3.1. Vapour fractions fields

The time-averaged planar vapour fraction field ($\overline {\beta (x, y)}$), along with the radial profiles of $\overline {\beta (x,y)}$ for the lowest cavitation number ($\sigma = 0.47$), are shown in figures 2(a) and 2(b), respectively. It can be seen that $\overline {\beta (x, y)}$ achieves a maximum value 0.43 near the wall (close to the throat), while the vapour fraction is 0.05 in the core of the venturi, a value comparable to the measurement uncertainty. The time-averaged vapour fraction gradually reduces away from the throat as the vapour cloud is convected downstream. This is consistent with the previous observations of Jahangir et al. (Reference Jahangir, Wagner, Mudde and Poelma2019), who measured time-averaged vapour fractions at $\sigma = 0.40$. Further, the instantaneous vapour fraction in a cycle (or a phase) can be as high as 0.68 at the same flow conditions; see figure 3. In the same figure, it can be seen that the maximum vapour fraction in a cycle is significantly higher than the maximum time-averaged vapour fraction. This is attributed to the temporal averaging of vapour fractions due to the periodic growth and collapse of the cavity. Further, the maximum vapour fraction decreases with increasing $\sigma$. This shows the necessity of time-resolved vapour fraction fields in such flows.

Figure 2. (a) Time-averaged back-projected planar vapour fraction field ($\bar {\beta }$). The white dotted outline indicates the vapour–liquid interface ($\bar {\beta }=0.1$), while the black solid lines indicate the venturi wall. (b) Profiles of $\bar {\beta }$ at different axial locations ($x_{a}$ to $x_{f}$: 0.12, 0.55, 1.1, 1.7, 2.2, 2.8 times $D_{t}$) in the venturi for $\sigma = 0.47$, where $R(x)$ is the local venturi radius.

Figure 3. The phase-averaged maximum vapour fraction in a shedding cycle ($\langle \beta \rangle _{max}$) and the maximum time-averaged vapour fraction ($\bar{\beta }_{max}$), at different cavitation numbers ($\sigma$).

3.2. Vapour cavity topology

The phase-averaged planar vapour fractions $\langle \beta (x, y, t/T) \rangle$ also allow us to quantify the shape and size of the attached vapour cavity in a venturi. The Froude number ($Fr_{L_{c}} = U_{t}/\sqrt {gL_{c}}$) defined on the cavity length is $\gtrsim$20. Hence the effect of gravity on the cavity shape can be expected to be negligible. Further, the Weber number ($We_{L_{c}} = \rho U_{t}^{2}L_{c}/\gamma$) is ${O}(10^{4})$, so the effects of inertia dominate the surface tension ($\gamma$) effects. The isocontour of $\beta = 0.1$ is used to demarcate the boundary of the cavity; see the inset in figure 4(a). Since the attached vapour cavity grows in a shedding cycle, the maximum cavity length ($L_{c}$) and maximum cavity height ($h_{c}$) are used to describe the topology of the attached cavity. The maximum cavity length for different $\sigma$ is shown in figure 4(a). It can be seen that $L_{c}/D_{t}$ measured by X-ray densitometry is systematically shorter than the results from shadowgraphy (Jahangir et al. Reference Jahangir, Hogendoorn and Poelma2018), with a deviation up to 50 % at $\sigma = 0.47$. This observation is not sensitive to our chosen threshold $\beta = 0.1$ to demarcate the cavity: $L_{c}$ estimated with threshold $\beta = 0.05$ is about 4 pixels (0.02$D_{t}$) higher than with threshold $\beta = 0.1$. Also, $L_{c}$ remains unaltered with an increase of threshold $\beta$ from 0.1 to 0.15. The observed deviation could be attributed to the difficulty in defining the vapour content (vapour fractions) and thus cavity in shadowgraphs (Dash, Jahangir & Poelma Reference Dash, Jahangir and Poelma2018). The vapour bubbles detached from the cavity may appear to be a part of it in shadowgraphs, overestimating cavity lengths. The maximum height of the attached cavity ($h_{c}$) is also quantified (see figure 4b). It is observed that $h_{c}$ increases monotonically with the cavity length. Further, cavity height scales directly with the cavity length, irrespective of the cavitation number.

Figure 4. (a) Maximum cavity length ($L_{c}$) normalised with the throat diameter ($D_{t}$) as a function of the cavitation number ($\sigma$). Square markers show shadowgraphy measurements of Jahangir et al. (Reference Jahangir, Hogendoorn and Poelma2018), while circles show current X-ray densitometry measurements. The white dotted outline in the inset shows the isocontour of $\beta = 0.1$, demarcating the cavity interface. (b) Maximum cavity thickness ($h_{c}$) normalised with $D_{t}$ for different cavity lengths and $\sigma$.

3.3. Vapour shedding dynamics

The time-synchronised X-ray imaging with dynamic pressure ($dp$) measurements allows us to visualise the shedding dynamics and ascertain the role of a pressure wave in the shedding cycle at different cavitation numbers ($\sigma$). The instantaneous projected vapour fractions ($\alpha$) are used to visualise the cavity dynamics. The colour indicates the vapour fraction: blue for a completely liquid phase, and yellow for a higher vapour content. Further, the cavity front evolution in a shedding cycle is studied with a space–time ($x\unicode{x2013}t$) plot of $\alpha$. The fixed axial locations of the pressure transducers make $dp$ measurement at the exact location of each cloud collapse cumbersome. Thus the $dp$ measurement closest to each cloud collapse location is used to measure the cloud implosion pressure peak. This peak arrives at the transducer with a minor delay, as discussed in the next paragraph. A single representative shedding cycle will be shown for $\sigma = 0.47$, 0.78 and 0.84 in the following discussion.

