3.1. Formation mechanism

The formation process of the SVR can be roughly divided into two stages based on its vorticity distribution: the rapid accumulation stage ($t_d^*\leq 1$) and the development stage ($t_d^*>1$). The azimuthal vorticity contours near the nozzle tip during the initial formation process in Cases 1 and 4 are shown in figure 7. Based on the right-hand rule, the azimuthal vorticity $\omega _\theta$ in the trailing shear layer and LVR of the starting jet is designated as positive (anticlockwise), whereas $\omega _\theta$ in SVR is negative (clockwise). For $t_d^*\leq 1$, the negative vorticity accumulates rapidly at the nozzle tip ($x=0$, $r=0.5D$) with a concentrated vorticity distribution, as shown in figures 7(a), 7(b), 7(d) and 7(e). During this stage, the vortex core of the SVR does not develop an independent maximum point of local vorticity magnitude; instead, the maximum point consistently remains located at the nozzle tip ($x=0$, $r=0.5D$). Subsequently, the development of the SVR rearranges the initially concentrated vorticity distribution into an approximately Gaussian distribution (Kaplanski et al. Reference Kaplanski, Sazhin, Fukumoto, Begg and Heikal2009) within the vortex core at $t_d^*>1$, as shown in figures 7(c) and 7(f). The maximum point of local vorticity magnitude then appears at the centre of the vortex core of the SVR. The vortex core of the SVR begins to move away from the nozzle tip. This, in turn, induces a small positive vorticity region near the nozzle wall. The two phenomena observed after the termination of the starting jet can also be interpreted as breaking away from the spatial confinement imposed by the axial flow of the starting jet. The two stages during the formation of the SVR can be roughly distinguished at $t_d^*=1$. This seems to be independent of the duration of the deceleration stage, i.e. $(L/D)_d$. The only variation observed is that SVR weakens as $(L/D)_d$ increases, comparing figures 7(c) and 7(f).

Figure 7.

Azimuthal vorticity contours in $x$–$r$ plane near the nozzle tip for the initial formation process of the SVR in (a–c) Case 1 with $(L/D)_d=0.125$ and (d–f) Case 4 with $(L/D)_d=1$ at $(L/D)_i=0.75$.

The vorticity contours and $Q$-criterion visualizations for Case 1 are further shown over a longer time period with larger spatial scale in figure 8. Also shown in the figure are the results obtained by the fluid tracer method (Yang, Jia & Yin Reference Yang, Jia and Yin2012). It can be seen from figure 8(a i) that before the starting jet begins to decelerate ($t_d^*=0$), a secondary boundary layer with negative vorticity, induced by the flow from the LVR (Irdmusa & Garris Reference Irdmusa and Garris1987), appears outside the nozzle tip. As the starting jet decelerates ($t_d^*>0$), the negative vorticity rapidly accumulates in this region, especially near the nozzle tip (figure 8b i). The $Q$-criterion also clearly indicates that the vortical structure begins to appear near the nozzle tip (figure 8b ii). This shows the existence of the SVR, i.e. the red region bounded by green. At the same time, the trailing shear layer of the starting jet also bends radially inward towards the centre line. The most critical is that for Case 1 with $(L/D)_i=0.75$, the LVR does not have enough time to translate away from the nozzle exit, where SVR is located. Consequently, the LVR and SVR form a pair of vortex rings with opposite signs, generating the mutually induced velocities (Zhu et al. Reference Zhu, Zhang, Gao and Yu2022). The SVR translates downstream and reduces the radial position of the vortex core due to the mutually induced velocity from the LVR (indicated by the red arrow on the SVR), even though the self-induced velocity of the SVR always points upstream (indicated by the black arrow on the SVR), as shown in figures 8(b i) and 8(c i). The resulting movement is determined by the relative strength and position between the SVR and the LVR. As the radial position of vortex core decreases, the SVR moves away from the nozzle wall, weakening the positive vorticity region induced by it on the nozzle inner wall. Meantime, the vortex core of the SVR continuously absorbs vorticity from the boundary layer induced by the LVR and rearranges the vorticity distribution, thereby expanding rapidly. The mutually induced velocity from the SVR in Case 1 also causes the radial position of vortex core for the LVR to decrease (Maxworthy Reference Maxworthy1977; Das et al. Reference Das, Bansal and Manghnani2017) and promotes its downstream motion (indicated by the red arrow on the LVR). The reduction in the radial position of vortex core for the LVR would also indirectly increase its self-induced velocity (Krieg & Mohseni Reference Krieg and Mohseni2021).

Figure 8.

(i) Azimuthal vorticity contours, (ii) $Q$-criterion visualizations (normalized by its maximum value with regions greater than 0.05 marked in red) and (iii) temporal evolution of fluid parcels near the nozzle tip during the formation process of the SVR in Case 1 with $(L/D)_i=0.75$ and $(L/D)_d=0.125$. The black and red arrows in subpanels (i) represent the self-induced and mutually induced velocities on the SVR and LVR, respectively.

The corresponding temporal evolution of fluid parcels is shown in figure 8(a iii,b iii,c iii). Tracers are released into the flow at the initiation of starting jet deceleration, i.e. $t_d^*=0$. These tracers are divided into four groups: group 1 (black for the region with $x<0$ and $r>0.5D$), group 2 (green for the region with $x>0$ and $r>0.5D$), group 3 (red for the region with $x<0$ and $0< r<0.5D$) and group 4 (blue for the region with $x>0$ and $0< r<0.5D$), as shown in figure 8(a iii). In the rapid accumulation stage ($t_d^*\leq 1$), the fluid tracers comprising SVR primarily originate from group 1 (figure 8b iii). The fluid entrainment from other groups is not obvious. This also shows the existence of radial inward flow from the outside of the nozzle behind the formation of the SVR. During the subsequent development stage ($t_d^*>1$), fluid tracers from group 3 are also entrained into SVR (figure 8c iii). The fluid tracers within SVR originate only from group 1 and group 3 with $x<0$. The absence of entrainment of fluid tracers from group 4 can be attributed to their inherent downstream velocity upon discharge from the nozzle prior to the deceleration of the starting jet. The absence of fluid tracers from group 2 can be explained by the induced velocity from the LVR, which causes them to move downstream faster and away from the SVR.

From the above discussion, it can be concluded that during the deceleration stage of the starting jet, there exists a radial inward flow near the nozzle exit from the outside of nozzle. This radial inward flow interacts with the nozzle wall resulting in the formation of the SVR. It is similar to the formation process of a vortex ring in the starting disk (Xu & Nitsche Reference Xu and Nitsche2015; Steiner et al. Reference Steiner, Morize, Delbende, Sauret and Gondret2023). Maxworthy (Reference Maxworthy1977) and Didden (Reference Didden1979) attributed this to the induced flow produced by the LVR. However, it is important to note that the LVR produces induced flow regardless of whether the starting jet is in the deceleration stage. Nevertheless, the formation of the SVR only initiates during the deceleration stage of a starting jet. Therefore, it can be considered that, in addition to the induced flow produced by the LVR, there must be another factor driving the fluid outside the nozzle radially inward. This factor should be crucial to the formation of the SVR.

