3.1. Mean geometry and variation of cavity topology

A set of still images depicting the development of cloud cavitation about the sphere with reduction in $\sigma$ from inception through to supercavitation for the nuclei deplete and abundant cases are shown in figure 6. Excerpts from the high-speed photography for both cases for $\sigma$ values from 1.0 to 0.3 in 0.1 increments are available online as supplementary movie 1 available at https://doi.org/10.1017/jfm.2022.511. The general occurrence of cloud cavitation about a sphere for the present test conditions in the deplete case has been described by Brandner et al. (Reference Brandner, Walker, Niekamp and Anderson2010) and de Graaf et al. (Reference de Graaf, Brandner and Pearce2017), as reviewed in the introduction. The following discussion includes results on the differences in cavitation occurrence between the deplete and abundant nucleation cases as well as new findings on cavity topology for both cases.

Figure 6. Cavitation development with cavitation number reduction. The left column (a,c,e,g,i,k,m,o) is with water deplete of nuclei, and photographs of the abundantly seeded cases are in the right column (b,d,f,h,l,n,p).

Inception in both cases occurs at about $\sigma = 1.0$ as the minimum value of $C_p \approx -1.1$ for a transcritical Reynolds number (as defined by Achenbach Reference Achenbach1972). For the deplete case, this was an intermittent process due to the lack of available nuclei. As noted above, inception in this case may be initiated due to a remnant bubble or from the background nuclei population which may require a period of time before a sufficiently weak nucleus is encountered. For the abundant case, there is a continuous supply of sufficiently weak nuclei such that inception is immediate once $\sigma \approx -C_{p,{min}}$ (Franc & Michel Reference Franc and Michel2005), which corresponds to a minimum pressure of vapour pressure.

The cavity leading edge or line of detachment in the deplete case is clearly defined and is associated with separation of the oncoming laminar boundary layer (Briançon-Marjollet et al. Reference Briançon-Marjollet, Franc and Michel1990). For the abundant case, the nuclei homogeneously dispersed in the free stream are activated at streamwise planes corresponding to the critical pressure of each nuclei (larger nuclei activating further upstream). Activated nuclei rapidly grow as discrete macroscopic cavitation bubbles with downstream advection. This discontinuous leading edge can then be defined in terms of the local streamwise void fraction. Continuous activation of nuclei and the subsequent growth and coalescence of cavitation bubbles leads to the formation of cavities of clustered or contiguous bubble volumes. Activated nuclei grow to maximum sizes of about 1 cm at area concentrations of order 1 cm$^{-2}$, both of which increase with decreasing cavitation number. The volumetric concentration of nuclei activations and the streamwise rate of increase of void fraction increases with decreasing $\sigma$. Cavities in the abundant case have a similar global extent to those in the deplete case, but with a locally lower void fraction. As described by Li & Ceccio (Reference Li and Ceccio1996) and Brandner et al. (Reference Brandner, Walker, Niekamp and Anderson2010), it is the local three-dimensional flow about activated bubbles that breaks up the oncoming laminar boundary layer, preventing the formation of a continuous cavity leading edge and detachment line, as occurs for the deplete case.

At high cavitation numbers ($\sigma \approx 1.0$ to 0.9) in the deplete case, the K-H instability in the separated laminar shear layer largely controls transition to turbulence and cavity breakup and condensation. Although developed cavitation bubbles tend to merge for the abundant case, the cavity breakup and condensation, with downstream pressure recovery, is of a discrete nature (see online movie 1). This behaviour can be attributed in part to discrete bubble collapse but also to the three-dimensional flows about each activated bubble that initially breakup the upstream laminar boundary layer. Both effects contribute to larger-scale vortical structures in the wake of the condensed cavity in the abundant case compared with the deplete case where interfacial instabilities are of a much finer scale. The cavity extent at these higher $\sigma$ values, both upstream and downstream, is greater for the abundant case due to earlier nuclei activation upstream of the detachment line for the deplete case. Additionally, the bubbles collapse further downstream in the abundant case, condensing as the pressure recovers in the wake.