The cavity dynamics at $\sigma = 0.47$ is illustrated in figure 5. The cavity is seen to grow from figure 5(a) to figure 5(c). As the cavity grows, there is a continuous re-entrant flow beneath the cavity at the closure region, as shown in figures 5(b) and 5(c). This is inferred from a fairly low vapour fraction close to the venturi wall ($\alpha \sim 0.2$) in comparison to the vapour cavity ($\alpha \sim 0.7$). The presence of a re-entrant jet for the considered range of $\sigma$ was also observed directly by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022) with near-wall velocimetry in the same flow geometry. At the instance marked by $d$ in figure 5(j), the imploding cavitation cloud from the previous shedding cycle gives rise to a pressure wave, evident from a strong pressure peak in $dp$ at $p_{6}$. The cloud collapse appears to have been coherent, as is evident from a single sharp peak of 340 kPa for a very short duration. The cloud collapse itself could not be visualised in the X-ray images at this $\sigma$, as the collapse is expected to occur at $x \sim 9D_{t}$, as observed in previous high-speed shadowgraphy (Jahangir et al. Reference Jahangir, Hogendoorn and Poelma2018). Note that the structure labelled ‘Shed cloud’ in figure 5(e) will result in a pressure peak in the next cycle. The pressure wave emanating from the collapse propagates through the low vapour fraction liquid: the high-frequency pressure transducers at $p_{5}$ and $p_{6}$ pick up the pressure wave propagating at velocity $u_{pw}\sim 450\ {\rm m\ s}^{-1}$. This is estimated using the time delay between the peaks and the distance between the transducers. There is a small time delay between the cloud implosion and the pressure wave reaching the pressure transducer ($\sim$0.04 ms, estimated using $u_{pw}$ and transducer locations). Additionally, there is a time delay ($\sim$0.24 ms) between the cloud collapse and the impingement (see next paragraph) on the cavity due to the finite distance between the cloud implosion and the impingement point. However, these time delays are negligible in comparison to the cavity shedding time period $T=17.4$ ms, due to the much higher speed of the shock front in the low vapour fraction region compared to its value in the attached vapour cavity. For all intents and purposes, the $x\unicode{x2013}t$ diagram and pressure signals can thus be considered to be synchronous, with no time delay between cloud collapse and impingement.

Figure 5. (a–h) A single shedding cycle at $\sigma =0.47$. (i) The $x\unicode{x2013}t$ evolution of the vapour fraction for this shedding cycle. (j) The time trace of the dynamic pressure recorded simultaneously at $p_{6}$. The bulk flow is from left to right. The colour indicates the $z$-projected vapour fractions.

The upstream travelling pressure wave impinges on the attached cavity and arrests the cavity growth. Here, impingement is defined as the first point of interaction between the pressure wave and the attached cavity. The impingement leads to a maximum cavity length, referred to as $L_{c}$. The ‘impingement point’ can be seen in the $x\unicode{x2013}t$ evolution of the cavity front in figure 5(i), where the growing cavity front is shown by a black dashed line, while the retracting cavity front is shown by a red dashed curve. Due to this impingement, a large part of the cavity gets shed (see figure 5e) and is then convected downstream with the bulk flow, where it will implode. After impinging, the upstream-travelling pressure wave immediately starts condensing the attached cavity. The condensation shock front is seen to travel upstream through the cavity, destabilising the attached cavity. This process is more clear in supplementary movie 1 available at https://doi.org/10.1017/jfm.2024.435. The shock front is also evident in the $x\unicode{x2013}t$ diagram as shown in figure 5(i): it is shown by a red dashed curve and is characterised by a sharp decrease in vapour fraction across it. The inverse of the slope of the condensation shock front in the $x\unicode{x2013}t$ plot is the shock front velocity.

It can be observed that the shock front accelerates significantly as it approaches the throat, as is evident from the changing slope of the shock front. This will be addressed in § 3.7. Further, some amount of vapour remains after the shock front has passed over the cavity and condensed it. This leads to low vapour fraction streaks in the $x\unicode{x2013}t$ plot and is also indicated by ‘Secondary shed cloud’ in figure 5(g). This is suspected to be a result of the interaction of the bubbly shock wave with the attached cavity, which is essentially composed of coalesced vapour bubbles. The condensation shock front travels upstream and reaches the throat, resulting in complete cavity fragmentation and destabilisation. This is in line with the previous observations of Jahangir et al. (Reference Jahangir, Hogendoorn and Poelma2018), Ganesh et al. (Reference Ganesh, Makiharju and Ceccio2016) and Budich et al. (Reference Budich, Schmidt and Adams2018) at low $\sigma$. This is followed by a brief period where the venturi throat is left ‘cavitation-less’. Meanwhile, the shed cloud is convecting downstream. This completes a shedding cycle until a new cavity starts growing at the throat as described above.

