3. Results and discussion

Figure 2(a) illustrates the representative deformation of the most flexible nozzle ($Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$) corresponding to the intermediate jet acceleration of $\varPi _{0} = 0.57$. The temporal variations of jet-exit velocity at the centre ($x=y=0$) are plotted with the polar angle $(\theta)$ (defined in figure 2a) for various $Eh$ (figures 2b–e). In figure 2(a), the side edges of the deformed nozzle are illuminated by the laser sheet on the $x$–$y$ plane. During the jet acceleration ($t^{*} < 1.0$), the cylindrical nozzle expands radially (see $t^{*} = 0.5$) and contracts ($t^{*} = 0.9$) to its original shape. After the jet starts to decelerate ($t^{*} > 1.0$), the nozzle contracts further by the negative pressure inside the nozzle, leading to a buckled state ($t^{*} = 1.2$), and subsequently, irregular foldings occur ($t^{*} = 2.0$). During the nozzle deformation before complete folding, the jet-exit velocity always aligns with the $y$-axis (i.e. $\theta = -90^{\circ }$) for all nozzle stiffnesses $Eh = \infty, 43.2, 14.4, 7.0\ {\rm N}\ {\rm m}^{-1}$ (and all jet conditions $\varPi _{0} = 0.48\unicode{x2013}0.64$), which indicates that the jet flow keeps the axisymmetry (the cases for $\varPi _{0} = 0.48$ are shown in figures 2b–e). A slight deviation occurs at the least flexible nozzle ($Eh = 43.2\ {\rm N}\ {\rm m}^{-1}$) because of the weak jet momentum at $t^{*} > 3.0$ (figure 2c), which is far after the end of the acceleration. However, for all cases in the present study, the flow and nozzle deformation remain axisymmetric for $t^{*} < 2.0$ and $t^{*} < 1.0$, respectively. Thus in the following analysis, it is assumed that the jet flow and nozzle deformation are approximately axisymmetric.

Typical evolution of the pulsed jet from the rigid nozzle for the fastest acceleration of $\varPi _{0} = 0.48$ is shown in figure 3(a). After the piston starts to accelerate at $t^{*}\ (= t/T_{acc}) = 0$, the shear layer is fed into the ambient fluid near the exit, and forms the vortex ring at $y/D = -0.2$. As the jet decelerates at $t^{*} = 1.0$, the vortex ring detaches from the nozzle exit (see $t^{*} = 1.5$). The negative pressure is developed inside the nozzle by the sudden flow stall inside the nozzle and the flow induced downstream by detached vortex ring, thereby resulting in the suction of ambient flow into the nozzle (Gao et al. Reference Gao, Wang, Simon and Chyu2020). Consequently, the secondary counter-rotating vortex ring appears at the nozzle exit ($t^{*} = 2.0$). When $t^{*} > 2.0$, the secondary vortex is entrained into the nozzle, and the primary vortex ring becomes completely isolated and moves downstream (see supplementary material and movie 1 for the detailed movements of the jet vortices). There is no trailing jet behind the primary vortex because the present formation number ($F = 2.7$) is sufficiently smaller than the critical value ($F = 4$), above which the vortex ring detaches spontaneously by the excessive supply and accumulation of vorticity (Gharib et al. Reference Gharib, Rambod and Shariff1998; Krueger & Gharib Reference Krueger and Gharib2003).

Figure 3. (a,b) Evolution of the vorticity ($\hat {\omega } = \omega D/v_{m,r}$) contour and velocity vectors in a pulsed jet with the fastest acceleration ($\varPi _{0} = 0.48$) for (a) rigid and (b) flexible ($Eh = 14.4\ {\rm N}\ {\rm m}^{-1}$) nozzles. (c–e) Temporal variation of: (c) the centreline velocity ($\hat {v}_{e} = v_{e}/v_{m,r}$) at the nozzle exit ($y = 0$) normalized with the maximum value for the rigid nozzle ($v_{m,r}$); (d) normalized total circulation ($\hat {\varGamma }=\varGamma /(v_{m,r}D)$); (e) normalized total jet volume ($\hat {V}=V/V_p$), where $V_p$ represents the displaced volume by piston. The dimensionless time is defined as $t^{*} = t/T_{acc}$.