With reduction of $\sigma$ below 0.9 cavity lengths just become sufficient in both deplete and abundant cases for re-entrant flow and coherent shedding to begin to develop. The high-speed imaging and measured spectra show the shedding to be uni-modal and of axisymmetric character, as discussed in § 1. Large-scale cavity shedding ensues for $\sigma$ values between approximately 0.8 and 0.6 where it becomes predominately bi-modal for the abundant as well as the deplete case. As discussed earlier, these are attributed to axisymmetric and asymmetric modes in attached and detached regimes. The differences in nucleation affect both frequencies and amplitudes of surface pressures which are analysed in detail below. This shedding is driven by coupled re-entrant jet formation and upstream shockwave propagation that at its most energetic creates large-scale circumferential cavity extinction with subsequent re-nucleation. In the deplete case as there are no free-stream nuclei these must be supplied by remnant bubbles from cavity extinction or from surface nuclei. Although detailed investigations have not been carried out to definitively demonstrate the source of nuclei, high-resolution photography and high-speed imaging suggest the former to be the case at least for this PVC sphere. Russell & Brandner (Reference Russell and Brandner2021) studied coherent cloud cavitation about a NACA 16 hydrofoil where shockwave-driven large-scale extinction and re-nucleation occurred for each cycle. They showed that nuclei were provided from the condensation and breakup of the previously shed cavities. Ram, Agarwal & Katz (Reference Ram, Agarwal and Katz2020) showed similar phenomena associated with the breakup of incipient cavities in adverse pressure gradients and further showed how remnant bubbles can be transported upstream by reverse flow or indeed buoyancy alone.

For the abundant case with a continuous supply of nuclei, the overall nature of the large-scale shedding is similar although the cavities are discontinuous as evident from the still and high-speed photography. Re-entrant jets develop as the cavities grow and with fluid accumulation and eventual penetration of the outer cavity surface shockwaves are initiated in a similar process to the deplete case. As the cavity is composed of discrete bubbles, the shockwave can be seen to propagate upstream in a discontinuous manner. Unlike the deplete case, shockwaves rarely cause large-scale extinction due to the continuous nuclei supply, particularly as $\sigma$ is reduced. At the lower $\sigma$ values bi-modal shedding in the detached regime persists but at decreasing frequencies until the supercavitation regime forms at $\sigma \approx 0.3$.

The global cavity geometry is generally similar for all $\sigma$ values and comparisons of basic geometric and kinematic properties can be derived from the still and high-speed photography. Measurements of the cavity upstream extent for the deplete and abundant cases and comparisons with basic theory are shown in figure 7. The angle of cavity detachment, $\theta$, measured from the front stagnation point for the deplete case has been derived from the still photography measured using only the two detachment locations on the vertical centre plane to avoid image distortion effects. Previous measurements of cavity detachment angle, in the deplete case, by Brandner et al. (Reference Brandner, Walker, Niekamp and Anderson2010) and de Graaf et al. (Reference de Graaf, Brandner and Pearce2017) for an arbitrary point gave a linear relationship between $\theta$ and $\sigma$. The present measurement shows that by eliminating distortion this relation is slightly nonlinear. The cavity detachment angle for the deplete flow appears to occur some 25 to 30$^{\circ }$ downstream of the angle at which the surface $C_p = -\sigma$ as derived from single-phase inviscid theory ($\theta ={\rm arccos}\frac {1}{3}\sqrt {5-4\sigma }$). This difference is in part attributable to boundary layer and surface tension effects but also that the pressure distribution on a cavitating sphere will be different from that for single-phase inviscid theory. For the abundant case, the angle at which activated nuclei first become visible is plotted. These were extracted from high-speed photography and defined as the angle at which the pixel intensity fluctuations increased above the background noise level. This location has a weak dependence on the camera magnification and lighting. More magnification would result in bubbles being visualised earlier in their growth, thus reducing the angle. These points are some 5 to 10$^{\circ }$ downstream of the angle at which $C_p = -\sigma$ as derived from single-phase inviscid theory. These curves would be expected to converge for high $\sigma$ values as the cavity volumes decrease except that, as noted above, the minimum $C_p$ for a transcritical Reynolds number is about 0.1 less than the inviscid solution which explains, in part, the lack of close convergence near inception. Brennen (Reference Brennen1969) obtained an inviscid solution for cavitating flow about a sphere for cavitation numbers between 0.6 and 0.2. This range corresponds closely with the detached regime observed in experiments (de Graaf et al. Reference de Graaf, Brandner and Pearce2017) where cavity closure occurs downstream of the sphere for which solutions could be obtained for the method used by Brennen. The imaging and surface pressure spectra obtained in the present work also show the detached regime to commence at cavitation numbers between about 0.6 and 0.7. The results obtained by Brennen are included in figure 7 and these show close agreement with the angle at which the activated nuclei first become visible in the abundant case. This result suggests that the boundary layer, being relatively thin compared with the diameter, has little effect and that the real flow is closely modelled using inviscid theory with the cavity included.