It is worthwhile to note that in a few (three in forty) shedding cycles at $\sigma = 0.47$, multiple bubbly shock waves are seen to travel through the growing cavity (see figure 6), exhibiting multi-step cavity shedding modes. Interestingly, the implosion of the secondary shed cloud (from the previous cycle, shown in figure 6a) produces a weaker pressure wave ($dp \sim 30$ kPa, ‘Collapse-1’ in figure 6k). This gives rise to an upstream-travelling bubbly shock wave as indicated by ‘CSF-1’ (see the yellow arrow in figure 6). This is inferred from collapse-1 coinciding in time with the emergence of CSF-1 in figure 6(j). However, the rate of vaporisation at the throat is expected to be high, dictated by the pressure at the throat. Thus the cavity continues to grow to assume its maximum length ($L_{c}$). During this time, the bubbly shock wave (CSF-1) appears frozen for a while before it moves upstream very slowly, by the yellow arrow in figures 6(c)–6(e). The rate of vapour production near the throat is likely sufficiently high to oppose the shock front propagation. This is followed by a stronger cloud implosion ($dp \sim 260$ kPa, ‘Collapse-2’) producing a ‘CSF-2’ (see the red arrow in figure 6) that ultimately destabilises the cavity similar to the shedding cycle shown in figure 5. Furthermore, it is clear from the $x\unicode{x2013}t$ plot (figure 6j) that the initial velocity of ‘CSF-1’ ($-2.2\ {\rm m\ s}^{-1}$) is substantially lower than that of ‘CSF-2’ ($-6.4\ {\rm m\ s}^{-1}$). Consequently, an earlier smaller cloud implosion does not have any consequence for the global shedding dynamics. Instead, the larger cloud implosion dictates the cloud shedding frequency.

Figure 6. (a–i) A shedding cycle at $\sigma =0.47$, showing multiple condensation shock fronts (CSF) in one cycle. (j) The $x\unicode{x2013}t$ evolution of the cavity front. (k) The $dp$ time trace recorded simultaneously at station $p_{6}$. The arrows show propagating condensation shock fronts: CSF-1 is marked by a yellow arrow with a red outline, while CSF-2 is marked by a full red arrow.

Similarly, in a limited number of shedding cycles (two in forty), it is observed that the cavity undergoes asymmetric cavity detachment. It is seen that the upper (figure 7a) and lower (figure 7b) parts of the cavity show different growth and shedding behaviour. It is seen that the cavity starts condensing due to CSF-1 emanating from collapse-1 (see ‘Impingement-1’ in figure 7a and ‘Collapse-1’ in figure 7c). The upper part of the cavity gets completely condensed, while the lower part is partially condensed. To this end, the vapour production rate is high enough for the cavity to keep growing near the throat. However, the pressure wave due to the implosion of the cloud from the previous cycle (‘Collapse 2’ in figure 7c) is already travelling upstream. The propagating shock front (CSF-2) impinges on the cavity at ‘Impingement-2’ and condenses the vapour it comes across (see figures 7a,b). Further, it encounters a growing cavity front leading to ‘Impingement-2.2’ as shown in figure 7(a). Consequently, the propagating velocity of the shock front reduces significantly from $-6.7\ {\rm m\ s}^{-1}$ to $-3.1\ {\rm m\ s}^{-1}$. This leads to the condensation shock fronts experiencing two distinct velocities, as evident from the two different slopes of CSF-2, shown in figure 7(a). This suggests that the condensation shock front, after Impingement-2.2 (see figure 7a) has to overcome the strong vaporisation (cavitation) brought about by the low pressure at the throat. The above anomalous shedding cycles may cause cycle-to-cycle variations; however, the condensation shock front due to the larger pressure pulse is seen to enforce a strong periodicity in the cavity shedding dynamics. This was confirmed by a matching sharp peak in the fast Fourier transform (FFT) of pressure and vapour fraction time series at such $\sigma$ (see figure 19 in Appendix A).

Figure 7. A shedding cycle at $\sigma =0.47$, showing multiple condensation shock fronts (CSF). (a) The upper part of the vapour cavity. (b) The lower part of the cavity. (c) The $dp$ time trace is recorded simultaneously at station $p_{6}$.

As the cavitation number increases ($\sigma = 0.78$), we observe slightly different cavitation dynamics with respect to the bubbly shock front. The vapour cavity grows linearly as shown in figures 8(a)–8(c). The cavitation cloud shed from the previous cycle, shown in figure 8(b), will implode in the downstream region. However, the cloud collapse appears less coherent, as is evident from multiple small peaks around the main peak, giving rise to a low-pressure rise ($dp \sim 120$ kPa) recorded at the $p_{5}$ location (see figure 8j). The cloud implosion gives rise to the upstream travelling pressure wave that impinges on the growing attached cavity, arresting the cavity growth. The impingement is evident from the change in the slope of the cavity front at the impingement point, as shown by the two black dashed lines in figure 8(i). The ‘bubbly shock wave’ is seen to propagate through the cavity without significant condensation, as shown by snapshots in figures 8(d,e) and a white dotted line in the $x\unicode{x2013}t$ plot (see figure 8i). The pressure wave can propagate through the cavity if the shock front is subsonic (Brandao et al. Reference Brandao, Bhatt and Mahesh2019). As it travels further upstream, it begins to condense the cavity (see again the red arrow in figures 8f,g). See also supplementary movie 2. The shock front can be seen more clearly by the emergence of a sharp change in the vapour fraction across the travelling front, as shown by the red dashed curve in figure 8(i). This coincides with the observed acceleration of the shock front as it approaches the venturi throat, inferred from the changing slope of this red dashed curve. The condensation front then travels upstream until it reaches the throat, resulting in complete cavity condensation and eventual shedding.

