3.1. Conical jet

Figure 3 illustrates the formation of a conical jet generated by two tandem bubbles with $\gamma =0.98$ and $\theta =1.10$ , comparing the results from numerical simulations with experimental observations. Driven by its initial high pressure, bubble 1 expands spherically (frame 1) and imparts significant inertia to the surrounding fluid. This inertial causes the bubble to grow and exceed its equilibrium radius, depressing both its internal pressure and the pressure in the adjacent field to a level significantly below the local hydrostatic pressure when the bubble reaches its maximum radius at $t$ = 2.33 ms (frame 2). Subsequently, bubble 2 forms and expands in a non-spherical shape because of the asymmetric pressure distribution in the flow field relative to the centre of bubble 2. The lower surface of bubble 2 is drawn toward bubble 1, forming a high-curvature tip (frames 3–5). The curvature at this tip is 5.87 times greater than that of the upper surface of bubble 2. Before the lower tip of bubble 2 fully contracts, the collapse of bubble 1 emits a shock wave that impacts the lower surface of bubble 2 (frame 6). This interaction generates a reflected rarefaction wave, which induces a cluster of micro-sized cavitation bubbles near the bottom of bubble 2, as highlighted by the red box in frame 6. The combined effect of the shock wave from the collapse of bubble 1 and the localised high-pressure region beneath bubble 2 further accelerates its lower surface, ultimately creating a distinct conical liquid jet (frame 7). As bubble 1 re-expands, its internal pressure drops rapidly (frame 7), consequently reducing the pressure in the surrounding flow field. However, the asymmetric collapse of bubble 2 triggers convergence of the surrounding fluid toward the initial contraction region, increasing the pressure at the jet base. This elevated pressure further accelerates fluid motion and bubble contraction, establishing a positive feedback loop. The resulting momentum-focusing effect (Lauterborn Reference Lauterborn1982; Philipp & Lauterborn Reference Philipp and Lauterborn1998; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016a ) maintains the high-pressure zone, thereby sustaining jet development. As the conical jet impacts the opposite surface of bubble 2, its velocity decreases very fast (frames 8–9). Meanwhile, a sharp protrusion forms at the top of bubble 2, travels a considerable distance and fragments into smaller pieces gradually (frame 10).

Notably, for cases where the jet tip can be clearly tracked across several consecutive frames, the velocity of this conical jet passing through the centre of bubble 2 in this experiment is approximately 32.3 m s⁻¹, measured using high-speed imaging by tracking the position of the jet across four consecutive frames. The error in space is the length of a pixel, corresponding to approximately 0.2 mm. The error in time due to camera jitter is on the scale of nanoseconds and thus negligible. Therefore, the error is calculated as $\pm$ 0.4 mm/0.069 ms = $\pm$ 5.8 m s⁻¹. To account for the refractive effect of the gas–liquid interface on displacement measurements, the distance from the jet tip to the bubble centre was multiplied by a correction factor of 1.33 (Philipp & Lauterborn Reference Philipp and Lauterborn1998), yielding a corrected jet velocity of approximately 43.1 $\pm$ 7.7 m s⁻¹. This value aligns closely with the numerical simulation, which predicts an instantaneous jet velocity of 42 m s⁻¹ at the same position. Despite the remarkable agreement illustrated in figure 3, the primary mechanism driving the formation of the conical jet, whether it is the high curvature of the jet tip or the collapse-induced high pressure from bubble 1, remains unanswered and will be further analysed in § 4.1.

Figure 4. Dynamics of umbrella-shaped jet from anti-phase bubble pair: comparison between the experiment (odd rows) and numerical simulation (even rows). The dimensionless times corresponding to the numerical results are 0.29, 1.31, 1.51, 1.64, 1.78, 1.87, 1.98, 2.09, 2.13 and 2.22, respectively. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.82$ , $\theta =1.38$ .

3.2. Umbrella-shaped jet

Decreasing the distance between the two tandem bubbles enhances their interaction, as shown in figure 4. This figure compares the experimental and numerical results for the umbrella-shaped jet with $\gamma =0.82$ and $\theta =1.38$ . Similar to the initial evolution of the conical jet, the lower surface of bubble 2 elongates under the influence of bubble 1 (frames 1–4), forming a tip with a curvature that is 7.33 times that of the upper surface, which represents a notable increase compared with the case shown in figure 3. Driven by this increased curvature, the tip contracts during the collapse of bubble 1 (frames 5–6). As shown in the numerical results corresponding to frame 6, the umbrella-shaped jet tip forms before the final collapse of bubble 1. Following this, the toroidal collapse of bubble 1 generates multiple pressure shock waves (Ohl et al. Reference Ohl, Kurz, Geisler, Lindau and Lauterborn1999; Gonzalez-Avila, Denner & Ohl Reference Gonzalez-Avila, Denner and Ohl2021). Given that the duration of these shock waves is substantially shorter than the characteristic time scale of jet evolution, they cumulatively accelerate the upstream fluid of the umbrella-shaped jet, causing it to widen or even form a second umbrella-shaped structure, which will be analysed in § 4.2. As shown in frames 7 and 8, the jet tip of bubble 2 takes on a clear umbrella-shaped morphology in the experiment, characterised by a flattened leading edge and a fragmented liquid film surrounding it. While similar jet morphologies have been documented in other systems (Shinjo & Umemura Reference Shinjo and Umemura2010; Karri et al. Reference Karri, Ohl, Klaseboer, Ohl and Khoo2012; Supponen et al. Reference Supponen, Kobel, Obreschkow and Farhat2015; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016b ; Zhang et al. Reference Zhang, Zhang, Zhang, Long, Han, Liu, Ohl and Li2025), the underlying mechanism behind this umbrella-shaped jet and its dynamics remain poorly understood. A more detailed and in-depth analysis of these phenomena will be presented in § 4.2.

In contrast to the conical jet, the umbrella-shaped jet achieves a significantly higher peak velocity, reaching approximately 92.3 m s⁻¹ in numerical simulations and 85.6 $\pm$ 7.7 m s⁻¹ in experiments. Remarkably, the jet maintains stable propagation after penetrating the upper surface of bubble 2, travelling a distance of approximately 4.5 times the maximum radius of the bubble. The unique capability of high-velocity liquid jets to achieve substantial penetration distance enables the tandem-bubble interaction to exhibit strong potential for needle-free injection and micro-pumping applications (Ohl et al. Reference Ohl, Arora, Dijkink, Janve and Lohse2006b ; Sankin et al. Reference Sankin, Yuan and Zhong2010; Robles et al. Reference Robles, Gutierrez-Herrera, Devia-Cruz, Banks, Camacho-Lopez and Aguilar2020). We will further explore the dependence of the jet’s penetration distance across a wide parameter space in § 6.