Figure 7. The location of the detachment line for the unseeded case (blue) as a function of cavitation number. The activation angle for the nucleated case is in orange. The inviscid solution for the location of where the pressure equals the vapour pressure is given in green, ignoring the presence of the cavity. The computational solution from Brennen (Reference Brennen1969) including the cavity is also included in red. All angles are measured from the front stagnation point.

The differences between the deplete detachment line and the abundant activation line can be explained by consideration of the combined effects of boundary layer separation and discrete bubble activations (Li & Ceccio Reference Li and Ceccio1996; Tassin Leger & Ceccio Reference Tassin Leger and Ceccio1998). As noted by Arakeri & Acosta (Reference Arakeri and Acosta1973) and Arakeri (Reference Arakeri1975), the cavity detachment without nuclei is associated with an upstream laminar boundary layer separation. For the boundary layer to separate, an adverse pressure gradient must exist upstream of the cavity detachment line, such that the liquid upstream is at a lower pressure and in a state of metastable tension as discussed by Briançon-Marjollet et al. (Reference Briançon-Marjollet, Franc and Michel1990), Tassin Leger & Ceccio (Reference Tassin Leger and Ceccio1998). For the abundant case, where there are free nuclei, these are activated in this region of tension upstream of the unperturbed detachment line. With sufficient rates of activation, the detachment line approaches the inviscid solution of Brennen (Reference Brennen1969), hence the excellent agreement as shown in figure 7.

3.2. Nuclei activation rates

Once the nuclei are activated and visible their advection speed may be measured from the high-speed photography and comparisons made with the inviscid velocity, $U = U_\infty \sqrt {\sigma +1}$, that occurs in the flow where $C_p = -\sigma$ for the abundant flow case. For the deplete case where there are no active nuclei, this velocity is the inviscid cavity surface speed. The measured activated bubble velocities and the inviscid velocity where $C_p = -\sigma$ are shown in figure 8. The bubble advection velocities are all within about 10 % of the inviscid value. This difference may in part be attributed to the bubbles not being on the surface but possibly also due to the flow modification as a result of bubble growth and their finite size.

Figure 8. Bubble velocity ($U_{bub}$) at the activation point as a function of the cavitation number for the nucleated cases. Also included in orange is the inviscid velocity found from theory as $\sqrt {\sigma +1}$.

High-speed imaging of the activated nuclei can also be used to measure the nucleation rate at each $\sigma$ and analyses performed to test how well these can be reconciled with the measured nuclei populations. Assuming the flow about the sphere is unaffected by the presence of cavitation, at least for the higher $\sigma$ values, inviscid theory may be used to estimate nucleation rates from the measured nuclei populations. These rates can then be compared with those derived from the high-speed imaging.

For single-phase inviscid flow about a sphere, the minimum pressure for any streamline occurs on the spanwise plane at the origin ($x=0$, in cylindrical polar coordinates as defined in figure 9). This plane can then be considered for estimating the nuclei activation rate for a particular $\sigma$ value, as shown in figure 9. The pressure at this plane is given from inviscid theory as

(3.1)\begin{equation} c_p = 2\frac{p_0-p_\infty}{\rho U_\infty^{2}} ={-}\frac{a^{3}}{r^{3}} - \frac{1}{4}\frac{a^{6}}{r^{6}}. \end{equation}

Figure 9. Coordinate system for the flow about a sphere. The streamtubes of opportunity ($T=0$) for selected cavitation numbers are indicated in grey.