The radial pressure gradient near the nozzle exit due to the deceleration of the starting jet may be considered as the additional factor. The profiles of gauge pressure at the nozzle exit for cases with different durations of the deceleration stage, i.e. different $(L/D)_d$, are shown in figure 9. It can be observed that a radial pressure gradient forms between the inner and outer sides of the nozzle during the deceleration stage ($t_d^*=0.08$ and $t_d^*=0.8$). This can be attributed to the negative gauge pressure within the nozzle exit caused by the deceleration of the starting jet (Gao et al. Reference Gao, Wang, Yu and Chyu2020). After a period of deceleration, an additional low-pressure region emerges near the nozzle wall ($r=0.5D$), i.e. at $t_d^*=0.8$, which corresponds to the low-pressure region induced by the formed SVR (Schlueter-Kuck & Dabiri Reference Schlueter-Kuck and Dabiri2016). After the termination of deceleration stage, i.e. at $t_d^*=1.2$, the radial pressure gradient disappears, and only the low-pressure region induced by SVR remains. Furthermore, as the duration of the deceleration stage increases, the rate of deceleration decreases, which in turn weakens the negative gauge pressure due to the deceleration of starting jet (Gao et al. Reference Gao, Wang, Yu and Chyu2020). This is also reflected in figure 9, where the radial pressure gradient decreases as $(L/D)_d$ increases. As the radial pressure gradient decreases, the weakening of the formed SVR leads to a corresponding weakening of the low-pressure region induced by it.

Figure 9.

The instantaneous profiles of gauge pressure at the nozzle exit ($x=0$) for Case 1 with $(L/D)_i=0.75$ and $(L/D)_d=0.125$, Case 3 with $(L/D)_i=0.75$ and $(L/D)_d=0.5$, and Case 5 with $(L/D)_i=0.75$ and $(L/D)_d=2$. The instants corresponding to these profiles in Cases 1, 3 and 5 are indicated on the right-hand side with the velocity programs.

The circulation growth processes of the SVR, calculated by (2.8), are shown in figure 10 for cases with $(L/D)_i=0.75$ and $(L/D)_d$ ranging from 0.125 to 2. Similar to the LVR (Rosenfeld et al. Reference Rosenfeld, Rambod and Gharib1998; Gao Reference Gao2011; Zhu et al. Reference Zhu, Zhang, Gao and Yu2023b), the circulation growth processes of the SVR are also independent of $Re$ (figure 10a). In addition, the circulation growth rate and the final circulation value of the SVR decrease as $(L/D)_d$ increases, which is consistent with the previous qualitative results. It can be found that the relationship between the final circulation of the SVR and $(L/D)_d$ can be approximately expressed as $\varGamma \sim [(L/D)_d]^{-0.5}$. To unify the circulation values of the SVR at different $(L/D)_d$, $[(L/D)_d]^{-0.5}$ should be taken into account for normalizing the circulation. For instance, the circulation can be normalized by $\bar {U}D[(L/D)_d ]^{-0.5}$. On the other hand, the duration of the circulation growth of the SVR increases with increasing $(L/D)_d$, with the major part occurring during the deceleration stage of the starting jet, i.e. $0< t_d^*\leq 1$. The duration of the circulation growth of the SVR can be approximated as $T_d$, although some may extend until $t_d^*=1.5$ for small $(L/D)_d$. The circulation growth of the SVR after $t_d^*=1$ can be attributed to the continuous absorption of negative vorticity near the nozzle exit generated during the deceleration stage of the starting jet. Of course, the additional contribution from the induced flow generated by the LVR near the nozzle exit cannot be neglected for cases with small $(L/D)_i$ and $(L/D)_d$. Considering the equation for the newly defined formation time $t_d^*$ in the present work, i.e. (2.7), $t_d^*$ may be more suitable than the traditional $t^*$ to illustrate the circulation growth process of the SVR. The circulation growth processes are also shown in figure 10(b) as $\varGamma [(L/D)_d]^{0.5}/(\bar {U}D)\sim t_d^*$. It can be found that the circulation growth processes with different $(L/D)_d$ collapse into a cluster of curves, i.e.

(3.1)\begin{equation} \frac{{\rm d}\{\varGamma [(L/D)_d]^{0.5}/(\bar{U}D)\}}{{\rm d} t_d^*}={\rm const.} \end{equation}

Introducing (2.7) into the above equation in place of $t_d^*$, it can be obtained that

(3.2)\begin{equation} \frac{{\rm d}\{\varGamma [(L/D)_d]^{0.5}/(\bar{U}D)\}}{{\rm d}[(t-T_a-T_c)/T_d]}={\rm const.},\end{equation}

where $\bar {U}$, $D$, $T_a$ and $T_c$ can be approximately regarded as constants for the same $(L/D)_i$ and $Re$, and only $T_d$ changes with $(L/D)_d$ with $T_d\sim (L/D)_d$. Equation (3.2) can then be rewritten as

(3.3)\begin{equation} \frac{{\rm d}\varGamma}{{\rm d}t}\sim [(L/D)_d]^{{-}1.5}. \end{equation}

Therefore, the circulation growth rate of the SVR ${\rm d}\varGamma /{\rm d}t$ can be scaled with $[(L/D)_d]^{-1.5}$ to unify it across different $(L/D)_d$.

Figure 10.

Circulation of the SVR against (a) $t^*$ and (b) $t_d^*$ for Cases 1-5 ($Re=2000$) and Cases 15-19 ($Re=4000$) with $(L/D)_i=0.75$, where $(L/D)_d$ ranges from 0.125 to 2.

3.2. Influence from the LVR

In order to analyse the influence from the LVR to the formation of the SVR, cases with larger $(L/D)_i=2.25$ are chosen for comparison. In these cases, the LVR has been positioned far away from the nozzle exit at $t_d^*=0$, resulting in a weaker induced flow generated by it near the nozzle exit (recall figure 6). The azimuthal vorticity contours and $Q$-criterion visualizations for Case 6 with $(L/D)_i=2.25$ are shown in figure 11. At $t_d^*=0$, comparing with Case 1 with $(L/D)_i=0.75$ (recall figure 8a i), figure 11(a i) reveals that the LVR has almost exited the region of $x\leq 0.6D$, resulting in a weaker and almost negligible secondary boundary layer. As the starting jet decelerates, similar to Case 1 with $(L/D)_i=0.75$, SVR with highly concentrated vorticity is formed near the nozzle tip during the initial rapid accumulation stage (figures 11b i and 11b ii). In the subsequent development stage, the vortex core area of the SVR expands by continuously absorbing vorticity from the boundary layer on the outer side of the nozzle and rearranging the vorticity, as shown in figures 11(c i) and 11(c ii). However, the difference from Case 1 with $(L/D)_i=0.75$ is that, as the mutually induced velocity from the LVR diminishes, the vortex core position of the SVR remains almost unchanged near the nozzle tip. The SVR continues to induce positive vorticity on the inner wall of nozzle, which would weaken SVR through the cancellation of vorticity (James & Madnia Reference James and Madnia1996; Zhu et al. Reference Zhu, Zhang, Gao and Yu2022). The temporal evolution of fluid parcels shown in figure 11(a iii,b iiic iii,) illustrates the influence of the LVR on the fluid entrainment during the formation of SVR. It can be observed that as the LVR translates away from the nozzle exit, fluid from group 2, in addition to the previously identified fluids from groups 1 and 3, would also be entrained into SVR, as shown in figures 11(b iii) and 11(c iii).