Figure 5. Dynamics of spraying jet from anti-phase bubble pair: comparison between the experiment (odd rows) and numerical simulation (even rows). The dimensionless times corresponding to the numerical results are 0.28, 1.21, 1.49, 1.53, 1.72, 1.77, 1.79, 1.83, 1.87 and 2.00, respectively. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.35 ms. The initial parameters are $\gamma =0.76$ , $\theta =1.23$ .

3.3. Spraying jet

Further enhancing the interaction between the two tandem bubbles leads to the transition of the jet morphology from an umbrella-shaped morphology to a spraying jet. Figure 5 presents the formation of the spraying jet with $\gamma =0.76$ and $\theta =1.23$ . In contrast to the conical and umbrella-shaped jets, the lower surface of bubble 2 extends into the interior of bubble 1 during its expansion phase (frames 3–4). Subsequently, the upper surface of bubble 1 impacts the lower surface, initiating an annular collapse that emits a shock wave. This shock wave induces neck breakup at the elongated tip of bubble 2 (frames 5–7). The high-pressure stagnation point generated by the neck breakup drives the rapid upward acceleration of the liquid jet (Fan et al. Reference Fan, Bußmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024), culminating in the formation of a high-speed jet with spray at its tip. The spraying jet, being significantly faster, completes its trajectory from formation (initiated by neck breakup) to impact the upper surface of bubble 2 within the time interval of three frames. The relative measurement error is caused mainly by the limited temporal resolution, with the temporal uncertainty reaching up to one inter-frame interval and the resulting relative error in velocity as high as 50 % (Gonzalez-Avila et al. Reference Gonzalez-Avila, Van Blokland, Zeng and Ohl2020). The measured maximum velocity of this regime of jet can exceed 1000 m s⁻¹ in the experiments. It is noteworthy that the tip of this jet is inherently unstable. Both experimental observations and numerical simulations reveal that the upper surface of bubble 2 is initially impacted by fine droplets, and then penetrated by the continuous jet. Here, we refer to the continuous jet as the uninterrupted liquid column behind the spraying droplets, characterised by a slower velocity compared with the unstable jet tip. The velocity of the continuous jet is particularly significant for practical applications that rely on coherent jet integrity. The experimentally measured velocity reaches approximately 375 $\pm$ 5.8 m s⁻¹, while the numerical simulation predicts a velocity of 417 m s⁻¹. Due to its narrower and faster continuous segment compared with conical and umbrella-shaped jets, this spraying jet is capable of achieving a propagation distance of up to 6.5 times the maximum bubble radius in this case. The mechanism of jet acceleration induced by neck breakup in two laser-induced bubbles has been demonstrated in Fan et al. (Reference Fan, Bußmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024). In this study, we similarly reproduced supersonic jets using centimetre-scale cavitation bubbles, but we observed a more detailed evolution process of the supersonic jet within bubble 2. We found that the jet is inherently unstable, characterised by a fragmented tip. The fragmented droplets sometimes follow an inclined trajectory, ultimately striking near the centre of the upper surface of bubble 2, as shown in later discussions. Interestingly, although the trajectory of the spraying jet tip deviates from the symmetry axis of the tandem-bubble system, the continuous jet quickly realigns itself, allowing it to continue penetrating and travelling along the axis.

3.4. Penetration of piercing jets

After the piercing jets penetrate the opposite surface of bubble 2 and exit into the ambient liquid, we consistently observe that each jet is enveloped by a gas cavity during its propagation, a phenomenon similar to the gas cavity in water-entry problems, as shown in figures 6, 7 and 8. The time coordinate is normalised to zero at the critical moment of jet penetration, establishing a consistent reference for comparative analysis.

The evolution of the bubbles and the penetration process of a conical jet is illustrated in figure 6, where the curve depicts the temporal evolution of the distance penetrated by the jet tip. In this case, the conical jet retains a velocity of only 24 m s⁻¹ after penetrating the bubble surface, with the jet tip advancing a distance of merely 2 times the maximum bubble radius before its collapse and fragmentation. Instants 1–2 reveal that the conical jet generates a continuous gas cavity trailing behind its tip upon penetrating the bubble surface. The cavity initially pinches off near the bubble (instants 3–5) and, after penetrating a stable distance (instants 5–6), disintegrates into bubble clusters (instant 7). As the velocity of the jet tip decreases, the cavity at the tip undergoes collapse followed by immediate expansion, resulting in a noticeable increase in the distance curve at instant 7. Owing to the absence of upstream jet replenishment, the cavity at the tip rapidly loses its forward momentum and gradually dissipates (instant 8). Hence, we only focus on the penetration distance at the instant of tip collapse, thereby excluding the subsequent cavity evolution from our study.

Figure 6. The evolution of the gas cavity after the penetration of a conical jet. The curve depicts the temporal evolution of the distance $S$ penetrated by the jet tip in water. In this and subsequent figures, all variables without units are dimensionless. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.86$ , $\theta =1.53$ .

Figure 7. The evolution of the gas cavity after the penetration of an umbrella-shaped jet. The curve depicts the temporal evolution of the distance $S$ penetrated by the jet tip in water. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.71$ , $\theta =1.53$ .

Figure 8. The evolution of the gas cavity after the penetration of a spraying jet. The curve depicts the temporal evolution of the distance $S$ penetrated by the jet tip in water. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.85$ , $\theta =0.95$ .

Compared with the conical jet, the umbrella-shaped jet exhibits a higher velocity and retains a velocity of approximately 63 m s⁻¹ after penetrating the bubble, which facilitates the formation of an extended cavity in the water (instants 2–4) and allows for a penetration distance exceeding 5 times the maximum bubble radius, as depicted in figure 7. Under this circumstance, bubble 2 undergoes significant elongation. The cavity adjacent to the toroidal bubble pinches off and collapses immediately after the toroidal collapse of bubble 2, followed by successive collapse propagating downstream along the cavity (instant 6). The cavity at the jet tip eventually collapses and its forward momentum gradually dissipates (instant 8).

The morphological evolution of the gas cavity after a spraying jet penetrates the bubble surface is shown in figure 8. In comparison with conical and umbrella-shaped jets, the spraying jet demonstrates a significantly higher velocity after penetrating the bubble surface (108 m s⁻¹ in the presented case), which results in the formation of a more elongated gas cavity, extending up to 6.5 times the maximum bubble radius. Bubble 2 undergoes significant elongation, resulting in markedly weaker secondary rebound compared with bubble 1 after collapse. A considerable amount of bubble 2’s energy converts into jet kinetic energy, resulting in a weak collapse. Meanwhile, bubble 1’s shockwave is shielded by bubble 2, thereby minimising indiscriminate damage to the upper target injection site.