Figure 8. (a–h) A single shedding cycle for the case $\sigma =0.78$. (i) The $x\unicode{x2013}t$ evolution of the vapour fraction of the cavity. (j) The dynamic pressure corresponding to this cycle recorded simultaneously at station $p_{5}$.

At higher cavitation numbers ($\sigma = 0.84$), two different types of shedding cycles are seen. In a few cycles, a pressure wave transformed into a condensation shock front is seen to cause cavity destabilisation, similar to $\sigma = 0.78$ (see figure 9). In other cycles, the cavity dynamics differ, as shown in figure 10. The vapour cavity grows linearly in time (figures 10a–d). At the instance marked $e$ in figure 10(j), the cloud implosion from the previous cycle gives rise to an incoherent and lower pressure rise ($dp \sim 60$ kPa). The dynamic pressure time trace is recorded at location $p_{4}$. The upstream travelling pressure wave cannot be seen impinging on the growing cavity in the $x\unicode{x2013}t$ diagram in figure 10(i). Moreover, the slope change of the cavity front growth is smoother and not abrupt as in the previously considered cases $\sigma = 0.47$ and $0.78$. This is shown by a black dashed curve in figure 10(i). There is no sharp upstream travelling vapour fraction discontinuity in the $x\unicode{x2013}t$ evolution of the cavity following the cloud implosion, as is clear from figure 10(i). It can be hypothesised that the pressure wave propagates upstream through the attached cavity without condensing it. Thus there is no significant effect of the cloud implosion on cavity growth and detachment. Instead, we observe that the cavity gets pinched off near the venturi throat (see figure 10h) and rolls up to form a convecting cloud, similar to shedding brought about by the re-entrant jet travelling beneath the cavity. This can be seen in supplementary movie 3. With further increase in $\sigma$ ($=0.90$), most shedding cycles are seen to be destabilised by the re-entrant jet. It was also shown previously by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022) that there is a strong upstream-travelling re-entrant jet below the cavity in the same flow geometry at these higher $\sigma$.

Figure 9. (a–h) A single shedding cycle for the case $\sigma =0.84$. (i) The $x\unicode{x2013}t$ evolution of the vapour fraction of the cavity. (j) The dynamic pressure corresponding to this cycle recorded simultaneously at station $p_{5}$.

Figure 10. (a–h) A single shedding cycle for the case $\sigma =0.84$. The layout is the same as in figure 9. The dynamic pressure is recorded at station $p_{4}$.

It is important to stress that these two types of cavity shedding mechanisms are fundamentally different. At low $\sigma$, the cavity gets condensed and fragmented by the condensation front before the new cavity starts growing. However, at high $\sigma$, a strong re-entrant jet below the vapour cavity pinches the cavity before the new cavity starts growing. It is confirmed in our study that the likelihood of the re-entrant jet causing cavity shedding increases with the increase in $\sigma$, as also shown recently by Bhatt, Ganesh & Ceccio (Reference Bhatt, Ganesh and Ceccio2023). Pressure waves due to cloud implosions are present at all $\sigma$ (Gawandalkar & Poelma Reference Gawandalkar and Poelma2022). These observations suggest that both pressure waves emanating from the cloud implosion and re-entrant jets are present to destabilise the attached cavity for a given $\sigma$. However, the way in which the pressure wave interacts with the attached cavity dictates the dominant shedding mechanism. At low $\sigma$ ($\lesssim$0.78), the pressure wave gets transformed into a condensation shock front in nearly every shedding cycle, condensing and fragmenting an attached cavity. However, at $\sigma \sim 0.84$, the pressure wave cannot transform into a condensation shock in all the shedding cycles. In these cycles, the attached cavity is destabilised by a re-entrant jet. With an even further increase in $\sigma$ ($\sim$0.97), the attached cavity is destabilised solely by the re-entrant jet, as shown previously by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). The correlation between the pressure peaks due to cloud collapse and cavity detachment is further established by computing the dominant frequency of the $\alpha$ and $dp$ time signals for a range of $\sigma$, as discussed in Appendix A. Further, the time scale of the condensation shock front is in agreement with the global shedding frequency at low to moderate $\sigma$, as presented in Appendix B.

3.4. Characteristics of bubbly shock waves

In order to understand the above-mentioned shedding behaviour(s), it is imperative to understand the interaction of the pressure waves with the attached vapour cavity. Thus we characterise the bubbly shock waves travelling through the cavity. In particular, we use the vapour fraction pre-shock front ($\alpha _{1}$), the post-shock front ($\alpha _{2}$), the pressure rise across the shock front ($\Delta p$), the velocity of the condensation shock front ($u_{sw}$), and the Mach number ($Ma_{sw}$) to characterise the bubbly shock waves. Further, all the quantities are phase-averaged using the approach detailed in Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). The standard deviation of the phase average is expressed as error bars. The error bars can be attributed mainly to the cycle-to-cycle variation in shedding dynamics along with random noise in the measurement.