The jet vortices from the flexible nozzle are significantly intensified, and their dynamics are considerably altered (figure 3b). During the acceleration ($t^{*} < 1.0$), the nozzle expands radially because of positive pressure inside, building tensile stress along the nozzle wall. Consequently, a portion of the work done by the piston is stored as elastic energy and used to increase the jet kinetic energy. The jet-exit velocity ($\hat {v}_{e}=v_{e}/v_{m,r}$) of the flexible nozzle with $Eh = 14.4\ {\rm N}\ {\rm m}^{-1}$ is temporally slower than the rigid one at $t^{*} < 1.0$ because of the redistribution of energy (figure 3c). The stored elastic energy is released to supplement the jet kinetic energy, and $\hat {v}_e$ increases by ${\sim }170\,\%$ (for $Eh = 14.4\ {\rm N}\ {\rm m}^{-1}$) compared to the maximum value for the rigid nozzle at $t^{*} = 2.0$, at which the jet-exit velocity of the rigid nozzle has been reduced substantially. Contrary to the rigid nozzle, the counter-rotating secondary vortex ring is not induced at the exit of the flexible nozzle (see $t^{*} = 2.0$ in figure 3b), which indicates that the negative pressure is negligible or suppressed by nozzle shrinkage. This collapse behaviour (figure 2a) is responsible for the difference in jet vortex dynamics, i.e. additional fluid induced towards the jet direction is entrained into the primary vortex ring ($t^{*} = 4.0$ in figure 3b) and further increases the circulation of the primary vortex ring at $t^{*} > 1.0$ (figure 3d). Here, the nozzle collapses inwardly after contracting beyond its original shape ($t^{*}>1.0$ in figure 2a), since the buckled structure attributed to the bending has a smaller energy state than maintaining the axisymmetric shape (Kraus Reference Kraus1967). Considering the considerably thin wall of the present nozzle ($h \sim {O}(10^{2})\ \mathrm {\mu }$m), the bending force (${\sim }h^{3}$) on the wall is far smaller than the tensile force (${\sim }h^{2}$), indicating that the collapsing motion would exert negligible fluid pressure to the surrounding fluid compared to the expansion–contraction behaviour. Thus the flexible nozzle acts as if it vanished during the deceleration stage, effectively suppressing the negative pressure that mitigates the jet velocity and vortex circulation (Krueger & Gharib Reference Krueger and Gharib2003; Gao et al. Reference Gao, Wang, Simon and Chyu2020).

In figures 3(c–e), we compared the jet characteristics in terms of the exit velocity, total circulation and total jet volume, according to $Eh$. The case $Eh = \infty$ denotes the rigid nozzle. In figure 3(d), the total circulation is defined as $\varGamma = \int _{S^+}\omega \, {\rm d} A$, where $S^+$ is the region $0 < x/D < 1.5$ and $-8 < y/D < 0$, in which half of the counter-rotating vortex ring moves downstream. The jet volume (figure 3e) passing through $y/D = -1.0$ is calculated as $V = \int _{0}^{t} \int _{-1.5D}^{1.5D} v(2{\rm \pi} x)\, {\rm d} x \, {\rm d} t$, where $v$ represents the vertical velocity, and is further normalized by the volume displaced by piston movement ($V_p$), as $\hat {V}=V/V_{p}$. Thus $\hat {V}$ shows the entrainment rate compared to the volume supplied by the piston motion. In figure 3(d), the flexible nozzle substantially increases the maximum centreline velocity ($v_e$), total circulation ($\varGamma$) and total jet volume ($V$), compared to the rigid nozzle. They are not monotonically enhanced with the flexibility. For the cases shown, the maximum increase in the exit velocity and circulation is achieved at $Eh = 14.4\ {\rm N}\ {\rm m}^{-1}$ (see inverted triangle symbols in figures 3c,d).