The cavity evolution after jet penetration resembles that produced by the water entry of a rigid body (Speirs et al. Reference Speirs, Pan, Belden and Truscott2018). The subsequent breakup of these cavities is commonly characterised using the Bond ( $Bo = {\rho g D^2}/{\sigma }$ ) and Weber ( $We = {\rho U^2 D}/{\sigma }$ ) numbers (Aristoff & Bush Reference Aristoff and Bush2009; Truscott, Epps & Belden Reference Truscott, Epps and Belden2014; Speirs et al. Reference Speirs, Pan, Belden and Truscott2018). Given that the tip radii span overlapping ranges across the three regimes (0.75–1.50 mm for the conical jet, 0.50–1.25 mm for the umbrella-shaped jet and 0.25–0.75 mm for the high-speed spraying jet), for these low Bond number ( $Bo \sim O(10^{-1}$ - $10)$ ) cases, the cavity shape is dependent on the Weber number (Aristoff & Bush Reference Aristoff and Bush2009; Truscott et al. Reference Truscott, Epps and Belden2014; Speirs et al. Reference Speirs, Pan, Belden and Truscott2018). At low Weber numbers, both conical jet and umbrella-shaped jet penetrate the bubble surface and undergo the shallow seal. In contrast, spraying jets, associated with higher $We$ , form cavities where inertia force prevails, resulting in a deep seal under hydrostatic pressure influenced by bubble 2. It should be noted that a splash occurs after spraying jets penetrate the bubble surface and enter the water, as shown in frame 10 of figure 5. However, since these jets typically form when the bubble is at its maximum radius, a stage characterised by relatively low internal gas density, the associated pressure drop at the splash interface remains inadequate to drive closure. As expected, a deep seal rather than a surface seal is observed. The pinch-off types of cavity vary significantly across jet types and are accompanied by distinct flow structures corresponding to different jet tip morphologies, but they negligibly influence the maximum penetration distance, which is governed primarily by jet velocity. This study will focus on the penetration capability of the jets in water, as detailed in § 6.

4.1. Conical jet formation: the role of the pressure wave from bubble 1

As illustrated in § 3.1, the formation of the conical jet is initiated by the transient high-pressure wave generated by the collapse of bubble 1, which accelerates the elongated tip of bubble 2 and drives its contraction into the bubble interior to form the liquid jet. One may argue that the high curvature at the tip of bubble 2 also contributes to the formation of a liquid jet (Lauterborn Reference Lauterborn1982; Tomita et al. Reference Tomita, Robinson, Tong and Blake2002; Sieber, Preso & Farhat Reference Sieber, Preso and Farhat2022; Zeng et al. Reference Zeng, Zhang, Tan, An and Ohl2024). However, whether the formation of the conical jet is driven by the high curvature or pressure wave remains to be elucidated.

Figure 9. Boundary integral simulation of the jet evolution of bubble 2. Bubble 1 is removed at the point when the downward velocity of bubble 2’s lower tip reaches zero. ( $a$ ) Morphological evolution of bubble 2. Instants 1–7 depict different snapshots during bubble evolution from BI simulation. ( $b$ ) Time evolution of the velocity along the axis of symmetry of bubble 2. The red and blue solid lines represent the velocity of the jet and north pole of bubble 2 in the BI simulation, respectively. The solid diamonds with error bars represent the corresponding experimental data. The velocity is measured using three consecutive frames from high-speed imaging. The error is primarily attributed to spatial resolution, where the positional error corresponds to the length of one pixel. Temporal error arising from camera jitter occurs on the nanosecond scale and is therefore negligible. The solid circles represent the results obtained from the VoF simulation. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.98$ , $\theta =1.10$ .

We first examine the role of the high curvature at the elongated tip of bubble 2 in driving the formation of the conical jet through BI simulation. To eliminate the effect of the collapse of bubble 1 on bubble 2, bubble 1 is removed from the simulation when the downward velocity of bubble 2’s lower tip reaches zero (Fong et al. Reference Fong, Adhikari, Klaseboer and Khoo2009; Dadvand et al. Reference Dadvand, Khoo, Shervani-Tabar and Khalilpourazary2012; Peters et al. Reference Peters, Tagawa, Oudalov, Sun, Prosperetti, Lohse and van der Meer2013; Han et al. Reference Han, Zhang, Tan and Li2022). The subsequent morphological evolution and the time evolution of the velocity along the axis of symmetry of bubble 2 are presented in figure 9. Here, the red line represents the jet velocity, and the blue line represents the velocity at its north pole in figure 9( $b$ ). Both velocities are defined as negative when the bubble surface expands outward and positive when it contracts inward to facilitate comparison. The experimental and VoF data, represented by solid diamond and square symbols, are also included. Instant 1 marks the moment when the expansion velocity at the lower tip of bubble 2 decreases to zero. At this point, bubble 2 expands into an egg-like shape under the influence of bubble 1, as shown in figure 9(a). Bubble 1 is then removed from the simulation to eliminate the influence of its pressure wave. As expected, the region with a higher curvature on the bubble surface undergoes rapid contraction (Lauterborn Reference Lauterborn1982; Tomita et al. Reference Tomita, Robinson, Tong and Blake2002), contributing to jet formation at the lower tip of bubble 2. However, the simulated velocity and acceleration of the jet are significantly lower than experimental observations and VoF results in figure 9(b). In BI simulation, the upper surface of the bubble and the jet tip contract at similar velocities (instants 5–7). The continuous widening of the tip transforms the conical jet into a cylindrical shape, which differs from our experimental observations. We can conclude that while the high curvature of the bubble surface may contribute to the initial jet formation stage, it does not primarily govern the subsequent evolution of the conical jet.

Following Ory et al. (Reference Ory, Yuan, Prosperetti, Popinet and Zaleski2000) and Peters et al. (Reference Peters, Tagawa, Oudalov, Sun, Prosperetti, Lohse and van der Meer2013), we adjust the pressure pulse from bubble 1 to investigate the role of the pressure wave in the formation of the conical jet. When the tip velocity of bubble 2 decreases to zero during its expansion phase, we hold the geometric profile of bubble 1 constant and apply a pressure pulse with a duration of 20 µs while varying its amplitude. The duration of the pressure impulse generated by the collapse of bubble 1 is experimentally determined to be approximately 20 µs through hydrophone measurements (Cui et al. Reference Cui, Zhang, Wang and Liu2020). The pressure amplitude is constrained within the range of 5–20 MPa (Cui et al. Reference Cui, Zhang, Wang and Liu2020) to ensure the accelerated jet velocity remains within physically realistic limits. After the pressure pulse, the internal pressure of bubble 1 is set to hydrostatic pressure to eliminate any subsequent influence.