The pressure wave emanates from cloud collapse/implosion in the downstream region of the venturi. This is inferred from the sharp peak in the dynamic pressure ($dp$) signal during the shedding process (see figures 5j, 8j, 10j). The pressure peak brought about by the cloud implosion is recorded by the transducer closest to the collapse location ($p_{4}$, $p_{5}$ or $p_{6}$; see figure 1(c) for a schematic). It can be seen that the phase-averaged pressure peak ($\overline {dp}$) decreases significantly with increasing $\sigma$, as shown in figure 11(a). The pressure wave propagating through the low vapour fraction region is detected as a pressure discontinuity by transducers at the $p_{4}$ to $p_{6}$ locations. The availability of multiple transducers allows us to track this pressure discontinuity in time. The time delay between the peaks and the distance between the transducers are used to estimate its velocity of propagation ($\overline {u_{pw}}$). The pressure wave is seen to propagate at a velocity that is an order of magnitude higher than the bulk velocity ($U_{t}$), as shown in figure 11(b). This pressure wave, after impinging on the cavity, interacts with it, i.e. condenses it. The pressure rise ($\Delta p$) associated with this bubbly shock wave is recorded using multiple pressure transducers, located at $p_{1}$, $p_{2}$ and $p_{3}$. The magnitude of the pressure rise is measured as the shock front propagates through the attached vapour cavity.

Figure 11. (a) The phase-averaged pressure rise due to the cloud implosion recorded in the pressure recovery section ($p_{4}$, $p_{5}$ or $p_{6}$, whichever is closest to the location of the cloud implosion) for a range of $\sigma$. (b) The phase-averaged velocity of the pressure wave travelling through the liquid measured using dynamic pressure transducers at $p_{4}$ to $p_{6}$.

An example of the pressure wave dynamics and its interaction with the growing cavity is shown for $\sigma = 0.67$ in figure 12. The collapsing cloud is shown in figures 12(a) and 12(b). Note that the collapse takes place to the right-hand side of the $p_{3}$ location, which in this case falls within our FOV. The pressure wave emanating from this implosion propagates upstream through the liquid and is first picked up by the pressure transducer at $p_{3}$; see the sharp peak (marked as $b$) in the red $dp$ signal in figure 12(g). Further, this discontinuity is seen to propagate through the vapour cavity as a condensation shock front (marked by red arrows), as identified near $p_2$ in figure 12(d) and $p_1$ in figure 12(e). This coincides with the pressure rise at $p_{2}$ (marked $d$ in the green time signal) and $p_{1}$ (marked $e$ in the blue time signal), as shown in figure 12(g). It is important to note that the condensation shock front appears somewhat obscure in these high-speed images, necessitating vapour fraction measurements as in this study. The pressure rise ($\Delta p$) at $p_{1}$ appears to be higher than the pressure rise at $p_{2}$. This will be addressed in § 3.7. It is observed that the pressure discontinuity ($\Delta p$) across the condensation front is weaker, i.e. the pressure rise is almost one order of magnitude lower than the pressure peak ($dp$) due to the cloud implosion. The phase-averaged $\Delta p$ across the condensation front decreases with increasing $\sigma$, as shown in figure 13(a). Thus the pressure rise caused by the condensation shock front at higher $\sigma$ is lower, indicating that the strength of the condensation shock front is decreasing with increasing $\sigma$. This then coincides with the condensation shock front becoming a less dominant shedding mechanism, as illustrated by the observed shedding dynamics.

Figure 12. (a–f) High-speed shadowgraphs showing the propagation of the condensation shock front through the vapour cavity at $\sigma =0.67$ (red arrows show the bubbly shock wave signature). (g) The corresponding dynamic pressure time series at three axial locations ($p_{1}, p_{2}, p_{3}$) exhibit the pressure rise brought by it.

Figure 13. Characteristics of the bubbly shock waves. (a) The phase-averaged pressure rise recorded across the condensation shock front travelling through the attached vapour cavity ($p_{1}$ to $p_{3}$). The red markers show the pressure rise predicted using the measured shock front velocity and vapour fractions using (3.1).(b) Phase-averaged vapour fraction, pre ($\alpha _{1}$) and post ($\alpha _{2}$) shock front. (c) Phase-averaged condensation shock front velocity ($u_{sw}$) normalised with the throat velocity ($U_{t}$). Blue markers indicate the velocity predicted with the Rankine–Hugoniot (R–H) equation ($u_{rh}$), red markers indicate the velocity computed by fitting a straight line to the condensation shock front in the $x\unicode{x2013}t$ evolution of $\alpha$, star markers indicate the shock front velocity obtained by $L_{c}/t_{ret}$, and the diamond marker indicates the velocity of the shock front measured directly using PIV by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022).

The vapour fraction is quantified using time-resolved X-ray densitometry, as mentioned with an uncertainty of less than 5 %. The vapour fractions on either side of the condensation shock front (red dashed curve in figures 5i, 8i, 9i) are expressed as $\alpha _{1}$ and $\alpha _{2}$ (see also inset in figure 13c). The vapour fraction ahead of the condensation shock front ($\alpha _{1}$) varies from 0.83 to 0.45, with a variation of $\sigma$ from 0.39 to 0.84. So $\alpha _{1}$ decreases monotonically with $\sigma$. However, the post-shock vapour fraction is almost constant at $\alpha _{2} \sim 0.1$, as shown in figure 13(b). Thus there is a strong gradient of vapour fraction across the shock front, suggesting rapid condensation of the vapour cavity by the propagating condensation shock front. Further, the measured vapour fraction values are in agreement with the vapour fractions reported by Ganesh et al. (Reference Ganesh, Makiharju and Ceccio2016). The velocity of the shock front in the laboratory frame of reference is estimated by computing the inverse of the gradient to the linear part of the shock front identified in the $x\unicode{x2013}t$ plot (see red markers in figure 13c). The linear fit to the shock front follows the previous studies of Ganesh et al. (Reference Ganesh, Makiharju and Ceccio2016) and Budich et al. (Reference Budich, Schmidt and Adams2018). Alternatively, the shock front velocity ($\sim L_{c}/t_{ret}$) is also determined using the length of the vapour cavity through which it travels ($L_{c}$) and the time it spends in the cavity ($t_{rec}$; see Appendix B). These are shown by ${star}$ markers in figure 13(c). They are consistently somewhat higher than the previous velocity estimates, suggesting that the condensation shock front velocity is not constant; rather, it accelerates as it approaches the throat. This is in agreement with direct PIV measurements by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). The acceleration will be addressed in detail in § 3.7. Despite a relatively large variation in $\alpha _{1}$, the shock front velocity at different $\sigma$ is almost constant when normalised by $U_{t}$, i.e. $\sim$0.55$U_{t}$. Interestingly, it is close to the maximum velocity of the re-entrant jet in a shedding cycle reported by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). Further, $u_{sw}$ is almost two orders of magnitude lower than $u_{pw}$, as shown in figures 11(b) and 13(c).