Thus the dependency on flexibility changes according to the input jet condition ($\varPi _0$). In figure 4, we compared the hydrodynamic impulse ($\hat {I}_{h} = I_{h}/(v_{m,r}D^{3})$), maximum centreline velocity at nozzle exit ($\hat {v}_{m} = v_{m}/v_{m,r}$), total circulation ($\hat {\varGamma }=\varGamma / (v_{m,r}D)$) and total fluid volume ($\hat {V} = V/V_{p}$) passing through $y/D = -1.0$, depending on $Eh$ and $\varPi _0$. The hydrodynamic impulse ($I_h$) was calculated as $I_{h} = 0.5\rho _{f}\int _{C}\boldsymbol {x}\times \boldsymbol {\omega }\, {\rm d} V$ (Saffman Reference Saffman1955). Here, $\boldsymbol {x}=(r,\theta,y)$ denotes the position vector originating from the centre of the nozzle exit (as shown in figure 1b), and $\boldsymbol {\omega }=(\omega _{r},\omega _{\theta },\omega _{y})$ represents the vorticity vector. Given the axisymmetric nature of the flow ($\omega _{r}=\omega _{y}=0$) and the differential volume ${\rm d} V=2{\rm \pi} r \times {\rm d} A$, the hydrodynamic impulse simplifies to $I_{h}=0.5{\rm \pi} \rho _{f}\int _{C}r^{2}\,|\omega _{\theta }(r,y)|\, {\rm d} A$. The control surface $C$ is in the ranges $-2 \le x/D \le 2$ and $-5.5 \le y/D \le 0$ in the measurement plane, excluding the area inside the nozzle located at $y/D>0$, and $\omega _\theta$ represents the vorticity component perpendicular to the measurement plane. Also, $\hat {I}_h$ and $\hat {\varGamma }$ are measured at $t^{*} = 4.0$, and $\hat {V}$ at $t^{*} = 20.0$, after which they converge. For $\varPi _{0} = 0.48$, when the pulsed jet accelerates the fastest (figure 3), the nozzle with $Eh = 14.4 {\rm N}\ {\rm m}^{-1}$ outperforms the nozzles that are more ($Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$) and less ($Eh = 43.2\ {\rm N}\ {\rm m}^{-1}$) flexible. The maximum centreline velocity shows a strong correlation with the impulse and circulation (figures 4a–c) because the growth rates of both hydrodynamic impulse ${\rm d} I_{h}/{\rm d} t \sim \rho _{f} v_{e}^{2} A$ (Gao et al. Reference Gao, Wang, Simon and Chyu2020) and circulation ${\rm d} \varGamma /{\rm d} t \sim v_{e}^{2}/2$ (Krieg & Mohseni Reference Krieg and Mohseni2013) scale with the jet-exit velocity ($v_{e}^{2}$). The jet volume has less variation ($\Delta \hat {V}<1.0$) among different flexibilities (except for the rigid nozzle) under the same jet condition (figure 4d). However, it increases with decreasing $\varPi _0$ because of a larger amount of entrainment into the vortex ring by the faster acceleration (Ruiz, Whittlesey & Dabiri Reference Ruiz, Whittlesey and Dabiri2011). As $\varPi _0$ increases to $0.57$ and $0.64$, the maximum velocity ($\hat {v}_m$) of the least flexible nozzle ($Eh = 43.2\ {\rm N}\ {\rm m}^{-1}$) becomes comparable to the rigid nozzle (figure 4b) because the weakened acceleration is not sufficient to expand the stiff nozzle. However, the impulse and circulation are still slightly higher than the rigid nozzle (figures 4a,c), which is attributed to the additional fluid supplied during the nozzle collapse at $t^{*} > 1.0$. For this weakly accelerated condition ($\varPi _{0} = 0.57\unicode{x2013}0.64$), softer nozzles with $Eh = 7.0$ and $14.4\ {\rm N}\ {\rm m}^{-1}$ are still sufficiently flexible to amplify the kinetic energy of the jet. For the weakest acceleration ($\varPi _{0} = 0.64$), the optimal flexibility transitions to $Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$, supporting our claim on the optimal flexibility coupled with the flow condition.

Figure 4. Variation of (a) hydrodynamic impulse ($\hat {I}_{h} = I_{h}/(v_{m,r}D^{3})$) at $t^{*} = 4.0$, (b) maximum centreline velocity ($\hat {v}_{m} = v_{m}/v_{m,r}$), (c) total circulation ($\hat {\varGamma }=\varGamma / (v_{m,r}D)$) at $t^{*} = 4.0$, and (d) total jet volume ($\hat {V} = V/V_{p}$) at $t^{*} = 20.0$, depending on $\varPi _0$ and $Eh$. The shaded symbols in (a) correspond to cases used to further examine the nozzle deformation in figure 5.