Figure 10. Boundary integral simulation of the jet evolution of bubble 2. A pressure pulse with a duration of 20 µs is applied within bubble 1. ( $a$ ) Morphological evolution of bubble 2. The dimensionless time for the morphology of bubble 1 is 1.58, while the morphologies of bubble 2 from the outermost to the innermost correspond to dimensionless times of 1.58, 1.64, 1.73, 1.82, 1.93 and 2.04, respectively. ( $b$ ) The maximum velocities of the conical jet of bubble 2 corresponding to different pressure pulses applied to bubble 1. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The horizontal coordinate, $\Delta p \Delta t$ , is non-dimensionalised by scaling with $R_{max} \sqrt{\rho P_{\rm \infty}}$ . The initial parameters are $\gamma =0.98$ , $\theta =1.10$ .

Figure 10 illustrates the tip evolution of bubble 2 under the impact of the pressure pulse from bubble 1. The temporal sequence of bubble 2’s morphology in figure 10(a) shows a significant change compared with the simulation in figure 9(a). The conical jet forms under the combined effects of tip curvature and the pressure wave from bubble 1, consistent with our experimental observations. As suggested by Ory et al. (Reference Ory, Yuan, Prosperetti, Popinet and Zaleski2000) and Peters et al. (Reference Peters, Tagawa, Oudalov, Sun, Prosperetti, Lohse and van der Meer2013), when the interface is predominantly governed by the pressure pulse, the jet velocity scales with the product of pressure pulse amplitude and duration, expressed as $U_{\textit{jet}} \propto \Delta p \Delta t$ . This relationship is well verified by our numerical simulation, as shown in figure 10(b). In summary, we conclude that the relative contributions of surface curvature and pressure wave vary across different stages of jet development. A high-curvature surface is crucial for jet initiation, as regions with greater curvature undergo retraction first (Lauterborn Reference Lauterborn1982; Tomita et al. Reference Tomita, Robinson, Tong and Blake2002). However, the subsequent jet evolution is more governed by the pressure wave, which is corroborated by our finding that the velocity of the conical jet exhibits a positive correlation with the product of the pressure pulse amplitude and its duration. Therefore, the interplay of curvature and pressure wave is key to the formation of conical jets.

4.2. Umbrella-shaped jet formation: bubble 1 collapses after jet formation

We have provided preliminary insights into the formation of the umbrella-shaped jet in figure 4, characterised by a flattened leading edge and a fragmented liquid film around the jet tip. The umbrella-shaped jet resembles the mushroom-shaped structure observed when a liquid jet impinges against stagnant gas (Shinjo & Umemura Reference Shinjo and Umemura2010). Air resistance is identified as the primary driver of mushroom-shaped structure formation. Interestingly, our simulations using the BIM, based on potential flow theory, successfully reproduced a similar umbrella-shaped tip, as depicted in figure 11(a). Given that the BIM does not account for air resistance and assumes uniform pressure within the bubble, this suggests that the air flow is not the primary cause of the umbrella-shaped structure formation. Through experimental observations, we find that the umbrella-shaped jet typically forms when the collapse of bubble 1 occurs after the initial jet formation stage of bubble 2. This finding demonstrates that the umbrella-shaped jet in the tandem bubbles system correlates with the flow structure within the water jet, which is governed by bubble-bubble coupling effects. This section will offer a more detailed analysis of the formation of the umbrella-shaped jet.

Figure 11. Boundary integral simulation of the umbrella-shaped jet evolution of bubble 2. ( $a$ ) Morphological evolution of bubble 2. The dashed lines indicate the position for velocity extraction. ( $b$ ) The evolution of axial velocity ( $U_{\textit{z}}$ ) along the axis of symmetry in jet column of bubble 2. Here, $U_{\textit{jet}}$ and $U_{\textit{max}}$ represent the velocity at the jet tip and the maximum velocity along the axis of symmetry in jet column, respectively. The same colour of solid lines and labels in ( $a$ ) and ( $b$ ) represents the same instants 1.68, 1.80, 1.86, 2.01. The circles mark the location of the maximum velocity along the jet column. ( $c$ ) Time evolution of pressure at a moving probe (0.05 dimensionless units below the jet base, solid line) and the time evolution of velocity at the jet tip (dashed line). The jet tip contracts at $\textit {t}\approx 1.64$ . The bubble profiles at times corresponding to labels 1, 2, 3 and 4 are shown in panel (a). The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.83$ , $\theta =1.10$ .

We present the evolution of axial velocity along the jet column from the BI simulation in figure 11(b). Meanwhile, the evolution of pressure below the jet base and the velocity of the jet tip are illustrated in figure 11(c). The tip of the lower surface of bubble 2 contracts at $\textit {t}\approx 1.64$ (the black dashed line in figure 11 c), accompanied by a short increase in velocity. During the initial formation of the jet, the peak velocity of the jet occurs at its tip driven by the local high curvature. However, due to the continuous increase in pressure at the jet base, which is induced by the focusing flow below bubble 2, the velocity of the upstream fluid (the fluid close to the jet base) gradually exceeds the tip velocity of the jet during its subsequent evolution. As the upstream fluid overtakes the jet tip, the liquid at the tip is displaced and spread outward, generating an umbrella-shaped structure, as shown by instant 2 in figure 11(a). The jet tip experiences a noticeable reduction in acceleration after the collapse of bubble 1. This reduction results from the collapse-induced pressure accelerating the fluid near the jet base and widening the liquid jet column, thereby increasing its fluid inertia and consequently enhancing its resistance to acceleration. With the rebound of bubble 1, the pressure at the jet decreases gradually. The location of the maximum velocity of the jet gradually approaches the jet tip, resulting in a decrease in the velocity difference between the upstream fluid and the tip, as shown between instants 3 and 4 in figure 11(b). A similar umbrella-shaped structure can also be observed in the downward Bjerknes bubble jet when the bubble is generated near a free surface under some circumstances (Supponen et al. Reference Supponen, Kobel, Obreschkow and Farhat2015; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016b ). The free surface functionally emulates bubble 1, thereby inducing the characteristic egg-like deformation of the beneath bubble. As the high-curvature tip of bubble 2 contracts, a momentum-focusing effect occurs (Lauterborn Reference Lauterborn1982; Philipp & Lauterborn Reference Philipp and Lauterborn1998; Koukouvinis et al. Reference Koukouvinis, Gavaises, Supponen and Farhat2016a , Reference Koukouvinis, Gavaises, Supponen and Farhatb ), causing the velocity of the trailing jet to exceed that of the jet tip, thereby generating a similar umbrella structure. These results demonstrate that umbrella-shaped jet formation can be initiated solely by pressure increase at the jet base during its evolution, with pressure waves being non-essential.