Which can be converted to tension, $T$, through the definitions of the tension, $T=p-p_{v}$, and the cavitation number $\sigma =2({p-p_{v}})/{\rho U_\infty ^{2}}$. The tension, therefore, as a function of radial coordinate, $r$, is

(3.2)\begin{equation} T_{0}=p_{0}-{p_{v}}{} = \frac{1}{2}\rho U_\infty^{2} \left(\sigma - \left(\frac{a}{r}\right)^{3} - \frac{1}{4} \left(\frac{a}{r}\right)^{6}\right).\end{equation}

This equation can be transposed to give the radial coordinate as a function of tension,

(3.3)\begin{equation} \left. \begin{array}{c} \dfrac{r}{a} = k^{-{1}/{3}},\\ k ={-}2 + 2\sqrt{1+\sigma-\dfrac{T}{\dfrac{1}{2} \rho U_\infty^{2}}}. \end{array}\right\} \end{equation}

This equation can be used to derive the streamtube of opportunity which encapsulates all streamlines that will be exposed to tensions equal to, or less than, zero, or equivalently, static pressures equal to, or less than, vapour pressure. Setting $T=0$ in (3.3) provides the radial coordinate at the origin at which vapour pressure occurs as a function of $\sigma$,

(3.4)\begin{equation} \frac{r_0}{a} = \frac{1}{\sqrt[3]{2(\sqrt{\sigma + 1} - 1)}}. \end{equation}

The upstream radius of this streamtube is found by equating the streamfunction and finding the limit as the streamwise coordinate goes to infinity, that is, far upstream

(3.5)\begin{equation} \frac{r_\infty}{a} = \frac{r_0}{a}\sqrt{3 - 2\sqrt{\sigma + 1}}. \end{equation}

Streamlines defining the streamtubes of opportunity for selected $\sigma$ values are shown in figure 9 and the streamtube radii at the origin and upstream as a function of $\sigma$ are shown in figure 10. As noted above, this analysis is only justifiable for high $\sigma$ values where the flow is least modified by the cavitation. Also noted earlier, the seeded plume for the nuclei abundant case is about 80 mm diameter, or about $r_\infty /a = 0.5$, and from figure 10 it can be seen that this is sufficient for the larger $\sigma$ values. For $\sigma$ values below 0.8 to 0.9, the curve shown in figure 10 indicates a much larger plume is required to seed the streamtube of opportunity. Although as discussed, this analysis is no longer justifiable with cavity volumes becoming large compared with the sphere volume such that the bulk flow will be significantly modified. It is arguable that the upstream diameter of the streamtube of opportunity will be significantly smaller than the single-phase result shown in figure 10, but it remains possible that the streamtube of opportunity for the lower $\sigma$ values is not fully seeded. It is also worth noting that the radius of the streamtube of opportunity at the origin for the higher $\sigma$ values only extends some 10 % of the sphere radius beyond the surface, which is of the order of typical activated bubble sizes.

Figure 10. Radius of the streamtube of opportunity at the sphere (blue) and far upstream (orange).

Nuclei transport in inviscid or viscous flows involves several forces demonstrated in recent examples of one-way coupled Lagrangian analyses by Hsiao, Ma & Chahine (Reference Hsiao, Ma and Chahine2018), Chen et al. (Reference Chen, Zhang, Peng and Shao2019), Beelen & van Rijsbergen (Reference Beelen and van Rijsbergen2018), Paul, Venning & Brandner (Reference Paul, Venning and Brandner2021). Nevertheless, some insight can be gained into activated nuclei sizes and rates from an albeit simple analysis assuming ideal nuclei advection given the small sizes involved.

A closed-form expression for the nuclei activation rate for a single tension and cavitation number can be derived using the non-cumulative nuclei distribution in tension (${-{{\rm d}C}/{{\rm d}T}}$). The activation rate is here defined as the number of nuclei activations per unit time, and has the physical units of ‘per second’. The activation rate through a differential streamtube for a single tension is the product of the concentration for that tension and the differential flow rate. The total activation rate for that tension is then the integral of all differential streamtubes with tensions less than or equal to the tension in question,

(3.6)\begin{align} \frac{{\rm d}A}{{\rm d}T} &= {-\frac{{\rm d}C}{{\rm d}T}} \int_a^{r_0(T)} U_r 2{\rm \pi} r \,\mathrm{d}r\nonumber\\ &= {-\frac{{\rm d}C}{{\rm d}T}}2{\rm \pi} U_\infty \int_a^{ak^{{-}1/3}}{r\left(1+\frac{a^{3}}{2r^{3}}\right) \,\mathrm{d}r}\nonumber\\ &={-}\frac{{\rm d}C}{{\rm d}T} {\rm \pi}U_\infty a^{2} (k^{{-}2/3} - k^{1/3}), \end{align}