Measurements of $\Delta p$ and vapour fraction ($\alpha _{1}$, $\alpha _{2}$) are validated with each other using a one-dimensional Rankine–Hugoniot (R–H) equation, given by (Brennen Reference Brennen1995)

(3.1)\begin{equation} u_{rh}^{2} = \frac{\Delta p}{\rho_{l}} \left[\frac{(1-\alpha_{2})}{(1-\alpha_{1})(\alpha_{1}-\alpha_{2})} \right] . \end{equation}

Here, all the symbols have previously defined meanings. The R–H equation is derived using mass and momentum conservation along with the binary mixture model for density. The shock front velocities predicted with measured $\alpha _{1}$, $\alpha _{2}$, $\Delta p$ using the R–H equation (blue markers in figure 13c) show excellent agreement with the estimated values from $x\unicode{x2013}t$ plots (red markers in figure 13c). Further, $\Delta p$ predicted with measured $\alpha _{1}$, $\alpha _{2}$ and $u_{sw}$ using (3.1) (red markers in figure 13a) agrees with direct $\Delta p$ measurement.

3.5. Cavity pressure

The absolute pressure inside the vapour cavity is cumbersome to measure experimentally. Thus to get a ballpark estimate of the pressure inside the cavity, we measure the pressure at the venturi wall, at a location $p_{1}$ (see figure 1). This pressure at a single point is used as a proxy for the whole cavity pressure. The unsteady total pressure measurement is time-synchronised with high-speed imaging. This is done to discount the pressure time signal where the pressure transducer is not covered by the cavity. Further, the periodic, sharp pressure peaks caused by the cloud implosion downstream are filtered out using a low-pass filter. This is followed by time averaging the pressure signal for 10 s to yield the cavity pressure ($P_{c}$). It is seen that $P_{c}$ can be higher than the vapour pressure ($P_{v}$). However, at a lower cavitation number ($\sigma$), the pressure is seen to approach $P_{v}$ (see figure 14). This can be attributed to the bubbly nature of the cavity, where the vapour fraction increases with a decrease in the cavitation number. For a given static pressure in the system, the pressure is seen to be decreasing monotonically with the increase in the free-stream velocity. Naturally, with the decrease in the global static pressure, the absolute pressure inside the cavity decreases. The absolute pressure, when normalised by the dynamic pressure at the throat, collapses on a curve, suggesting that $\sigma$ and $U_{t}$ are sufficient to estimate the pressure inside the cavity (see the inset in figure 14). The pressure of the bubbly cavity mixture will be used to estimate the speed of sound in such a mixture.

Figure 14. Conditionally phase-averaged static pressure inside the cavity ($P_{c}$) measured at location $p_{1}$ as a function of the cavitation number ($\sigma$) for different global static system pressures; $P_{v}$ refers to the vapour pressure. The inset shows the pressure normalised with the incoming dynamic pressure.

3.6. Mach number of the condensation shock front

It is well known that the speed of sound is significantly lower in a bubbly mixture than in pure liquid or vapour (Prosperetti Reference Prosperetti2015; Shamsborhan et al. Reference Shamsborhan, Coutier-Delgosha, Caignaert and Abdel Nour2010). The cavity having substantial vapour fractions (0.4–0.8; see figure 13b) can reduce the speed of sound, rendering the medium compressibility important. Thus with the measured vapour fraction ($\alpha _{1}$) and the pressure of the bubbly mixture of the cavity ($P_{c}$), we can now make a rough estimate of the speed of sound in the cavity ($c_{m}$) using (Brennen Reference Brennen1995)

(3.2)\begin{equation} c_{m} = \left\{ \left[\frac{\alpha_{1}}{kP_{c}} + \frac{1-\alpha_{1}}{\rho_{l} c_{l}^{2}} \right] [\alpha_{1} \rho_{v} + (1-\alpha_{1}) \rho_{l}] \right\} ^ {-{1}/{2}}. \end{equation}

It is assumed that the cavity is a homogeneous bubbly mixture, and water vapour behaves as a perfect gas with the polytropic gas constant $k$. Further, the bubble dynamics in the cavity such as coalescence and surface tension are omitted for simplicity. Here, $\rho$ is the mass density, and $c$ is the speed of sound in the medium; subscripts $v$ and $l$ are used for vapour and liquid, respectively. The mixture pressure ($P_{c}$) is estimated in the previous subsection. The Mach number of the shock front ($Ma_{sw}$) can now be estimated with the speed of sound ($c_{m}$) and $u_{sw}$, as