Figure 12. Evolution of the double umbrella-shaped jet. In panels (a) to (d), the lower bubble is bubble 1 and the other is bubble 2. The blue solid line boxes indicate tip contraction, reflected rarefaction wave and double umbrella-shaped structure. Panel (e) presents the boundary integral simulation of the double umbrella-shaped jet evolution of bubble 2. A 20 µs pressure pulse of 60 MPa is applied within bubble 1 at instant 1. The dimensionless times corresponding to instants 1, 2 and 3 are 1.76, 1.80 and 1.88 respectively. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.83$ , $\theta =1.06$ .

Interestingly, we observed the formation of a double umbrella-shaped jet under specific conditions in our system, as illustrated in figure 12. In panel (a), the tip of bubble 2 begins to contract inward. A cluster of cavitation bubbles in panel (b) marks the complete collapse of bubble 1, which triggers the emission of the pressure waves. Following this shock, two umbrella-shaped structures can be observed simultaneously, one at the tip and another in the middle of the jet, as illustrated in panels (c) and (d). The first umbrella-shaped structure originates from flow focusing, demonstrating that pressure waves are not essential for its formation. Conversely, the second structure develops from the liquid column that was widened by pressure waves emitted during bubble 1’s collapse. This double umbrella-shaped jet aligns with findings by Geschner, Obermier & Chaves (Reference Geschner, Obermier and Chaves2001) and Srinivasan, Salazar & Saito (Reference Srinivasan, Salazar and Saito2011), who demonstrated that imposing a pressure excitation after one umbrella structure on a liquid jet led to the formation of the second umbrella-shaped structure (see figure 3 in Geschner et al. Reference Geschner, Obermier and Chaves2001).

Through the BI simulation, we reproduced the evolution of this distinct jet, as shown in panel (e). In this simulation, the geometric profile of bubble 1 is held constant when bubble 1 collapses to its minimum volume. At this moment (instant 1), the internal pressure of bubble 1 is modulated into a transient pressure pulse (duration of $20\, {\unicode{x03BC}} \textrm {s}$ , amplitude of 60 MPa), inducing a transient shock at the jet base of bubble 2. After the pressure pulse, the internal pressure of bubble 1 is set to hydrostatic pressure to eliminate any subsequent influence. Following the pressure pulse, an axial protrusion emerges at the jet midsection (instant 2), ultimately developing into a secondary umbrella-shaped structure (instant 3) that matches experimental observations. This further clarifies two critical aspects of the umbrella-shaped jet formation. First, the bubble has to develop a high-curvature tip to initiate the tip contraction. Second, it is the pressure increase at the jet base of bubble 2 (resulting from the focusing flow or pressure wave) that drives the formation of the umbrella-shaped jet. Meanwhile, we systematically vary the initial gas density parameters in the equation of state ( $\rho _{\textit{g0}}$ = 0.26, 1.29 and $6.45 \,{\textrm{kg m}}^{-3}$ ) to represent different levels of gas content inside the bubble (the corresponding numerical results are not included here). The results indicate that gas density has only a minor influence on jet velocity (within 0.2 %), and the tip morphology of the umbrella-shaped jet remains generally consistent. These simulations demonstrate that the jet structure originates primarily from the liquid flow and is largely independent of the gas flow inside the bubble. This finding advances our fundamental understanding of how flow structure governs the tip morphology of jets.

Figure 13. Four types of spraying jets: (a) needle-like spraying jet. The initial parameters are $\gamma =0.76$ , $\theta =1.25$ and the corresponding times for each frame are 4.14, 4.19 and 4.25 ms. (b) Inclined spraying jet. The initial parameters are $\gamma =0.77$ , $\theta =1.33$ , and the corresponding times for each frame are 4.25, 4.36 and 4.53 ms. (c) Inclined sheet-like spraying jet. The initial parameters are $\gamma =0.72$ , $\theta =1.35$ , and the corresponding times for each frame are 4.19, 4.32 and 4.49 ms. (d) Mist-like spraying jet. The initial parameters are $\gamma =0.7$ , $\theta =1.21$ , and the corresponding times for each frame are 4.12, 4.54 and 5.58 ms.

Figure 14. Evolution details of neck breakup process of bubble 2 following the impingement of bubble 1’s jet on the lower surface. The initial parameters are $\gamma =0.79$ , $\theta =1.11$ .

5.1. Spraying jet caused by neck breakup

Figure 14 illustrates the detailed evolution of the neck breakup process of bubble 2. The tip of bubble 2 goes deeply into bubble 1 without coalescence during the contraction of bubble 1 (panel a). Consequently, the high pressure generated by the collapse of bubble 1 acts directly on the neck of the elongated tip of bubble 2 (panel b), initiating the neck breakup (panels c). Driven by the high pressure generated by the flow focusing during the neck breakup, the supersonic jet in centimetre-scale tandem bubbles can achieve velocities exceeding 1000 m s⁻¹ (panel d), comparable to those in millimetre-scale experiments (Fan et al. Reference Fan, Bußmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024). Although the spraying jet accelerated by neck breakup is very fast, we observed that the tip of the spraying jet is always fragmented into mist droplets, as illustrated in figure 13(a–c). In these experiments, the droplets first impact the bubble surface, forming a cluster of tiny craters, while the continuous jet subsequently penetrates the upper surface. This behaviour may be explained by the Worthington jet drop-off mechanism, as illustrated by Gordillo & Gekle (Reference Gordillo and Gekle2010). Following the neck breakup, the jet tip attains significantly high kinetic energy and breaks up due to the capillary deceleration of the liquid. The detached droplets are then further fragmented into smaller droplets due to instability, a process known as secondary atomisation (Guildenbecher, López-Rivera & Sojka Reference Guildenbecher, López-Rivera and Sojka2009; Choi, Byun & Park Reference Choi, Byun and Park2022; Huck et al. Reference Huck, Osuna-Orozco, Machicoane and Aliseda2022). This explains the fine droplet cloud observed at the tip of the spraying jets in the experiments.