where $T$ and $\sigma$ are encapsulated within $k$. By choosing a single tension the concentration term (${-{{\rm d}C}/{{\rm d}T}}$) is constant and can be excluded from the integration, the limits of integration are from the sphere surface to the radius where the tension is equal to the value chosen $r_0(T)$. Equation (3.6) may then be evaluated for all measured tensions for both the deplete and abundant nuclei populations, for the higher $\sigma$ values, as shown in figure 11. Here the activation rates per unit tension have been converted to rates per unit diameter using (2.2). For the deplete case, the activation rates are small in comparison with the abundant case as expected and biased to the small nuclei sizes, as low as 5 ${{\rm \mu} }$m, where concentrations are much higher. This plot also shows the nonlinear increase in activation rates with decreasing $\sigma$ again due to smaller nuclei with higher concentrations becoming susceptible. For the abundant flow, the plots are truncated at 18 ${{\rm \mu} }$m reflecting the lower bound of the MSI measurement. Here the rates are some $10^{6}$ times greater with similar trends but less increase with decreasing $\sigma$ due to the distribution not varying with tension or diameter as greatly as for the deplete case.

Figure 11. Activation rate per unit diameter as a function of bubble diameter from (3.6 and 2.2). Plot (a) is with the deplete nuclei population, and (b) is with the abundant. Note that there are six orders of magnitude difference between the vertical scales of the two figures.

While (3.6) is useful in showing the dependencies involved to estimate the total nuclei activation rate its closed-form integration is cumbersome. Alternatively, the total rate may be numerically integrated directly by including the cumulative nuclei distribution $C(T)$ in tension (or diameter) in the integration in a similar way as for (3.6) but integrating across all differential streamtubes with tensions less than zero,

(3.7)\begin{align} A &= \int_a^{r_0}2{\rm \pi} r U_r C(T(r))\,\mathrm{d}r\nonumber\\ &= 2{\rm \pi} U_\infty\int_a^{r_0}r \left(1+\frac{a^{3}}{2r^{3}}\right) C(T(r))\,\mathrm{d}r. \end{align}

The results from (3.7) are shown in figure 12 as a cumulative plot starting from the largest nuclei. For the nuclei deplete case (left), the rates increase as a power law reflecting the deplete nuclei distribution. The curves reach a constant value at the smallest nuclei that can be activated for each $\sigma$ which is about 4 ${{\rm \mu} }$m for the $\sigma = {0.8}$ case. In all cases the estimated total activation rates are small and are of the same order of magnitude observed in the experiments (this is a qualitative estimation since the interval between activations is some 10 times longer than the duration of the high-speed video recordings, precluding the calculation of an activation rate). As discussed in § 3.1, at high cavitation numbers it may take several minutes for a transient activation or inception to occur.

Figure 12. Total nuclei activation rate about a sphere for deplete (a) and abundant (b) free-stream nuclei levels. The horizontal lines indicate the measured activation rate from the high-speed videos.

For the nuclei abundant case, the nuclei activation rates measured from the high-speed photography are included as horizontal lines in figure 12(b). These are total measurements across all nuclei sizes as the initial size before activation could not be measured. The growing microbubbles were counted manually in the video frames. Each activation was tagged and tracked with downstream advection ensuring double counting did not occur. The activation rate was relatively constant with $\sigma$, increasing from about $0.86\times 10^{5}$ s$^{-1}$ at $\sigma =1.0$ to $1.02\times 10^{5}$ s$^{-1}$ at $\sigma =0.8$. The estimated activation rates are some one order of magnitude greater than those actually observed.

There are several assumptions to which these differences may be attributed. The inviscid minimum pressure coefficient of $-$1.25 is lower than the value for the finite Reynolds number tested which, as noted above, is actually about $-$1.1. Although only large cavitation numbers, where cavity volumes are small, are considered, the presence of cavitation will further alter the pressure distribution from the inviscid case. The boundary layer, although thin, has not been accounted for and the decreased mass/nuclei flux would also reduce the activation rate compared with the pure inviscid case.