Figure 11.

Azimuthal vorticity contours (i), $Q$-criterion visualizations (ii) and temporal evolution of fluid parcels (iii) near the nozzle tip during the formation process of the SVR in Case 6 with $(L/D)_i=2.25$ and $(L/D)_d=0.125$.

Comparing figures 8 and 11, one obvious variation is that the influence of the LVR on the vortex core trajectories of the SVR changes as $(L/D)_i$ increases. This is particularly evident in the radial position development of the vortex core, where the vortex core position of the SVR can be determined by the negative vorticity peak in the vortex core. The radial position development of vortex core for the SVR is shown in figure 12 for cases with both $(L/D)_i=0.75$ and $(L/D)_i=2.25$, where $(L/D)_d$ ranges from $0.125$ to $2$. In the rapid accumulation stage ($t_d^*\leq 1$), the negative vorticity peak remains near the nozzle tip ($x=0$, $r=0.5D$), so that this stage is excluded from the present comparison. As the flow develops with increasing $t_d^*$, the radial position of vortex core for SVR decreases for all cases. This reduction can be attributed to two factors: the expanding vortex core area during the development stage is constrained by the nozzle wall, and the induced velocity from the LVR moves the vortex core of the SVR towards the symmetry axis ($r=0$). Therefore, for $(L/D)_d$ equal to $0.125$, $0.25$ and $0.5$, the reduction rate in the radial position of vortex core for the SVR is higher at $(L/D)_i=0.75$, comparing with that at $(L/D)_i=2.25$. As for $(L/D)_d=1$ and $2$, the rate of reduction in the radial position between different $(L/D)_i$ is almost the same. This is because, even with a smaller $(L/D)_i$, the LVR is still far away from the nozzle exit at $t_d^*>1$ after the deceleration stage with a larger $(L/D)_d$, thereby losing its effective influence on SVR. Furthermore, the rapid decrease in the circulation of the SVR immediately after reaching its peak when $(L/D)_d$ is relatively large at $(L/D)_i=0.75$ (recall figure 10b) can be explained by the aforementioned radial position development of the vortex core for the SVR. With a large $(L/D)_d$, SVR remains near the nozzle wall due to its losing the mutually induced velocity from the LVR, and weakens further through the typical ‘vortex–wall’ interaction (Orlandi & Verzicco Reference Orlandi and Verzicco1993; Zhu et al. Reference Zhu, Zhang, Gao and Yu2022).

Figure 12.

Comparison of Cases 1–5 with $(L/D)_i=0.75$ and Cases 6–10 with $(L/D)_i=2.25$ about the radial position development of vortex core for the SVR at $Re=2000$.

The influence from the LVR on the formation of the SVR is further analysed by the circulation growth shown in figure 13 for cases with $(L/D)_i=2.25$. The weakening of the influence from the LVR does not qualitatively alter the circulation growth process of the SVR. The conclusion obtained at $(L/D)_i=0.75$ that the final circulation value and circulation growth rate of the SVR can be scaled by $[(L/D)_d]^{-0.5}$ and $[(L/D)_d]^{-1.5}$, respectively, can still be found in figure 13 for cases with $(L/D)_i=2.25$. In addition, through the detailed comparison of figures 10 and 13, it can be seen that weakening the influence from the LVR has two effects on the circulation growth process of the SVR. Firstly, at $(L/D)_i=2.25$, the maximum circulation of the SVR is approximately $10\,\%$ lower than that at $(L/D)_i=0.75$. This suggests that the induced flow formed by the LVR and the possibly trailing vortex near the nozzle exit (Maxworthy Reference Maxworthy1977; Didden Reference Didden1979) indeed promotes the formation of the SVR. Recalling figure 6, it is evident that as $(L/D)_i$ increases from 0.75 to 2.25, the induced flow formed by the LVR and the possibly trailing vortex near the nozzle exit is weakened far more than the decrease in the maximum circulation of SVR. This further confirms that the pressure difference caused by the deceleration of the starting jet, which is dominated by $(L/D)_d$, is mainly responsible for the formation of the SVR. Secondly, after the rapid growth, the circulation of the SVR either increases slowly ($(L/D)_d=0.125$ and $0.25$) or almost remains unchanged ($(L/D)_d=0.5$) at $(L/D)_i=0.75$, whereas it decreases continuously for all $(L/D)_d$ at $(L/D)_i=2.25$. This can be attributed to the fact that SVR is being unable to move away from the nozzle wall when the induced velocity from the LVR is eliminated, resulting in continuous ‘vortex–wall’ interaction and a subsequent decrease in circulation (recall figure 11c i). In addition, the loss of the induced flow formed by the LVR and the possibly trailing vortex would also be responsible for this.

Figure 13.

Circulation of the SVR against (a) $t^*$ and (b) $t_d^*$ for Cases 6–10 ($Re=2000$) and Cases 20–24 ($Re=4000$) with $(L/D)_i=2.25$, where $(L/D)_d$ ranges from $0.125$ to $2$.

Under the influence of the LVR, the secondary boundary layer with negative vorticity (same as that in SVR) is enhanced. This may be a crucial section in the influence of the LVR on the formation of the SVR. In order to explore the role of the secondary boundary layer during the formation process of the SVR, a vertical wall is added in the nozzle exit plane, i.e. the nozzle with a tip angle of $90^\circ$, as shown in figure 2(b). This configuration allows the induced flow to generate a stronger secondary boundary layer (Irdmusa & Garris Reference Irdmusa and Garris1987; Gharib et al. Reference Gharib, Rambod and Shariff1998; Rosenfeld et al. Reference Rosenfeld, Rambod and Gharib1998).