Through extensive experiments, we have observed that supersonic jets with straight needle-like tips are relatively rare, while inclined jets (figure 13 b) or inclined sheet-like jets (figure 13 c) are significantly more prevalent. We attribute this prevalence to the inherent instability of the neck breakup process (Gekle & Gordillo Reference Gekle and Gordillo2010). Any initial perturbations during the breakup process can cause asymmetric impingement and non-singular collapse of the neck (Burton, Waldrep & Taborek Reference Burton, Waldrep and Taborek2005; Keim et al. Reference Keim, Møller, Zhang and Nagel2006; Enriquez et al. Reference Enriquez, Peters, Gekle, Schmidt, Lohse and van der Meer2012). Even minor initial perturbations persist throughout the neck breakup process due to memory effects (Keim et al. Reference Keim, Møller, Zhang and Nagel2006). As a result, initial azimuthal asymmetries during neck breakup can easily affect the morphology and direction of the jet in the subsequent process, forming inclined spraying jets in figure 13(b). Schmidt et al. (Reference Schmidt, Keim, Zhang and Nagel2009) and Enriquez et al. (Reference Enriquez, Peters, Gekle, Schmidt, Lohse and van der Meer2012) revealed that, when the initial elliptic cavity is nearly circular, the neck does not collapse into an infinitesimal singularity but instead impinges from two opposing sides, leading to the sheet-like jet, which closely resembles the jet observed in figure 13(c).

Although the tip of the spraying jet is fragmented into droplets due to instability, the subsequent continuous jet remains highly stable. The jet travels at a significantly high velocity with excellent directionality, well aligned with the axis of the two tandem bubbles. The good stability of jet direction is maintained due to the support by the jet base instead of the stagnation point of the neck breakup (Gekle & Gordillo Reference Gekle and Gordillo2010). Further details on the penetration capability of this jet in liquid will be provided in §§ 6 and 7.

5.2. Spraying jet caused by bubble coalescence

The mist-like spraying jet formation involves deep penetration of bubble 2’s elongated tip into bubble 1’s interior, making experimental observation challenging. We therefore employ OpenFOAM simulations to complement the investigation of this jet evolution process. The overall evolution of the mist-like spraying jet is illustrated in figure 15, with velocity and pressure fields provided. The elongated tip of bubble 2 coalesces with the surface of bubble 1 due to the small $\gamma$ (instant 1). At instants 2–3, the tip of bubble 2 forms a fragmented jet during its contraction process, and the subsequent evolution of the fragmented jet leads to the formation of a cluster of dispersed tiny bubbles (instant 4). The dispersed tiny bubbles are drawn by bubble 2 due to its continuous contraction (instant 5). The fragmented liquid jet propagates further into the bubble interior, drawing a trailing gas tail at its base as visualised (instants 5–6). The jet exhibits a fragmented, radially splashing morphology (instant 6), which is quite different from the needle-like spraying jets. These jet evolution details as shown above prove experimentally inaccessible since bubble 2’s elongated tip remains either obscured by bubble 1 at instants 1–3 or manifests as a misty gas cluster at instants 4–5. Although we employ axisymmetric simulations rather than computationally intensive three-dimensional modelling, the numerical results successfully capture the dominant physical processes in the experiment, thereby complementing the limitations of experimental observations and providing critical insights into the formation of mist-like spraying jets.

Figure 15. Dynamics of mist-like spraying jet from anti-phase bubble pair: comparison between the experiment (odd rows) and numerical simulation (even rows). The dimensionless times corresponding to the numerical results are 1.38, 1.44, 1.56, 1.67, 1.74 and 1.87, respectively. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms. The initial parameters are $\gamma =0.65$ , $\theta =1.19$ .

Figure 16. The evolution of the mist-like spraying jet shown in figure 15. ( $a$ – $b$ ) Zoom-in details of the flow field at $t={\textrm{1.68, 1.78}}$ . The small bubbles below the jet base undergo collapse and generate localised high-pressure regions. ( $c$ ) Time–space map of pressure along the axis of symmetry of the two tandem bubbles. The black solid and dashed lines represent the axial position of the upper surface and jet tip of bubble 2, respectively. The yellow dashed and solid lines correspond to two typical moments in ( $a$ ) and ( $b$ ), respectively. The time scale $ R_{\textit{max}} \sqrt {\rho / P_{\infty }}$ is 2.25 ms.

Enlarged views of the flow field surrounding the tiny bubbles at $t={\textrm{1.68, 1.78}}$ are presented in figure 16( $a$ – $b$ ), respectively. These small bubbles undergo collapse and generate localised high-pressure waves to accelerate the fragmented jet. Additionally, the time–space map of the pressure along the symmetry axis during the formation and development of the jet is presented in figure 16(c) to offer a more comprehensive perspective on the jet evolution. The black solid and dashed lines represent the axial position of the upper surface and jet tip of bubble 2, respectively. It can be seen that the upper surface of bubble 2 shows negligible changes during jet development. The yellow dashed and solid lines correspond to two typical moments, which are associated with ( $a$ ) and ( $b$ ), respectively. The intermittent collapse of these bubbles induces continuous pressure fluctuations at the jet base during the time interval $t=1.6$ – $1.8$ . Subject to these high-pressures, the fragmented jet continues to accelerate, and its instability is further amplified, leading to radial splashing (instants 5–6 in figure 15).

In contrast to the localised high-pressure stagnation point generated by neck breakup, the collapse of the dispersed tiny bubbles induces continuous high-pressure fluctuations that accelerate broader liquid jet formation. This results in a jet velocity substantially lower than that of needle-like spraying jets. From the perspective of needle-free injection applications, such mist-like jets, characterised by relatively low velocities and splashing tips, are undesirable and should be avoided.

6.1. Preliminary remarks

The evolution of piercing jets within bubble 2 exhibits notable velocity transitions across three distinct regimes, with maximum velocities exceeding 1200 m s⁻¹ in the spraying jet regime. However, the piercing jet experiences velocity attenuation after penetrating the bubble surface. Particularly for spraying jets, while the high-speed fragmented droplets at the jet tip can only disturb the bubble interface, the jets exhibit pronounced velocity differences between their internal (within bubble 2) and external (after penetrating bubble 2) velocities. To further elucidate the penetration performance of these three jet regimes, we present the dependence of jet velocity and penetration distance on the initiation bubble distance $\gamma$ in figure 17. Here, the internal velocity $U_{\textit{jet}}$ is the maximum velocity of the jet in bubble 2, the external velocity $U_{w\textit{ater}}$ represents the velocity of the jet as it enters the liquid after penetration and $S_{\textit{max}}$ denotes the maximum penetration distance by the jet in the liquid at the moment of cavity protrusion collapse at its tip. The relative initiation bubble time difference, $\theta$ , is held constant in both experiments and numerical simulations. The jet velocities obtained from the BIM and VoF show good agreement with experimental results in figure 17( $\textit{a}$ ). Due to its limitations in handling mesh topology on bubble surfaces, the BIM is used exclusively to simulate scenarios involving conical and umbrella-shaped jets.