In the present analysis the critical spanwise plane at the origin, at which the inviscid minimum pressure occurs, is used to calculate the number of activated nuclei. Whereas in the real flow, larger nuclei will be activated at planes upstream of the minimum and these growing bubbles locally affect the flow. That is, for the present analysis, no interaction or coupling between phases is assumed. This concept is discussed by Franc & Michel (Reference Franc and Michel2005) and Lecoffre (Reference Lecoffre1999) and relates to flows in which negative pressures occur such that once vapour bubbles develop, with effectively vapour pressure internally and at the surface, then at each bubble surface local pressures are raised from the single-phase negative pressures to vapour pressure. Furthermore, they explain that this effect can lead to a condition of saturation where an increase in the nuclei concentration produces no further increase in activation rate. Although local, these effects are distributed and introduce the potential for suppression of further nuclei activation and bubble-bubble interactions. Given that the thickness of the streamtube of opportunity about the sphere is of the order of the size of the cavitation bubbles this effect could be significant.

Finally, if all forces acting on the nuclei are accounted for, larger nuclei may not be carried ideally on the fluid streamlines but be deviated by pressure gradients and other effects. This phenomena known as ‘screening’ (Liu, Kuhn de Chizelle & Brennen Reference Liu, Kuhn de Chizelle and Brennen1993; Beelen & van Rijsbergen Reference Beelen and van Rijsbergen2018; Hsiao et al. Reference Hsiao, Ma and Chahine2018; Chen et al. Reference Chen, Zhang, Peng and Shao2019; Paul et al. Reference Paul, Venning and Brandner2021) may lead to nuclei being transported from the inside to the outside of the streamtube of opportunity reducing the number of activations about the sphere compared with the potential available nuclei in the streamtube upstream. Liu et al. (Reference Liu, Kuhn de Chizelle and Brennen1993) considered the screening problem for an axisymmetric headform and used potential flow about a sphere to derive an analytical model. Their results showed that screening, boundary layer and size effects can account for a reduction in activation rates of an order of magnitude. Although ultimately they showed that taking these effects in account predicted activation rates were one order of magnitude greater than those observed experimentally. Paul et al. (Reference Paul, Venning and Brandner2021) considered nuclei transport about a sphere and showed that screening and other effects are small. These results would imply that the order of magnitude differences between predicted and observed results is most likely due to two-way coupling between the disperse and continuous phases. That is, as described above, activated finite growing bubbles eliminate local tension and, hence, suppress activation rates below expected values such that the activation rate has reached saturation.

3.3. Unsteady surface pressure and high-speed imaging

The standard deviation of the measured unsteady pressures from the near and far sensors, for the deplete and abundant nucleation cases, for all $\sigma$ values are shown in figure 13. The near and far results show nominally the same amplitudes and trends as expected due to symmetry. The variation of amplitudes reflect the intensity and location of the shedding. For the deplete case at high $\sigma$ values, the intensity is relatively low due to short cavity breakup driven by small-scale interfacial instabilities. The intensity increases rapidly with $\sigma$ reduction as cavity lengths grow and the re-entrant jet instability develops. A maximum is reached within the attached shedding regime at about $\sigma =0.85$ which is attributable to the nature of the shedding as well as the proximity of the re-entrant jet location or shedding zone moving over the sensor location. Beyond the maximum, the amplitude reduces in a linear trend with reducing $\sigma$ through the detached regime, where re-entrant jet formation is off-body. The standard deviation ultimately reaching zero with the formation of supercavitation where any unsteady flow occurs well downstream.

Figure 13. The effect of free-stream nucleation, cavitation number and pressure sensor location on the magnitude of the pressure fluctuations. The near sensor is in (a) and the far is in (b). The blue is using water deplete of nuclei, and the orange is with abundant nuclei injection.

For the nuclei abundant case at high $\sigma$ values, the amplitude increases immediately after inception to values much greater than the deplete case due to the unsteadiness created by continuous nuclei activation and collapse. For $0.65 < \sigma < 0.9$, corresponding to the attached shedding regime, amplitudes are reduced by about 10–20 %. This reduction is due to the continuous nuclei activation breaking up or reducing the intensity and coherence of the large-scale shedding that would otherwise occur in the deplete case. As discussed in § 1, for the deplete case when the shedding is most energetic, the coupled re-entrant jet and shockwave formation lead to large-scale cavity extinction. That is, large regions of the flow may oscillate between 0–100 % void fraction. For the abundant case where there is continuous nuclei activation and bubble growth, the flow does not reach such extremes, hence the reduction in amplitude in the attached regime. With reduction of $\sigma$ into the detached and supercavitation regimes where shedding is moved downstream, differences in nucleation have a significantly reduced effect and amplitudes of the standard deviation for the two cases tend to converge.