(3.3)\begin{equation} Ma_{sw} = \frac{\overline{u_{sw}}}{c_{m}}. \end{equation}

Since we are interested in the interaction of the pressure wave with the bubbly cavity when it impinges on it, the phase-averaged condensation shock front velocity estimated in figure 13(c) is used to estimate the Mach number ($Ma_{sw}$). Figure 15 shows that $Ma_{sw}$ decreases with increasing $\sigma$, further corroborating that the strength of the condensation shock travelling through the cavity is decreasing with increasing $\sigma$. Such behaviour of the condensation shock front was also reported by Ganesh et al. (Reference Ganesh, Makiharju and Ceccio2016) and Bhatt & Mahesh (Reference Bhatt and Mahesh2020) for a two-dimensional wedge and backward-facing step geometry, respectively. We see that $Ma_{sw}$ is greater than unity, i.e. it is supersonic for lower $\sigma$. However, it falls below or hovers close to unity for higher $\sigma$ ($\sim$0.90). At even higher $\sigma$, the condensation shock fronts were not observed in our study. Thus $Ma_{sw} \sim 1$ coincides with the disappearance of condensation shock fronts. This suggests that the pressure and vapour fraction fields at higher $\sigma$ render the bubbly shock wave weaker and likely subsonic. This weakening of the shock front is likely to mitigate the complete condensation of the cavity at higher $\sigma$, as opposed to lower $\sigma$, where the pressure wave completely condenses the cavity. This is in line with observed cavity dynamics at $\sigma \sim 0.84$ and 0.90, where most shedding cycles begin to show re-entrant jet-driven shedding. With further increase in $\sigma$ ($\sim$0.97), all the shedding cycles were destabilised by the re-entrant jet, as shown previously by Jahangir et al. (Reference Jahangir, Hogendoorn and Poelma2018) and Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). Thus the shedding mechanism switches from a condensation shock front to a re-entrant jet as the pressure wave emanating from the cloud implosion becomes weaker.

Figure 15. The Mach number of the condensation shock front ($Ma_{sw}$) as a function of the cavitation number ($\sigma$). The black dashed line shows $Ma_{sw}=1$.

In summary, it is evident that the Mach number of the bubbly shock front ($Ma_{sw}$) decreases as $\sigma$ increases, suggesting that the pressure wave travelling through the vapour cavity appears to get weaker with increasing $\sigma$. This is corroborated by the observed cavity shedding dynamics, i.e. at $\sigma = 0.47$: the impinged pressure wave starts immediately condensing the cavity. At intermediate $\sigma = 0.78$, a condensation shock front emerges as the pressure wave propagates upstream. However, at $\sigma = 0.84$, the pressure wave appears to be too weak to condense the cavity readily. Thus the pressure wave is expected to propagate through the cavity without condensing it for the majority of the cycles. At this $\sigma$, a strong re-entrant jet is also present below the cavity, as shown by Gawandalkar & Poelma (Reference Gawandalkar and Poelma2022). Therefore, the re-entrant jet takes over as the dominant cavity detachment mechanism for higher $\sigma$. The velocity of the condensation shock front and re-entrant jet is of the same order of magnitude. If the pressure and the vapour fraction of the bubbly cavity allow the condensation of the cavity, then the phase change from vapour to liquid will dictate the cavity destabilisation. Alternatively, the cavity pinch-off by the re-entrant jet is favoured. Therefore, it is seen that the pressure field and the shock wave characteristics have a strong influence on the prevalence of the dominant shedding mechanism.

3.7. Kinematics of the condensation shock front

As stated in the previous subsection, the condensation shock front is seen to accelerate as it travels upstream towards the venturi throat. This can also be seen in previous studies, for instance, in the $x\unicode{x2013}t$ diagram showing the cavity dynamics in the experiments of Wu et al. (Reference Wu, Ganesh and Ceccio2019) and the large eddy simulations of Bhatt & Mahesh (Reference Bhatt and Mahesh2020). However, due to the low flow blockage in those studies, the acceleration is suspected to be not severe. As a result, it has not received much attention. In the current flow geometry, the flow blockage ratio is $\sim$9, which is significantly higher. Further, it is seen that at intermediate cavitation numbers ($\sigma \sim 0.78\unicode{x2013}0.84$), the emergence of a sharp condensation shock front coincides with the observed acceleration of the shock front. Thus it is imperative to account for the effect of acceleration of the condensation shock front in the present context. The velocity of the condensation shock front as a function of axial distance ($x$) is evaluated by computing the phase-averaged gradient to the measured condensation shock front in the $x\unicode{x2013}t$ diagram (see figure 16). The phase averaging of $u_{sw}(x)$ is performed over so-called ‘clean cycles’ ($\sim$80 %), omitting shedding cycles with local variations such as sudden acceleration or multiple condensation shock fronts. The error bars indicate the cycle-to-cycle variation in the velocity. The acceleration appears significant: the velocity doubles in half the cavity length. The acceleration experienced by the shock front while travelling upstream could be attributed to the axial variation of vapour fraction across the shock front ($\alpha _1(x)$), the change in the cross-section of the venturi as seen by the shock front, or the change in pressure of the cavity. Due to the difficulty in measuring experimentally the pressure inside the cavity, the pressure is assumed to be constant. The axial variation of the vapour fraction ($\alpha _{1}(x)$) encountered by the travelling shock front is shown for three $\sigma$ values (see figure 17a). The axial variation is negligible ($\lesssim$0.07) and within the measurement uncertainly. Further, we check if the shock front is accelerating due to the converging walls of the venturi (from the shock front frame of reference, given that it is travelling upstream). In that case, the measured shock wave velocity must vary to obey the mass conservation imposed by the geometry. This can be used to construct a simple model based on one-dimensional mass conservation and the initial shock front velocity to predict the velocity variation. The measured phase-averaged shock front velocity ($\overline {u_{sw}(x)}$) variation with $x$ is shown by markers in figure 17(b). It is observed that the agreement with predictions (dashed lines in figure 17b) is good. However, closer to the throat, the agreement appears worse. This can be attributed to assumptions made in the model such as constant pressure and density, and the difficulty in defining the condensation shock front close to the throat. Considering this, we can safely attribute the acceleration to the converging walls of the venturi as seen by the shock front.