As depicted in figure 17( $\textit{a}$ ), a distinct transition is observed with decreasing $\gamma$ : the jet morphology evolves sequentially from a conical jet to an umbrella-shaped jet and ultimately to a spraying jet. Generally, the internal velocity $U_{\textit{jet}}$ exhibits a non-monotonic dependence on $\gamma$ , first increasing then decreasing as $\gamma$ is reduced. When the value of $\gamma$ decreases to approximately 0.77, $U_{\textit{jet}}$ abruptly increases to exceed 120, corresponding to a dimensional value of 1200 m s⁻¹, as shown in figure 21 provided in the Appendix. This increase is attributed to the extreme acceleration caused by the neck breakup at the elongated tip of bubble 2, which closely aligns with the velocity reported by Fan et al. (Reference Fan, Bußmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024). Further reducing the initial bubble–bubble distance leads to the coalescence between the two bubbles and contributes to the rapid decrease in $U_{\textit{jet}}$ . The strong correlation between maximum penetration distance $S_{\textit{max}}$ and external jet velocity $U_{w\textit{ater}}$ in figure 17( $\textit{b}$ ) suggests that penetration performance is dominantly controlled by the external jet velocity $U_{w\textit{ater}}$ rather than the internal jet velocity $U_{\textit{jet}}$ . Particularly for needle-free injection applications, the external jet penetration characteristics constitute the primary focus of this investigation.

Figure 17. Dependence of the velocity and maximum penetration distance of piercing jets on $\gamma$ at a fixed $\theta = 1$ . The velocity is measured using several consecutive frames from high-speed imaging. The error is primarily attributed to spatial resolution, where the positional error corresponds to the length of one pixel. For the fastest jets (spraying jets), the experimentally measured velocity represents a lower bound of the real velocity due to the inherent temporal resolution limitations of high-speed imaging. The jet completes its trajectory from formation (initiated by neck breakup) to impact on the bubble surface within three frames, resulting in a maximum relative uncertainty as high as 50 % (Gonzalez-Avila et al. Reference Gonzalez-Avila, Van Blokland, Zeng and Ohl2020). ( $a$ ) Dependence of the internal velocity $U_{\textit{jet}}$ of piercing jets on $\gamma$ . The solid symbols represent experimental data. The hollow symbols denote numerical results obtained from BIM and OpenFOAM. Here, $U_{\textit{jet}}$ is the maximum velocity of the jet in bubble 2. ( $b$ ) Dependence of the external velocity $U_{w\textit{ater}}$ and maximum penetration distance $S_{\textit{max}}$ of piercing jets on $\gamma$ , where $U_{w\textit{ater}}$ represents the velocity of the jet as it enters the liquid after penetration and $S_{\textit{max}}$ denotes the maximum penetration distance by the jet in the liquid at the moment of cavity protrusion collapse at its tip. All variables in the figure are dimensionless. The velocity scale $ \sqrt {P_{\infty } / \rho }$ is 10 m s⁻¹.

6.2. Penetration performance of piercing jets

To describe the piercing jet dynamics after entering the liquid, we develop a simplified model by conceptualising the jet as a liquid bullet with initial velocity $U_{w\textit{ater}}$ and effective mass $m$ . When such a liquid bullet propagates through water, the entrained cavity envelops its main body, leaving only the tip in contact with the surrounding liquid. Consequently, the dominant drag force $F$ arises primarily from hydrodynamic pressure at the advancing tip, expressed as $F = -C_{\textit{D}} \rho A u^2$ , where $\rho$ denotes the water density, $A$ denotes the jet’s cross-sectional tip area (the cross-sectional area of the liquid phase at the cavity tip) and $u$ is the jet velocity. The drag coefficient $C_{\textit{D}}$ (primarily determined by the tip geometry) is taken as $C_{\textit{D}} = 0.5$ (Soh, Khoo & Yuen Reference Soh, Khoo and Yuen2005; Mallick et al. Reference Mallick, Kumar, Tamboli, Kulkarni, Sati, Devi and Chandar2014; Kuwata et al. Reference Kuwata, Abe, Yokota, Nonomura, Sawada, Yakeno, Asai and Obayashi2021) throughout this study. Neglecting viscous and surface tension effects, the jet’s momentum equation reduces to

(6.1) \begin{equation} m \frac {\textrm{d}u}{\textrm{d}t} = -C_{\textit{D}} \rho A u^2, \end{equation}

where $m$ represents the effective jet mass. By separating variables and integrating (6.1), we obtain $u(t) = {U_{w\textit{ater}}}/{(1 + K U_{w\textit{ater}} t)}$ , where $K = {C_{\textit{D}} \rho A}/{m}$ . Thus, the penetration $S$ can be expressed as

(6.2) \begin{equation} S(t) = \int _0^t u(t) \, {\textrm{d}}t = \frac {1}{K} \ln (1 + K U_{w\textit{ater}} t). \end{equation}

Since the piercing jet undergoes rupture during propagation, only the fluid downstream of the rupture point contributes energy to penetration. Typically, the length of this jet segment measures approximately $0.5S_{\textit{max}}$ , where $S_{\textit{max}}$ denotes the maximum penetration distance. Assuming this segment of the jet enters the water at a constant velocity $U_{w\textit{ater}}$ , and considering the experimental observation that the cross-sectional area of this jet segment remains nearly unchanged, the liquid bullet’s effective mass simplifies to $m = 0.5 \rho AS_{\textit{max}}$ , yielding $K = 1/S_{\textit{max}}$ . We measured $K$ and $U_{w\textit{ater}}$ for each case in figures 6, 7 and 8 and the predicted curves from (6.2) are compared with experimental curves in figure 18. The liquid-bullet model predictions agree well with experimental results prior to cavity collapse at the jet tip. After the collapse, the jet’s advancing energy dissipates significantly, reaching maximum penetration distance $S_{\textit{max}}$ . Extensive experimental data reveal that collapse times for all piercing jets consistently occur within $0.8 \lt t \lt 1.2$ . We uniformly adopt $t = 1$ and substitute it into (6.2) to obtain $S_{\textit{max}} = U_{w\textit{ater}}/{(\textit{e}-1)}$ , which represents a general conclusion applicable to most piercing jets. Once the jet velocity $U_{w\textit{ater}}$ is determined, we can estimate the maximum penetration distance $S_{\textit{max}}$ and further obtain the temporal curve of the penetration distance $S$ through (6.2).

Figure 18. Temporal curves of the penetration distance $S$ for three regimes of piercing jets. The hollow symbols denote experimental results and the dashed lines represent predictions of the liquid-bullet model given by (6.2) for different values of $K$ and $U_{w\textit{ater}}$ . Here, $U_{w\textit{ater}}$ represents the velocity of the jet as it enters the water. All variables in the figure are dimensionless.