The azimuthal vorticity contours and $Q$-criterion visualizations for Case 11 with the vertical wall are shown in figure 14. Comparing with Case 1 without the vertical wall at $t_d^*=0$ (recall figure 8a i), figure 14(a i) reveals that the secondary boundary layer indeed becomes more pronounced, with a portion of it being directly entrained by the LVR. At $t_d^*=0.8$ (within the rapid accumulation stage), SVR with highly concentrated vorticity is formed near the nozzle tip (figures 14b i and 14b ii). Furthermore, a portion of negative vorticity from the vicinity of the SVR is being entrained by the LVR at this time (figure 14b i). During the development stage, i.e. $t_d^*>1$, the vortex core area of the SVR expands by continuously absorbing vorticity from the secondary boundary layer and rearranging the vorticity (figures 14c i and 14c ii). Comparing figures 8(c i) and 14(c i) reveals that, due to the obstruction of the vertical wall, SVR does not move downstream under the influence of the LVR, and the reduction in the radial position of the vortex core for the SVR is also less pronounced. The radial position of the vortex core for the LVR is significantly reduced and it translates away from the nozzle exit slowly. This allows the induced velocity from the LVR to continuously enhance the secondary boundary layer and transport more negative vorticity to SVR. The fluid entrainment process of the SVR in the case with the vertical wall is qualitatively changed. The fluid particles in group 1 ($x<0$ and $r>0.5D$), originally entrained by SVR (recall figure 8c iii), are completely shielded, as shown in figure 14(a iii). On the contrary, the fluid particles in group 2 ($x>0$ and $r>0.5D$), which cannot be entrained into SVR in the case without the vertical wall, expands radially inward and upstream to form SVR (see figures 14a iii and 14b iii). In addition, the ability of the SVR to entrain the fluid from group 3 ($x<0$ and $0< r<0.5D$) has almost been eliminated, and almost all the fluid that constitutes SVR comes from group 2 (figure 14c iii).

Figure 14.

Azimuthal vorticity contours (i), $Q$-criterion visualizations (ii) and temporal evolution of fluid parcels (iii) near the nozzle tip during the formation process of the SVR in Case 11 with the vertical wall at $(L/D)_i=0.75$ and $(L/D)_d=0.125$.

The circulation growth processes of the SVR with different $(L/D)_i$ and $(L/D)_d$ are shown in figure 15 to quantitatively analyse the influence from the secondary boundary layer. Due to the variation in geometry, the circulation of the SVR formed by the nozzle with a vertical wall is significantly reduced. However, the relationship $\varGamma [(L/D)_d]^{0.5}/(\bar {U}D)\sim t_d^*$ still collapses into a cluster of curves for different $(L/D)_d$, as shown in figure 15(b). This reflects that the conclusion that the final circulation value and circulation growth rate of SVR can be scaled by $[(L/D)_d]^{-0.5}$ and $[(L/D)_d]^{-1.5}$, respectively, remains valid. It can also be found from the red lines in the figure that the LVR would move away and cannot effectively affect the flow near the nozzle exit at larger $(L/D)_d$, resulting in a convergence of circulation growth processes at different $(L/D)_i$. Only when $(L/D)_d$ is small, are there differences between the circulation growth processes with different $(L/D)_i$ (see the black lines). The difference with different $(L/D)_i$ mainly occurs after $t_d^*=1$. This can be explained by the fact that for $t_d^*<1$, the diameter of the LVR is large, so that its induced velocity cannot effectively transport the negative vorticity from the secondary boundary layer to SVR. On the contrary, the LVR entrains part of the negative vorticity, which offsets the enhancement to the secondary boundary layer due to its induced velocity, as shown in figure 14(b i). As the diameter of the LVR gradually decreases after $t_d^*=1$, its induced velocity could enhance the transport of negative vorticity from the secondary boundary layer to SVR (figure 14c i). For cases with a vertical wall, combining with the induced velocity from the LVR, the influence of the LVR increases the circulation of the SVR by $50\,\%$ as $(L/D)_i$ decreases from 2.25 to 0.75 (see the black lines in figure 15). This increment is much greater than the $10\,\%$ observed in the cases without a vertical wall (comparing figures 10 and 13).

Figure 15.

Circulation of the SVR in the cases with the vertical wall for Case 11 with $(L/D)_d=0.125$ and Case 12 with $(L/D)_d=1$ at $(L/D)_i=0.75$, as well as Case 13 with $(L/D)_d=0.125$ and Case 14 with $(L/D)_d=1$ at $(L/D)_i=2.25$ against (a) $t^*$ and (b) $t_d^*$.

4.1. Influence to the development of starting jet

4.1.1. Leading vortex ring

The influence of the SVR on the LVR can be considered by examining the radial position development of vortex core for LVR (Blondeaux & De Bernardinis Reference Blondeaux and De Bernardinis1983; Das et al. Reference Das, Bansal and Manghnani2017). However, this has not been quantitatively analysed in conjunction with the formation process of the SVR. The radial position development of vortex core for the LVR are shown in figure 16. For $(L/D)_i=0.75$, three types of variations are shown for different $(L/D)_d$. First, the radial position of vortex core only reduces for $(L/D)_d=0.125$; second, the radial position of vortex core increases initially and then reduces for $(L/D)_d=0.25$, $0.5$ and $1$; and finally, the radial position of vortex core only increases for $(L/D)_d=2$ (figure 16a). As $(L/D)_i$ increases from $0.75$ to $2.25$, the radial position of vortex core for the LVR becomes larger after a longer period of growth, but with a weaker decreasing trend, comparing figures 16(a) and 16(b). The radial position of vortex core for the developing LVR is primarily influenced by several factors. Growth through the absorption of vorticity in the early stage (Gao & Yu Reference Gao and Yu2010) and interaction with the trailing vortex in the later stage (Gao et al. Reference Gao, Yu, Ai and Law2008) both contribute to an increase in the radial position. Conversely, only the induced velocity from the SVR reduces the radial position of vortex core for LVR. Therefore, as shown in figure 16, it is evident that as SVR weakens or the LVR moves away from the nozzle exit, the decreasing trend in the radial position of vortex core for the LVR is correspondingly weakened. In addition, the decrease in the radial position of vortex core for the LVR usually starts near $t_d^*=1$, i.e. close to the completion of the rapid accumulation stage. This can be attributed to the fact that SVR has not yet established its own induced velocity field before this instant (Zhu et al. Reference Zhu, Zhang, Gao and Yu2022), and the LVR is still able to absorb the vorticity from the trailing jet and grow.

Figure 16.

Radial position development of vortex core for the LVR during and after the deceleration of starting jet in (a) Cases 1–5 with $(L/D)_i=0.75$ and (b) Cases 6–10 with $(L/D)_i=2.25$.