Figure 5. (a) Schematic of nozzle displacement ($w_{n,r}$ for $x > 0$, and $w_{n,l}$ for $x < 0$) perpendicular to the original surface in the cross-sectional ($x$–$y$) plane. (b,d,f) Overlay of nozzle edge profiles ($\hat {w}_n$) at various $t^*$, and (c,e,g) spatio-temporal variation of the nozzle deformation ($\hat {w}_n$) depending on $Eh$ and $\varPi _0$. Here, (b,c) $(Eh, \varPi _0) = (14.4\ {\rm N}\ {\rm m}^{-1}, 0.64)$, (d,e) ($7.0, 0.48$), (f,g) ($14.4, 0.57$). The normalized nozzle deformation is defined as $\hat {w}_n = 0.5(w_{n,r}+w_{n,l})/w_{n,m}$, in which the maximum value ($w_{n,m}$) is $0.016D$, $0.153D$ and $0.038D$ for (b,c), (d,e), and (f,g), respectively. In (b,d,f), the variation of $\hat {w}_n$ along the $y$-direction is shown at both sides for visualization.

To understand the underlying fluid–structure interaction, the radial displacement ($w_n$) of the nozzle is measured along the nozzle length ($y$) and time ($t$) with resolution $0.004D$ ($= 60\ \mathrm {\mu }$m) (figure 5a). Figures 5(b,d,f) show the overlay of nozzle edge profiles at various instants ($t^{*} < 2.0$), and figures 5(c,e,g) present the corresponding spatio-temporal contours of nozzle deformation ($\hat {w}_n$) depending on $Eh$ and $\varPi _0$. The nozzle tends to deform axisymmetrically until buckling ($t^{*}<1.0$), therefore the displacements measured at two sides (i.e. $w_{n,r}$ and $w_{n,l}$ in figure 5a) were averaged and normalized by the maximum value ($w_{n,m}$), as $\hat {w}_n=0.5(w_{n,r}+w_{n,l})/w_{n,m}$. The white area in figures 5(c,e,g) represent the data removed because of the noise caused by light refraction from the wrinkled surface. For clear comparison, figures 5(b,c) and 5(d,e) show cases with lower jet impulse, while the optimal case for maximum impulse is shown in figures 5(f,g) (all values are highlighted with a shadow in figure 4(a), and the nozzle deformations in each case are shown in the supplementary material and movie 2). For $(Eh, \varPi _0) = (14.4\ {\rm N}\ {\rm m}^{-1}, 0.64)$ (figures 5b,c), the nozzle expansion wave (red region) propagates from the support ($y/D = 2.0$) to the tip ($y/D = 0$) with a constant speed (visualized in figure 5(b) at $t^{*}<0.6$ and indicated by a solid arrow in figure 5c), and it bounces back at $t^{*} = 0.8$ (dashed arrow in figure 5c), along which the contraction occurs (blue region in figure 5c). The wave speed ($c$) on the nozzle surface increases with $Eh$ but decreases with the fluid inertia ($\rho _{f}D$), expressed as $c = (Eh/\rho _{f}D)^{0.5}$ (Avrahami & Gharib Reference Avrahami and Gharib2008; Choi & Park Reference Choi and Park2022). Choi & Park (Reference Choi and Park2022) suggested that the thrust of the continuous jet is maximized when the travel time of the wave along the nozzle matches the jet acceleration time, resulting in the constant dimensionless wave speed ($\hat {c}=cT_{acc}/L \simeq 3.0$ from their measurements). In figures 5(b,c), $\hat {c} = 5.3$ indicates that the surface wave moves faster than the jet, thereby agreeing with the earlier contraction (at $t^{*} = 0.8$) rather than the end of jet acceleration ($t^{*} = 1.0$). The elastic potential energy is released before the jet reaches its maximum velocity, therefore the resulting jet impulse is lower than that in the other cases (figure 4a). On the other hand, the travel time of the wave is relatively longer than $T_{acc}$, i.e. $\hat {c} = 1.1$, for $(Eh, \varPi _0) = (7.0 {\rm N}\ {\rm m}^{-1}, 0.48)$ (figures 5d,e). In this case, the nozzle expands slowly because of its lower stiffness, and the contraction occurs after the jet has decelerated, producing the lower impulse (figure 4a). The contraction wave propagates from the nozzle support at $t^{*} > 1.23$ in figure 5(d) (see dashed arrow in figure 5e), instead of bouncing back from the nozzle tip. Contrary to these two off-optimal conditions, $(Eh, \varPi _0) = (14.4\ {\rm N}\ {\rm m}^{-1}, 0.57)$ in figures 5(f,g) shows that the nozzle contraction ends at $t^{*} = 1.0$ when the jet has reached the maximum velocity. This condition ($\hat {c} = 2.6$) produces the maximum impulse (figure 4a) close to the optimal condition suggested by Choi & Park (Reference Choi and Park2022).