Figure 16. The shock front velocity ($\overline {u_{sw}}$) plotted against $x$ normalised by the maximum cavity length ($L_{c}$).

Figure 17. (a) The pre-shock vapour fraction ($\alpha _{1}$) plotted against $x$. (b) The axial variation of the phase-averaged shock wave velocity ($\overline {u_{sw}}$, markers) compared with velocity predictions by mass conservation (dashed lines) at different $\sigma$ (same colour coding as in (a)). Every third data point is shown for clarity.

Further, we examine if this increasing velocity of the shock front has an effect on the dynamics of the condensation shock wave. We measure $\Delta p$ brought about by the condensation shock front as it travels upstream. The $\Delta p$ measurements are done at multiple axial locations, $p_{1}$, $p_{2}$ and $p_{3}$ (see figure 1(c) for the locations). Interestingly, at $\sigma = 0.47$, the pressure rise of the shock front ($\Delta p$) at $p_{2}$ is 10.38 kPa, which increases to 17.01 kPa at $p_{1}$. Similarly, at $\sigma = 0.67$, the pressure rise increases from 10.32 kPa (at $p_{2}$) to 13.60 kPa (at $p_{1}$). Thus the pressure rise across the bubbly shock front is seen to increase as it propagates upstream. Finally, we estimate the Mach number of the accelerating condensation front, $M_{sw}(x)$, as it travels upstream through the bubbly vapour cavity (see figure 18). It is assumed that the pressure of the bubbly mixture through which the condensation front travels is constant. Further, the vapour fraction that it encounters is nearly constant, as shown in figure 17(a). Thus the speed of sound is also assumed to be constant, in accordance with (3.2). It is seen that at the lowest cavitation number ($\sigma \sim 0.47$), the bubbly shock front is supersonic as it impinges on the cavity and remains supersonic as it travels through the cavity. This results in complete condensation of the cavity, as also observed in the time series shown in figure 5. The complete condensation is evident from the sharp jump in the vapour fraction over the entire shock front, as shown in figure 5(i). As the cavitation number increases ($\sigma = 0.78$), the bubbly shock wave is initially subsonic as it impinges on the cavity, until $x/L_{c} \sim 0.89$, as shown in figure 18. This is also seen in the vapour fraction time series (see figure 8), where the pressure wave front is seen to travel through the cavity without condensing it. However, as it accelerates while travelling upstream, it becomes supersonic at $x/L_{c} \sim 0.84$ (corresponding to $x = 21.2$ mm in the $x\unicode{x2013}t$ diagram shown in figure 8i) and starts complete condensation of the vapour cavity. Thus the axial location where the pressure wave becomes subsonic agrees well within the measurement uncertainty. At even higher $\sigma$ ($\sim$0.84–0.90), the pressure wave cannot become supersonic readily (see figure 18). This can also be seen in the shedding dynamics shown in figure 9(i): the shock front inferred from a sharp change in $\alpha$ appears further upstream. At such a flow condition, the pressure wave after impinging on the vapour cavity is less likely to condense it, allowing the re-entrant jet below the cavity to pinch it off. This shows the effect of the kinematics of the shock front on the cavity shedding dynamics.

Figure 18. The Mach number ($Ma_{sw}$) variation of the bubbly shock front with the axial distance ($x$) normalised by cavity length ($L_{c}$) for different cavitation numbers ($\sigma$). The direction of propagation of the shock fronts is upstream, i.e. to the left in this figure.

The phenomenological description of the vapour cavity shedding in an axisymmetric venturi for intermediate $\sigma$ can be given as follows. The pressure wave emanating from the previous cloud implosion impinges on the growing cavity. However, the velocity of the pressure wavefront ($u_{sw}$) and the pressure rise across it ($\Delta p$) are not enough to condense the cavity, i.e. it is likely subsonic. As the pressure wave travels upstream, its velocity increases due to the converging geometry of the venturi (blockage). Consequently, the pressure rise across it also increases. This makes the shock front stronger or even supersonic, favouring condensation. At lower $\sigma$, the impinging pressure discontinuity is supersonic. Hence the condensation shock front is seen to exist over the entire cavity length. However, at high $\sigma$, the pressure discontinuity is weak and remains subsonic, despite the acceleration. Thus it appears that the effect of such a flow geometry with a large blockage is to accelerate the pressure wave, increasing its strength as it travels upstream. This, in turn, explains the large range of cavitation numbers ($\sigma \sim 0.30\unicode{x2013}0.78$) for which the condensation shock front is the dominant cavity destabilising mechanism, as opposed to the re-entrant jet, which is seen to occur only for $\sigma \sim 0.95\unicode{x2013}1.1$.