The influence of the SVR on the development of dynamic characteristics, i.e. the circulation, for the LVR is to be discussed. The circulation development processes of the LVR, calculated using (2.8) (without applying the negative sign after integration), are shown in figure 17 after the deceleration of starting jet in Cases 1–10 with different $(L/D)_i$ and $(L/D)_d$. The SVR reduces the ability of the LVR to absorb vorticity from the trailing jet. It should be noted that for two starting jets with different $(L/D)_i$ (but the same $(L/D)_d$), the velocity program after deceleration would be the same. This means that the trailing jet formed after deceleration should have similar characteristics, particularly in the distribution of vorticity flux. In cases with larger $(L/D)_i$, the LVR is closer to saturation and therefore has a weaker capacity to absorb vorticity (Krieg & Mohseni Reference Krieg and Mohseni2021). However, as shown in figure 17, the circulation growth rate of the LVR is actually lower in cases with $(L/D)_i=0.75$ than that in cases with $(L/D)_i=2.25$. Even in cases with $(L/D)_d=0.125$ and $(L/D)_d=0.25$ at $(L/D)_i=0.75$, the circulation of LVR decreases in the later stages of development. This can be attributed to two underlying mechanisms. First, the induced velocity from the SVR transports a portion of the nearby vorticity upstream (Wakelin & Riley Reference Wakelin and Riley1997), away from the LVR, thereby reducing the vorticity flux that the trailing jet can supply. This might also cause the circulation of LVR to decrease due to the stripping of its vorticity when the distance between the SVR and the LVR is small enough. Secondly, the presence of the SVR alters the relative radial position between the trailing shear layer and the LVR, mainly by reducing the radial position of the former more. Inspecting figures 8(b i), 11(b i) and 11(c i), it can be observed that the presence of the SVR significantly reduces the radial position of the nearby trailing shear layer. This directly reduces the radial position of the trailing shear layer at the tail of the LVR for a shorter trailing jet at a smaller $(L/D)_i$. This reduction in the radial position has been considered in the investigations on the pinch-off of the LVR (Mohseni, Ran & Colonius Reference Mohseni, Ran and Colonius2001; Allen & Naitoh Reference Allen and Naitoh2005; Dabiri & Gharib Reference Dabiri and Gharib2005), which would have weakened the ability of the LVR to absorb vorticity from the trailing shear layer. In addition, for $(L/D)_d=0.25$, $0.5$, $1$ and $2$ at $(L/D)_i=2.25$ and $(L/D)_d=0.5$, $1$ and $2$ at $(L/D)_i=0.75$, there is a step reduction in the circulation of the LVR in the later stage of its development. This can be attributed to the formation of a trailing vortex in the trailing shear layer that strips away the vorticity from the LVR (Gao et al. Reference Gao, Yu, Ai and Law2008). The most obvious evidence is that this step reduction occurs earlier in the case with smaller $(L/D)_i$, where the ability of the LVR to absorb vorticity is weakened. The faster vorticity accumulation in the trailing shear layer and the larger relative thickness of the trailing shear layer (caused by the smaller radial position) lead to the earlier formation of the trailing vortex (Zhao, Frankel & Mongeau Reference Zhao, Frankel and Mongeau2000; Gao & Yu Reference Gao and Yu2012). In addition, the above-mentioned effect of the SVR on suppressing the absorption of vorticity by the LVR and even stripping vorticity from the LVR can also be manifested by the influence of the trailing vortex on the LVR (Gao et al. Reference Gao, Yu, Ai and Law2008; Gao Reference Gao2011). It can be found that these phenomena of weakening the LVR are more obvious at $(L/D)_i=0.75$ than at $(L/D)_i=2.25$. Recalling figure 8, the trailing vortex cannot be generated at $(L/D)_i=0.75$ with $(L/D)_d=0.125$, but the circulation of the LVR in figure 17 still decreases. In addition, the decrease in the circulation growth rate of the LVR with decreasing $(L/D)_i$ also occurs in the stage before the formation of the trailing vortex, i.e. before the instant of the circulation step reduction. This effectively distinguishes the influence of the SVR from that of the trailing vortex. Therefore, it can be considered that the weakening of the circulation growth process and even the phenomenon of circulation reduction shown in figure 17 can be caused by SVR.

Figure 17.

Circulation development of LVR during and after the deceleration stage of starting jet in Cases 1–5 with $(L/D)_i=0.75$ and Cases 6–10 with $(L/D)_i=2.25$.

4.1.2. Trailing jet

As the LVR travels downstream, the trailing jet connecting it to the nozzle exit emerges and becomes a significant feature in the development of a starting jet (Gharib et al. Reference Gharib, Rambod and Shariff1998; Gao & Yu Reference Gao and Yu2010; Krieg & Mohseni Reference Krieg and Mohseni2021). The trailing jet interacts with SVR near the nozzle exit. A critical characteristic of the trailing jet, i.e. the instantaneous axial distribution of vorticity flux (Gao & Yu Reference Gao and Yu2012) calculated by

(4.1)\begin{equation} \dfrac{{\rm d} \varGamma}{{\rm d} t} (x)=\int_0^\infty\omega_\theta(x,r)u(x,r)\, {\rm d} r, \end{equation}

is shown in figure 18(a). The analysis is performed for the starting jet during and after the deceleration stage at $(L/D)_i=2.25$ with $(L/D)_d=0.125$ and $2$. The absence of the results for the smaller $(L/D)_i=0.75$ can be attributed to the limited travel distance of the LVR, which results in a relatively short and challenging-to-identify trailing jet. Based on the maximum point and the local minimum point near the tail of the LVR of vorticity flux, the vortex core position of LVR and the boundary between the LVR and the trailing jet are marked, respectively (Gao & Yu Reference Gao and Yu2012). Different from the axially uniform distribution of vorticity flux in the trailing jet of a starting jet with constant velocity (figure 7 in Gao & Yu (Reference Gao and Yu2012)), there is a decrease in the vorticity flux distribution along the axial direction upstream towards the nozzle exit, with the vorticity flux near the nozzle exit ($x=0$) continuously decreasing. The decreasing slope is steeper for $(L/D)_d=0.125$. In addition, at the same $t_d^*$ (lines with the same colour), the vorticity flux near the nozzle exit in case with $(L/D)_d=2$ (dashed line) is slightly higher than that with $(L/D)_d=0.125$ (solid line). Even at $t_d^*=0.9$ and $1.2$, before and after the termination of the starting jet, the vorticity flux near the nozzle exit becomes negative in case with $(L/D)_d=0.125$.

Figure 18.

(a) The instantaneous axial distribution of vorticity flux. The star marks the boundary between the LVR and the trailing jet, while the circle denotes the vortex core position of the LVR. (b) The development of vorticity flux at the nozzle exit in Case 6 with $(L/D)_d=0.125$ and Case 10 with $(L/D)_d=2$ at $(L/D)_i=2.25$, where A represents the initiation of deceleration, while B and C represent the termination of starting jets with $(L/D)_d=0.125$ and $(L/D)_d=2$, respectively.