Figure 19. Variation in the external jet velocity $U_{w\textit{ater}}$ as a function of the maximum penetration distance $S_{\textit{max}}$ . Here, $U_{w\textit{ater}}$ is the velocity of the jet as it enters the water and $S_{\textit{max}}$ is the distance travelled by the jet in water when the the jet tip cavity collapse. The data collapse onto the red solid line $S_{\textit{max}} = U_{w\textit{ater}}/{(\textit{e}-1)}$ , which is derived from (6.2). The yellow shaded region represents the distribution range of the spraying jet.

We provide a comprehensive insight into the penetration capabilities of the three types of jets in figure 19. In general, the penetration capability of a piercing jet is predominantly governed by its external jet velocity $U_{w\textit{ater}}$ , in agreement with our liquid-bullet model, as demonstrated by the solid curves in the figure. The conical and umbrella-shaped jets exhibit a concentrated data distribution. For identical external jet velocities, conical and umbrella-shaped jets achieve comparable maximum penetration distances. However, the umbrella-shaped jet can achieve higher external jet velocities, thereby extending its penetration range. The penetration capability of the spraying jets at low velocity, particularly mist-like spraying jets with pronounced instability, is unstable and slightly lower than that of the conical and umbrella-shaped jets. This may be due to the excessive measurement of the external jet velocity $U_{w\textit{ater}}$ of the mist-like spraying jets. The discontinuous cavity formed by the impact of the fragmented jet tip disturbs the velocity measurement of the upstream continuous jet, leading to the overestimation of the true velocity of the penetration-capable continuous phase. In contrast, high-velocity spraying jets still adhere to the liquid-bullet model. Their neck breakup acceleration mechanism enables external velocities exceeding those achievable by umbrella-shaped jets. Regarding the upper limit of jet penetration capability, spraying jets outperform the other types, making them highly suitable for applications requiring long-distance penetration and efficient material transport. For scenarios demanding shorter penetration distances, umbrella-shaped jets, owing to their superior stability and controllability, represent a more optimal choice. The methods for controlling initial parameters to achieve umbrella-shaped jets or optimise the penetration performance of spraying jets will be discussed in the following section.

6.3. Maps of jet regime, velocity and penetration distance in $\theta -\gamma$ parameter space

This section examines the effect of initial parameters $\theta$ and $\gamma$ of the tandem bubbles system on the different regimes of the piercing jet. Approximately 300 experimental data points are selected and interpolated to obtain the phase diagrams, where distinct symbols differentiate the three piercing jet regimes across the investigated parameter space ( $0.7 \lt \theta \lt 1.7$ and $0.6\lt \gamma \lt 1$ ). Particular emphasis is placed on the external jet velocity $U_{w\textit{ater}}$ and maximum penetration distance $S_{\textit{max}}$ of the piercing jet after penetrating the bubble surface.

Figures 20(a) and 20( $b$ ) present phase diagrams of three jet regimes according to $\theta$ and $\gamma$ . Our experimental data primarily focus on parameter space generating high-velocity spraying jets and umbrella-shaped jets, given their critical relevance to needle-free injection applications. As the relative initiation time difference $\theta$ increases from 0, the interaction between bubble 1 and bubble 2 transitions from attraction to repulsion, causing the jet direction to reverse from pointing toward bubble 1 to penetrating through bubble 2. When $\theta$ exceeds approximately 2, bubble 2 becomes hydrodynamically decoupled from bubble 1’s primary oscillation. The dynamics of bubble 2 is governed solely by pressure waves from bubble 1’s collapse and subsequent re-expansion. Therefore, this decoupled regime falls outside the scope of our investigation. The background colour in each figure represents the magnitude of $U_{w\textit{ater}}$ and $S_{\textit{max}}$ , with different symbols used to distinguish three piercing jet regimes. The boundaries between these regimes are denoted by dashed lines: the black dashed line separates conical jets from umbrella-shaped jets, while the blue dashed line distinguishes umbrella-shaped jets from spraying jets.

Figure 20(a) demonstrates that the jet regime is governed by the interplay of the parameters $\theta$ and $\gamma$ . Larger $\theta$ and $\gamma$ , which correspond to greater relative initiation distances and time differences, facilitate the formation of conical jets, while smaller values predominantly result in spraying jets. The stable umbrella-shaped jets are confined to a narrow intermediate region between these two regimes. The upper right and lower left regions of figure 20(a) correspond to regimes of minimal and maximal bubble–bubble coupling, respectively. These regions exhibit lower jet velocities, characteristic of conical jets and mist-like spraying jets. The spraying jets and umbrella-shaped jets with high velocity, which are suitable for achieving underwater directional energy transfer, are widely distributed in the region with moderate values of $\theta$ and $\gamma$ . The high-speed jet region of our phase diagram reproduces the findings reported by Han et al. (Reference Han, Köhler, Jungnickel, Mettin, Lauterborn and Vogel2015) and Fan et al. (Reference Fan, Bußmann, Reuter, Bao, Adami, Gordillo, Adams and Ohl2024). With substantially more experimental data points, our phase diagram achieves higher resolution than prior works. We find that the high-speed jet region in the phase diagram exhibits a band-like distribution along the left side of the boundary between umbrella-shaped jets and spraying jets. Parameter regions generating internal velocities exceeding 1200 m s⁻¹ are mapped, while their corresponding external velocity variations are simultaneously quantified. This enhanced resolution enables higher precision in controlling piercing jets.

Since the external jet velocity $U_{w\textit{ater}}$ cannot fully characterise the penetration capability of the piercing jets, we further investigate the distribution of their maximum penetration distance $S_{\textit{max}}$ within the $\theta -\gamma$ parameter space. Its distribution pattern within the $\theta -\gamma$ parameter space resembles that of the external velocity, as shown in figure 20(b). Spraying jets exhibit optimal penetration performance for long-range applications within the red-highlighted parameter region, which corresponds to the needle-like spraying jet. Meanwhile, the umbrella-shaped jets adjacent to the red-highlighted region represent superior short-range penetration capability. These findings establish a comprehensive control framework for utilising tandem-bubble jets as a viable alternative to conventional needle-based injection systems.

Figure 20. Phase diagram of the external jet velocity $U_{w\textit{ater}}$ ( $a$ ) and the maximum penetration distance $S_{\textit{max}}$ ( $b$ ) as a function of $\gamma$ and $\theta$ , with background colour interpolated from experimental data. The dashed lines represent the fitted boundaries between different jet regimes: the black dashed line separates conical jets and umbrella-shaped jets, while the blue dashed line distinguishes umbrella-shaped jets from spraying jets.