Gamma _{stop}$ . The strength of these stopping vortices is calculated using the circulation formulation described in § 3.3 and the temporal variation is plotted in figures 19(a) and 19(b). In figure 19(a) ${\Gamma _{stop}}/{\Gamma _{0}}|_{t/t_{cycle}=1}$ is plotted, where $\Gamma _0$ is the circulation produced by the rigid nozzle throughout fluid expulsion for ${L}/{D}=1$ . This normalisation is selected to illustrate the importance of accounting for these stopping vortices relative to the circulation generated by the rigid nozzle. In figure 19(b) ${\Gamma _{stop}}/{\Gamma _{x}}$ is plotted where $\Gamma _x$ is the maximum circulation achieved by each respective flexible nozzle in expelling fluid for ${L}/{D}=1$ . Note that the bounding area for the stopping vortex calculation was altered such that vorticity more than $0.1D$ below the nozzle tip was not included in the calculation to avoid including vorticity produced in the jetting phase for the $54\,\rm N\,m^-{^1}$ nozzle as this nozzle starts to form stopping vortices rapidly after jetting. Additionally, measurements were cut off at $t-t_{open} = 3$ s, for $Eh = 29$ and $54\,\rm N\,m^-{^1}$ , due to the earlier occurrence of maximum $\Gamma _{stop}$ . Alternatively, measurements were continued to $t - t_{open}=6$ s for the much slower reopening $Eh=19\,\rm N\,m^-{^1}$ nozzle. The strength of the stopping vortex scales with the stiffness of the collapsing flexible nozzles with the $Eh = 54\,\rm N\,m^-{^1}$ nozzle creating the strongest and $Eh = 19\,\rm N\,m^-{^1}$ generating the weakest, as the stiffer nozzle reopens much faster, i.e. shorter $t_{refill}$ , generating a stronger stopping vortex. The strength of the vortex is not affected by ${L}/{D}$ as the same initial condition is provided to each nozzle at the end of flow generation with the same deceleration of fluid, or pressure change resulting in a collapsed nozzle shape.

Figure 20(a) shows the maximum stopping vortex strength ( $\Gamma _{stop}$ ) for each nozzle stiffness alongside the maximum total circulation generating while expelling fluid ( $\Gamma _{expulsion}$ ) at ${L}/{D} = 1$ , normalised by the maximum rigid nozzle circulation ( $\Gamma _0$ ) at ${L}/{D} = 1$ . Here, $\Gamma _{stop}$ is averaged across all ${L}/{D}$ , as there is negligible difference between the strength of the vortices produced with $L/D$ (nozzle starts with the same, collapsed initial condition for all $L/D$ ; changing the ejected volume did not alter the nozzle reopening rate). As shown in figure 20(a), the $Eh = 54\,\rm N\,m^-{^1}$ nozzle produced the strongest stopping vortex, whereas the other nozzles either do not collapse ( $Eh = 76\,\rm N\,m^-{^1}$ ) or reopen slowly after collapse ( $Eh = 29$ and $19\,\rm N\,m^-{^1}$ ), resulting in weaker stopping vortices. Notably, the peak for $\Gamma _{stop}$ does not align with the peak (optimal nozzle stiffness for increased expulsion circulation) for $\Gamma _{expulsion}$ . This suggests that the full-cycle can be further optimised in the case of pulsed vortex rings (maximum circulation generated in both expulsion and in stopping) by ensuring these peaks align.

However, in addition to the vortex strength, consideration must also be given to the time for the nozzle to finish refilling after the expulsion process (i.e. completely generate the stopping vortex and return to initial open state). It is noted that the time to refill ( $t_{refill} - t_{cycle}$ ) for $Eh = 29\,\rm N\,m^-{^1}$ is six times longer than the $Eh = 54\,\rm N\,m^-{^1}$ nozzle. As the propulsive benefits of the flexible nozzle depend on its initial open starting condition (to ensure maximum inward deflection during vortex ring expulsion), it is important that the nozzle return to this state quickly to begin the next vortex expulsion cycle. Although the $Eh = 29\,\rm N\,m^-{^1}$ nozzle may outperform the other nozzles during the expulsion process in the present study, the $Eh = 54\,\rm N\,m^-{^1}$ nozzle would likely be more applicable to a bio-inspired propulsor as the time to complete a expulsion–refill cycle is greatly reduced. This highlights the importance of tuning the acceleration of the fluid with the damped natural frequency of the stiffest nozzle that will still collapse under deceleration to generate a strong PV ring, a strong stopping vortex and all within a relatively short duration.

Figure 20. Circulation and time for the flexible nozzles to fully reopen; (a) $\Gamma _{stop}/\Gamma _{0}$ averaged across all ${L}/{D}$ values and $\Gamma _{expulsion}/\Gamma _{0}$ for ${L}/{D}=1$ , (b) $t_{refill} - t_{cycle}$ , or time after the pump turns off for the nozzles to completely reopen generating the stopping vortex.

To summarise, the present study, in alignment with previous studies (Choi & Park Reference Choi and Park2022, Reference Choi and Park2024), indicates that the optimal stiffness nozzle in the jetting phase should have peak tip deformation synchronised with the input fluid acceleration for maximal momentum transfer. However, we also now observe that it is equally important that the nozzle reopens as quickly as possible during the refilling cycle, which adds a second condition to define the optimal nozzle stiffness for a complete expulsion–refill cycle. This secondary aspect of nozzle stiffness is characterised by a nozzle that is compliant enough to collapse from fluid deceleration, yet remains as stiff as possible to reopen quickly. Despite forming a stopping vortex, this is expected to benefit propulsion by creating net positive momentum and a favourable pressure gradient. This suggests that tuning the system ( $t_{acc}$ ) to maximise the jetting conditions for the stiffest nozzle capable of collapse would generate the largest net thrust over a full expulsion-refill cycle by additionally forming the strongest stopping vortices.