The vorticity flux at the nozzle exit is equal to the circulation growth rate of the total starting jet, which is related to the profiles of velocity and pressure at the nozzle exit from the Helmholtz vorticity transport equation (Krieg & Mohseni Reference Krieg and Mohseni2013; Zhu et al. Reference Zhu, Zhang, Gao and Yu2023a). The development of vorticity flux at the nozzle exit is shown in figure 18(b), along with the results calculated by the slug model (Gharib et al. Reference Gharib, Rambod and Shariff1998). The velocity programs of Case 6 and Case 10, with the same $(L/D)_i$, should be identical before the initiation of deceleration, but the development of circulation growth rates in these two cases differs. This is mainly because the different $(L/D)_d$ values result in variations in the $\bar {U}$ used to normalize the results. Before the initiation of deceleration at A, the slug model underestimates the circulation growth rate due to the presence of over pressure (Krueger Reference Krueger2005; Zhu et al. Reference Zhu, Zhang, Gao and Yu2023a). Especially during the acceleration stage, the circulation growth rate reaches its peak at the end of acceleration. This underestimation persists during the deceleration stage, especially for the case with $(L/D)_d=2$. The presence of the SVR has a positive effect on the circulation growth rate. The presence of the SVR can be equivalent to an obstacle that reduces the effective outward flow area of the starting jet at the nozzle exit (Zhu et al. Reference Zhu, Zhang, Xia, Yu and Gao2024), which is comparable to the influence of the starting jet produced by an orifice configuration (Zhu et al. Reference Zhu, Zhang, Gao and Yu2023a). As a result, the pressure and velocity within the nozzle exit are elevated, both contributing to the increase in circulation growth rate. It should be noted that the deceleration of the starting jet produces negative gauge pressure at the nozzle exit (Gao et al. Reference Gao, Wang, Yu and Chyu2020), which would reduce the circulation growth rate. The negative gauge pressure becomes more obvious as $(L/D)_d$ decreases, so the underestimation of the circulation growth rate by the slug model can be almost ignored during the deceleration stage in the case with $(L/D)_d=0.125$. On the other hand, the induced velocity from the SVR transports a portion of the trailing jet shear layer upstream, thereby reducing the circulation growth rate (Wakelin & Riley Reference Wakelin and Riley1997). This may explain why the circulation growth rate in the case with $(L/D)_d=0.125$ becomes negative near and after the termination of the starting jet, i.e. near B.

4.2. Effects on the propulsive performance of starting jet after deceleration

The propulsive performance of starting jets varies with different $(L/D)_i$ and $(L/D)_d$ during the deceleration stage, and this can be analysed through the generation of thrust. The development of total thrust $F_T(t)$ is shown in figure 19(a), as in Krueger & Gharib (Reference Krueger and Gharib2003), Gao et al. (Reference Gao, Wang, Yu and Chyu2020) and Zhang et al. (Reference Zhang, Wang and Wan2020), calculated based on the profiles of axial velocity $u(r,t)\vert _{x=0}$ and gauge pressure $P(r,t)\vert _{x=0}$ at the nozzle exit, i.e.

(4.2)\begin{equation} F_T(t)=\int_{r=0}^{r=D/2}2{\rm \pi} r[\rho u^2(r,t)\vert_{x=0}+P(r,t)\vert_{x=0}]\, {\rm d} r.\end{equation}

As shown on the right-hand side of figure 19, $F_T(t)$ is either positive or negative when the propeller receives thrust opposite to (required for propulsion) or in the same direction as the starting jet, respectively. Both the outward ($u(r,t)\vert _{x=0}>0$) and inward ($u(r,t)\vert _{x=0}<0$) flows at the nozzle exit contribute positively to $F_T(t)$ because, through control volume analysis, they both increase the momentum within the control volume containing the external fluid in the same direction as the starting jet. This contributes to the positive $F_T(t)$. There is also a definition of sign for the second term on the right-hand side of (4.2), which is determined by the gauge pressure $P(r,t)\vert _{x=0}$. When the pressure at the nozzle exit is greater than the ambient pressure, $P(r,t)\vert _{x=0}$ is positive and contributes positively to $F_T(t)$, and vice versa. Before the initiation of deceleration for $(L/D)_i=0.75$ (at A), $F_T(t)$ is the same for all cases. Subsequently, $F_T(t)$ in cases with $(L/D)_i=0.75$ exhibits a step reduction, while in cases with $(L/D)_i=2.25$, it continues to increase until B and then exhibits the step reduction. The continuous increase of $F_T(t)$ during the constant velocity stage has also been found in Zhang et al. (Reference Zhang, Wang and Wan2020) for an impulsive starting jet with $L/D = 6$ (figure 13b in Zhang et al. (Reference Zhang, Wang and Wan2020)). The influence of different $(L/D)_i$ on $F_T(t)$ is the same as that in Gao et al. (Reference Gao, Wang, Yu and Chyu2020) for the thrust contributed by pressure in the starting jets with different velocity programs. The thrust contributed by pressure is the second term on the right-hand side of (4.2), referred to as pressure thrust, $F_P(t)$. For velocity program transitioning later from the constant velocity stage to the deceleration stage, $F_P(t)$ can continue to increase for a longer time. Due to the downstream translation of the LVR, its effect of inducing low pressure near the nozzle exit is weakened, resulting in an increase in $F_P(t)$ (Gao et al. Reference Gao, Wang, Yu and Chyu2020). This is responsible for the continuous increase of $F_T(t)$ during the constant velocity stage. For smaller $(L/D)_i$, the earlier deceleration induces negative gauge pressure, resulting in a negative $F_P(t)$. Consequently, $F_T(t)$ with $(L/D)_i=0.75$ first exhibits a step reduction.

Figure 19.

(a) The development of total thrust $F_T(t)$ generated by starting jets and (b) the average value of total thrust $\bar {F}_T$ during the deceleration stage for Cases 1–10 with different $(L/D)_i$ and $(L/D)_d$, where A and B represent the initiation of deceleration for $(L/D)_i=0.75$ and $(L/D)_i=2.25$, respectively. The red dashed lines indicate the line of zero thrust.

For both $(L/D)_i=0.75$ and $2.25$, $F_T(t)$ starts to bifurcate at the initiation of deceleration due to different $(L/D)_d$. During the deceleration stage, which ends with a step rise, $F_T(t)$ decreases monotonically. The magnitudes of the step reduction at the initiation and the step rise at the termination both decrease with increasing $(L/D)_d$. Furthermore, $F_T(t)$ is consistently negative for cases with $(L/D)_d\leq 0.25$. In contrast, for cases with $(L/D)_d>0.25$, $F_T(t)$ transitions from positive to negative as the starting jet slows down, and it remains positive overall at $(L/D)_d=2$ due to the sufficiently small rate of deceleration. After the termination of the starting jet, the flow near the nozzle exit is dominated by SVR, and $F_T(t)$ continues to develop further. For cases with $(L/D)_d\leq 0.25$ and $(L/D)_i=0.75$, in which the LVR has a significant influence on the formation process of the SVR, $F_T(t)$ remains negative for a period of time. Otherwise, $F_T(t)$ is close to or slightly greater than 0 under the action of the SVR. The average value of $F_T(t)$ during the deceleration stage $\bar {F}_T$ is illustrated in figure 19(b). Deceleration in general is unfavourable for the generation of thrust. As the rate of deceleration increases, i.e. decreasing $(L/D)_d$, $\bar {F}_T$ decreases. Even, for the smaller $(L/D)_d=0.125$ and $0.25$, $\bar {F}_T$ is less than 0. As $(L/D)_i$ increases from $0.75$ to $2.25$, the generation of $F_T(t)$ is enhanced for all $(L/D)_d$, i.e. the larger $\bar {F}_T$. This increment decreases as $(L/D)_d$ increases. The total thrust $F_T(t)$ can be decomposed into the velocity thrust component $F_U(t)$ and the pressure thrust component $F_P(t)$ based on the two terms on the right-hand side of (4.2), i.e.