In figure 6(a), we compared the hydrodynamic impulse ($I_h$) normalized by that ($I_{h,rigid}$) of the rigid nozzle depending on $\hat {c}$ for the present pulsed jets, together with the results for continuous jets (Choi & Park Reference Choi and Park2022). Interestingly, $I_h$ of the pulsed jet is also maximized at $\hat {c}\simeq 3.0$. The impulse decreases sharply at $\hat {c}>3.0$ because the nozzle becomes too stiff so that the nozzle contracts earlier than the end of acceleration (figure 5a). When $\hat {c}>9.0$, the flexible nozzle behaves almost similar to the rigid one (but the impulse is still increased by the flexible nozzle at this range). When the nozzle becomes too flexible ($\hat {c}<3.0$), the nozzle contraction is far delayed beyond the instant of the maximum jet velocity (figures 5f,g). Therefore, the optimal flexibility is irrelevant to the presence of jet deceleration, which is attributed to the fact that the meaningful transfer of elastic energy to the jet kinetic energy occurs at $t^{*} < 1.0$ before the jet deceleration (figure 5c).

Figure 6. Variation in (a) $I_h$ and (b) $V$ passing through $y/D = -1.0$ with $\hat {c}$: circle, $Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$; triangle, $14.4\ {\rm N}\ {\rm m}^{-1}$; and square, $43.2\ {\rm N}\ {\rm m}^{-1}$. Closed and open symbols correspond to the pulsed jet (present study) and continuous jet (Choi & Park Reference Choi and Park2022), respectively, and $I_h$ and $V$ are normalized with values for the rigid nozzle.

Although the presence of the jet deceleration does not affect the optimal condition, the achievable impulse is significantly higher for the pulsed jet, i.e. the advantage of flexibility is more pronounced for the pulsed jet. This can be understood from two aspects that originate from jet deceleration. First, the expanded nozzle is fully contracted to transfer its entire elastic energy to the jet during jet deceleration ($t^{*} > 1.0$), and second, the nozzle collapse (figure 2a) greatly suppresses the negative pressure inside the nozzle. These two effects are limited for the continuous jet because of the subsequent supply of the fluid. Therefore, the maximum $400\,\%$ increase of the impulse is achieved, which is remarkable compared to that reported in previous studies ($70\,\%$ with the orifice nozzle (Krieg & Mohseni Reference Krieg and Mohseni2013) and $5\,\%$ with the fast acceleration (Gao et al. Reference Gao, Wang, Simon and Chyu2020)). In addition, our preliminary proof-of-concept investigation showed that the flexible nozzle (as a thruster) attached to a small-scale underwater vehicle results in a ${\sim }60\,\%$ increase in the translation speed, under the same optimal condition, compared to the body with a rigid nozzle (see supplementary material and movies for the details).

The modified vortex dynamics behind the flexible nozzle is associated with the enhanced entrainment of the surrounding fluid. The large jet entrainment is closely relevant to the enhancement of propulsive efficiency by reducing the loss of kinetic energy in the wake (Linden Reference Linden2011; Ruiz et al. Reference Ruiz, Whittlesey and Dabiri2011). Figure 6(b) shows the jet volume generated by the flexible nozzle relative to the rigid one. Similar to the impulse, the jet volume from the flexible nozzle is increased greatly, which becomes the maximum at $\hat {c} \simeq 3.0$. Only $45\,\%$ of the jet volume enhancement was achieved for the cylindrical rigid nozzle with an extremely small formation number $F = 0.5$ (Olcay & Krueger Reference Olcay and Krueger2008).

Figure 7 shows how jet entrainment is augmented for the flexible nozzle. For the rigid nozzle (figure 7a), the vortex ring stops being fed immediately after the jet ends and moves downstream ($t^{*} > 1.0$). In contrast, the vortex ring from the flexible nozzle ($Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$) is fed for a longer time duration (up to $t^{*} = 3.2$) during the nozzle collapse, and it grows in higher strength (faster velocity) compared to the rigid nozzle (see $t^{*} = 5.0$). Similar to the increase in the impulse (figure 6a), the entrainment rate of the pulsed jet is considerably higher than the continuous jet because it is also related to the nozzle collapse, which mitigates the negative pressure. For the present conditions, flexible nozzles with $\hat {c}<6$ are found to successfully subdue the negative pressure and attain a large entrainment. However, the dependency on the dimensionless wave speed is less salient than that of the hydrodynamic impulse. The entrainment is weakly affected by the jet acceleration, as observed in figure 4(d), therefore other factors such as the nozzle-tip movement (Dabiri & Gharib Reference Dabiri and Gharib2005) and radial velocity at the nozzle exit (Krieg & Mohseni Reference Krieg and Mohseni2013) will be involved, which are less correlated with $\hat {c}$.