(4.3)$$\begin{gather} F_U(t)=\int_{r=0}^{r=D/2}2{\rm \pi} r\rho u^2(r,t)\vert_{x=0} \, {\rm d} r, \end{gather}$$

(4.4)$$\begin{gather}F_P(t)=\int_{r=0}^{r=D/2}2{\rm \pi} rP(r,t)\vert_{x=0}\, {\rm d} r. \end{gather}$$

The development of $F_U(t)$ and $F_P(t)$ are shown in figure 20 after the initiation of deceleration. During the deceleration stage, $F_U(t)$ decreases monotonically and is almost identical for different $(L/D)_i$ and $(L/D)_d$ at the initial stage, but the differences become apparent as the starting jet decelerates and SVR grows (figure 20a). Overall, $F_U(t)$ increases with increasing $(L/D)_i$ from 0.75 to 2.25 and the decrease of $(L/D)_d$, which corresponds to the weakening of the influence from the LVR and the enhancement of the SVR, respectively. After the termination of the starting jet, due to the induced flow from the SVR at the nozzle exit, $F_U(t)$ always remains positive with a non-negligible magnitude. Comparing with the results calculated based on the slug model (indicated by the grey dashed line in figure 20a), the disturbance to $u(r,t)\vert _{x=0}$ by SVR caused an increase by at least $10\,\%$ in the average value of $F_U(t)$ during the deceleration stage. The slug model assumes that the velocity at nozzle exit is uniform and exactly equal to the velocity of starting jet (Didden Reference Didden1979; Gharib et al. Reference Gharib, Rambod and Shariff1998; Zhu et al. Reference Zhu, Zhang, Gao and Yu2022).

Figure 20.

The development of (a) velocity thrust $F_U(t)$ and (b) pressure thrust $F_P(t)$ generated by the starting jet after the initiation of deceleration for Cases 1–10 with different $(L/D)_i$ and $(L/D)_d$. Stages I and II correspond to the deceleration stage ($0< t_d^*\leq 1$) and the termination stage ($t_d^*>1$) of starting jet, respectively.

In comparison, the variation of $(L/D)_i$ and $(L/D)_d$ has a greater influence on $F_P(t)$ than on $F_U(t)$. Moreover, $F_P(t)$ shows significant variations with respect to $(L/D)_d$ during $0< t_d^*\leq 1$ (figure 20b). The gauge pressure at the nozzle exit is negative due to the deceleration of starting jet and further diminishes with increasing the rate of deceleration (Gao et al. Reference Gao, Wang, Yu and Chyu2020). Therefore, $F_P(t)$ diminishes as $(L/D)_d$ decreases with a magnitude much greater than that in $F_U(t)$. This is responsible for the decrease in $\bar {F}_T$ as $(L/D)_d$ decreases in figure 19(b). As $(L/D)_i$ increases from 0.75 to 2.25, $F_P(t)$ shows an overall increase for small $(L/D)_d$, while it only increases in the early stage and then remains the same for large $(L/D)_d$. This also explains the observed increase in $\bar {F}_T$ as $(L/D)_i$ increases, but the magnitude of increment decreases with increasing $(L/D)_d$ (recall figure 19b). At the instant of the termination of starting jet, there is a step rise in $F_P(t)$ for all cases due to no longer decelerating and generating negative gauge pressure. After the termination of the starting jet, $F_P(t)$ remains negative for a longer duration for $(L/D)_d\leq 0.25$ with $(L/D)_i=0.75$ (stage II in figure 20b), which results in negative $F_T(t)$.

Re-examining $F_P(t)$ during the deceleration stage, it is negative since $P(r,t)\vert _{x=0}$ is dominated by the deceleration of the starting jet in each case. The negative gauge pressure at the nozzle exit should be related to the rate of deceleration (Zhu et al. Reference Zhu, Zhang, Gao and Yu2023a). However, for the constant deceleration rate in the present work, $F_P(t)$ still increases at first ($0< t_d^*<0.5$) and then decreases ($0.5< t_d^*<1$) during the deceleration stage. This can be attributed to the disturbance caused by the formation of the SVR. In general, SVR is beneficial to the generation of $F_P(t)$ during the deceleration stage. To quantitatively calculate the influence of the SVR on $F_P(t)$, it can be assumed as a reference that there is no process in which $F_P(t)$ increases at first ($0< t_d^*<0.5$) and then decreases ($0.5< t_d^*<1$). The reference based on the above assumptions for calculating the influence of the SVR on $F_P(t)$ is inserted as a grey dashed line in figure 20(b). For simplicity, the reference is given only for Case 1 with $(L/D)_i=0.75$ and $(L/D)_d=0.125$. By designating the constant pressure thrust, assuming without the influence of the SVR, as $F_{P\_ref}$, the variation in the average value of $F_P(t)$ during the deceleration stage caused by the formation of the SVR can be calculated as follows:

(4.5)\begin{equation} \Delta \bar{F}_P=\frac{\displaystyle \int_{T_a+T_c}^{T_a+T_c+T_d}[F_P(t) -F_{P\_ref}]\, {\rm d} t}{T_d}, \end{equation}

where $F_{P\_ref}$ can be approximately selected as $F_P(t)$ at the initial instant when the starting jet has entered the deceleration stage but the influence of the SVR can still be ignored. When $\Delta \bar {F}_P$ is positive, it means that SVR has a positive influence on the generation of $F_P(t)$. After calculation, $\Delta \bar {F}_P$ varies from $0.02\rho AU_{max}^2$ to $0.13\rho AU_{max}^2$ for starting jets with different $(L/D)_i$ and $(L/D)_d$. Furthermore, the deceleration of the starting jet reduces the average value of $F_P(t)$ during the deceleration stage by the magnitude of $F_{P\_ref}$. However, the formation of the SVR could reduce the magnitude of the negative $F_P(t)$, expressed as a ratio as follows:

(4.6)\begin{equation} \eta=\frac{\Delta \bar{F}_P}{\vert F_{P\_ref} \vert}. \end{equation}

After calculation, $\eta$ varies from $1\,\%$ to $60\,\%$ for starting jets with different $(L/D)_i$ and $(L/D)_d$. Therefore, it can be considered that the formation of the SVR compensates for approximately $1\,\%$ to $60\,\%$ of the reduction in $F_P(t)$ caused by the deceleration.

The instantaneous axial velocity profiles $u(r,t)\vert _{x=0}$ and gauge pressure profiles $P(r,t)\vert _{x=0}$ at the nozzle exit are shown in figures 21 and 22 to discuss the relationship between the propulsive performance and the influence from the vortical structures. The existence of the SVR is beneficial for the generation of $F_U(t)$. It can be seen from figure 21(a) that the induced flow from the SVR directly changes $u(r,t)\vert _{x=0}$. On one hand, the induced flow transports the external fluid into the nozzle, corresponding to the region of $u(r,t)\vert _{x=0}<0$. By control volume analysis, this increases the momentum within the control volume containing the external fluid in the same direction as the starting jet, contributing positive $F_U(t)$. On the other hand, due to the fixed volume inside the nozzle, this portion of inward fluid needs to be discharged by the region of $u(r,t)\vert _{x=0}>0$ into the control volume containing the external fluid. This is equivalent to increasing the mass flux discharged by the starting jet and also contributes positively to $F_U(t)$. This benefit becomes more pronounced as $(L/D)_i$ increases from 0.75 to 2.25, with the higher velocity magnitude for both the negative and positive $u(r,t)\vert _{x=0}$. For smaller $(L/D)_i=0.75$, the induced velocity from the LVR, which has not yet moved away from the nozzle exit, carries SVR downstream (recall figure 8b i). The SVR is then offset from the nozzle exit, diminishing the influence of the induced flow from it. As the starting jet approaches termination or completely terminates, the flow near the nozzle exit is dominated by the induced flow from the SVR. The SVR transports external fluid back into the nozzle (Gao et al. Reference Gao, Wang, Yu and Chyu2020). An equal amount of fluid should be discharged due to continuity, as observed at $t_d^*=1.1$ in figure 21(b). Both the flow entering and discharging through the nozzle produce positive $F_U(t)$ by increasing the momentum within the control volume containing the external fluid in the same direction as the starting jet. This can explain why $F_U(t)$ remains positive with a non-negligible magnitude after the termination of starting jet (recall figure 20a).

Figure 21.

The instantaneous profiles of axial velocity at the nozzle exit $u(r,t)\vert _{x=0}$ for (a) $t_d^*=0.5$ and (b) $t_d^*=1.1$ in Cases 1–10 with different $(L/D)_i$ and $(L/D)_d$.

Figure 22.

The development of gauge pressure profiles at the nozzle exit $P(r,t)\vert _{x=0}$ for (a) Cases 1 and 6 at $(L/D)_d=0.125$ and (b) Cases 4 and 9 at $(L/D)_d=1$ but with different $(L/D)_i$.

Shifting our attention to the development of gauge pressure profiles at the nozzle exit $P(r,t)\vert _{x=0}$ that produces $F_P(t)$, as shown in figure 22 for five critical instants in cases with different $(L/D)_d$ and $(L/D)_i$. The cases with different $(L/D)_i$ at small $(L/D)_d$ are compared in figure 22(a). When the starting jet has not yet begun to decelerate at $t_d^*=0$, it can be observed that for $(L/D)_i=2.25$, $P(r,t)\vert _{x=0}$ is entirely 0, while it is below 0 for $(L/D)_i=0.75$. This is because, for $(L/D)_i=0.75$, the LVR is not far away from the nozzle exit and induces low pressure there. Therefore, $F_P(t)$ produced at $(L/D)_i=2.25$ is always larger than that at $(L/D)_i=0.75$ during the deceleration stage at small $(L/D)_d$ (recall figure 20b). As the deceleration of the starting jet occurs, $P(r,t)\vert _{x=0}$ decreases rapidly from close to 0 to an overall negative value with a large magnitude, as shown by the $P(r,t)\vert _{x=0}$ at $t_d^*=0.04$. This corresponds to the step reduction of $F_P(t)$ immediately after $t_d^*=0$ (recall figure 20b). As $P(r,t)\vert _{x=0}$ develops further, from $t_d^*=0.04$ to $t_d^*=0.2$, it shows an overall increase. This can be associated with the formation of the SVR, which causes the starting jet to converge at the nozzle exit. The reduction in the outflow area of starting jet leads to its acceleration, requiring an additional pressure difference to drive the process (Krieg & Mohseni Reference Krieg and Mohseni2013; Zhu et al. Reference Zhu, Zhang, Gao and Yu2023a, Reference Zhu, Zhang, Xia, Yu and Gao2024). The gauge pressure at the nozzle exit thus increases. As a result, $F_P(t)$ generated increases during $t_d^*<0.5$. However, SVR also induces a low pressure region as a vortical structure (Schlueter-Kuck & Dabiri Reference Schlueter-Kuck and Dabiri2016; Gao et al. Reference Gao, Wang, Yu and Chyu2020) that expands as it develops, shown by the region near the nozzle wall ($r=0.5D$) of $P(r,t)\vert _{x=0}$ at $t_d^*=0.8$. Therefore, $F_P(t)$ decreases during $0.5< t_d^*<1$ because the low pressure induced by the SVR offsets the pressure increase caused by the reduction in outflow area and dominates. In general, the existence of the SVR is beneficial for the generation of $F_P(t)$ during the deceleration stage, although only for a short period of improvement. After the termination of the starting jet at $t_d^*=1.2$, the negative gauge pressure caused by deceleration disappears, and $P(r,t)\vert _{x=0}$ is only dominated by SVR. The induced velocity from the SVR tends to transport fluid back into the nozzle, but additional fluid cannot be added due to the fixed volume inside the nozzle. This results in a positive gauge pressure in the centre part of nozzle exit to repel fluid entry. For $(L/D)_d=0.125$ with $(L/D)_i=0.75$, the LVR moves SVR downstream (recall figure 8b i), weakening this process, and the pressure in the central part cannot be effectively increased. Furthermore, the LVR, which has not yet moved far away, together with SVR, induces low pressure region near the nozzle exit. Therefore, $F_P(t)$ is negative for a longer duration but continues to increase as the LVR moves away during $t_d^*>1$. As $(L/D)_d$ increases from 0.125 to 1 shown in figure 22(b), the above discussion about $t_d^*=0$, $t_d^*=0.04$ and $t_d^*=0.2$ remains unchanged, only the portion for $t_d^*>0.5$ is altered. For larger $(L/D)_d=1$, since the LVR at small $(L/D)_i$ has also moved far away from the nozzle exit after a longer duration of deceleration, the differences between $(L/D)_i=0.75$ and $(L/D)_i=2.25$ gradually disappear as $t_d^*$ approaches and exceeds 1. In addition, as $(L/D)_d$ increases, indicating a longer duration of the deceleration stage, the magnitude of negative gauge pressure at the nozzle exit caused by the deceleration of starting jet also decreases.