Figure 7. Contours of the vertical flow velocity ($\hat {v}=v/v_{m,r}$) and velocity vectors for (a) rigid and (b) flexible ($Eh = 7.0\ {\rm N}\ {\rm m}^{-1}$) nozzles at the same jet condition $\varPi _{0} = 0.57$.

We discuss briefly the effect of the formation number ($F$) on hydrodynamic thrust. Figures 8(a) and 8(b) show instantaneous jet flows at various formation numbers for the rigid and flexible nozzles, respectively, with the nozzle length $L = 0.8D$ (see supplementary material and movie 3 for the jet evolution), and figure 8(c) compares time histories of the jet-exit velocity for $F = 2.2\unicode{x2013}8.9$ ($\hat {c} = 16$). Here, the flexible nozzle with $L/D = 0.8$ generates the maximum thrust at the optimal condition ($\hat {c}\simeq 3.0$), which is detailed in the supplementary material and movies. For the rigid nozzle (figure 8a), a single vortex ring forms at $F < 4.0$, and a trailing jet with subsequent vortex rings forms above the leading vortex ring at $F > 4.0$, which is consistent with the literature (Gharib et al. Reference Gharib, Rambod and Shariff1998; Gao et al. Reference Gao, Wang, Simon and Chyu2020). For the flexible nozzle (figure 8b), the trailing jet occurs at all $F$ values, indicating the lower critical $F < 3.3$, at which the vortex ring solely occurs. For $F = 3.3$, the trailing jet looks obscure at the instant shown ($t^{*} = 4.0$) because it is dissipated considerably (the trailing jet for $F = 3.3$ can be seen in the supplementary material and movie 3). The decreased critical formation number was reported in a previous study (Choi & Park Reference Choi and Park2022) such that the pinch-off location of the jet vortices moved upwards by $54\,\%$ compared to that of the rigid nozzle for $\hat {c}=7.5$ and $F = 37$ (continuous jet) attributed to the early growth of the jet exit velocity caused by nozzle deformation. Figure 8(d) shows the hydrodynamic impulses ($I_h$) of the flexible nozzles normalized by those of the rigid nozzle ($I_{h,rigid}$) depending on $F$. Here, $I_h$ was measured at $t^{*} = 4$. It is observed that $I_{h}/I_{h,rigid} > 1.0$ for $F < 5.6$ because of the timely nozzle deformation and suppression of negative pressure (i.e. the same fluid–structure interaction mechanism holds); it converges to $1.0$ for $F > 5.6$. The value of $F$ at which $I_{h}/I_{h,rigid}$ saturates to $1.0$ is expected to increase as $\hat {c}$ becomes closer to the optimal value ($\simeq 3.0$) (not tested in the present study). In addition, $I_{h}/I_{h,rigid}$ increases as $F$ decreases because the thrust-augmenting mechanism works at the early stage of jet generation and affects the initial volume of the jet fluid. For the flexible nozzle, the clear relevance between the critical $F = 4$ (for rigid cylindrical nozzle) and thrust enhancement is yet to be found; instead, the maximum thrust is dictated by the optimal condition (figure 6). Although the formation number significantly affects the vortex interaction and its formation, its contribution (up to $10\,\%$) to the total jet impulse (Gao et al. Reference Gao, Wang, Simon and Chyu2020) is considerably less than that ($400\,\%$) achieved through the optimal flexibility condition of the present study (figure 3b).

Figure 8. (a,b) Instantaneous jet flow (velocity vectors and vorticity $\hat {\omega } = \omega D/v_{m,r}$ contour) of (a) rigid and (b) flexible ($Eh = 14.4\ {\rm N} {\rm m}^{-1}$) nozzles, depending on the formation number ($F$), measured at $t^{*} = 3.6$. (c) Time history of the jet-exit velocity ($v_e$), depending on $F$. The data were taken at $y/D = -0.2$. (d) Hydrodynamic impulse from the flexible nozzle, depending on $F$. Here, $I_h$ was measured at $t^{*} = 4$ and normalized by the value of the rigid nozzle ($I_{h,rigid}$).