The power spectral density (PSD) and CPSD of the near and far unsteady pressures, for deplete and abundant cases, are presented as spectrograms of cavitation number in figures 14 and 15, respectively, derived as described in § 2. Here the Strouhal number has been defined as $St=fD/U_\infty$. The amplitude scales of the power spectrograms in figure 14 are the same for deplete and abundant cases to show the relative magnitude of response of the various modes. The amplitude of the CPSD or coherence in figures 15 are also the same for each nucleation condition, but also coloured to show the phase angle between the near and far pressure signals.

Figure 14. Spectrograms showing the cavitation dynamics as a function of the cavitation number. The influence of water quality is shown by comparing the (a,c) figures for the deplete case to the (b,d) figures with abundant nuclei injection. The (a,b) corresponds to the near pressure tap, while the (c,d) is the far. Neither the spectra nor the initial data have been normalised to enable a direct comparison. The vertical lines indicate the nominal positions of the three different shedding regimes.

Figure 15. Phase difference of the CPSD between the two pressure sensors as a function of cavitation number. The phase is given by the colour, and the power sets the transparency. The flow is deplete of nuclei in (a) and abundant in (b). The vertical lines indicate the three different shedding regimes.

In both cases just after inception for $0.9 < \sigma < 1.0$ there is a uni-modal response in the frequency range $0.25 < St < 0.50$. For the deplete case, as discussed above, cavity breakup is driven by small-scale interfacial instabilities that lead to roll-up of larger-scale vortices shed within the frequency range noted above. This phenomena can be seen from the high-speed imaging in the supplementary material for $\sigma =0.90$ (see supplementary movie 1). The unsteady pressures show moderate coherence and to be out-of-phase which could be attributed to the random nature of the local shedding. For the abundant case, continuous nuclei activation at close circumferential spacing around the sphere leads to contiguous cavity growth and downstream collapse. The discontinuous bubble or cavity collapse events appear to initiate local shockwave phenomena that momentarily propagate upstream contiguously. That is, there appears to be a dynamic equilibrium between downstream advection of growing/collapsing bubbles and upstream propagation of contiguous shockwaves from local distributed collapse events. In this case the unsteady pressures show weak coherence.

For $\sigma < 0.90$, a bi-modal response commences in both nucleation conditions for which three regimes are apparent with selected $\sigma$ values labelled I, II and III in figures 14 and 15 with corresponding frequencies for each mode listed in table 1. Similar spectrograms previously derived from a single surface pressure measurement by de Graaf et al. (Reference de Graaf, Brandner and Pearce2017) identified attached and detached regimes at higher and lower $\sigma$ values, respectively. This observation is confirmed with the present higher resolution spectrograms although two sub-regimes, I and II, within the attached regime are apparent, although regime I is limited to a range or band of cavitation numbers of 0.1.

Table 1. Strouhal number for each of the shedding modes in each cavitation regime. For each regime, I, II and III, there are two shedding frequencies, indicated by ‘$L$’ for the low-frequency mode and ‘$H$’ for the high-frequency mode. The seeding level is indicated with a ‘$d$’ for deplete or an ‘$a$’ for abundant.

Figure 16. Power spectral density of the pressure signals in regime I. $\sigma =0.825$ for the deplete (blue) and $\sigma =0.800$ for the abundant (orange) cases. The high-frequency modes from regime II are also indicated.

For the deplete case, the regime I band is centred at about $\sigma =0.825$ and for the abundant case about $\sigma =0.800$. The PSD for the selected values of $\sigma =0.825$ and $\sigma =0.800$ in regime I for the deplete and abundant cases, respectively, are compared in figure 16. Here, frequencies are annotated as follows: $f_{dL}^{\rm I}$ indicates the deplete ($d$), low frequency ($L$), regime I shedding mode, while $f_{aH}^{\rm II}$ indicates the abundant (a), high frequency (H), regime II shedding mode. The high- and low-frequency modes for the deplete case ($f_{dH}^{\rm I}$ and $f_{dL}^{\rm I}$) are similar in amplitude whereas the high mode for the abundant case is nearly an order of magnitude greater than the low mode. The high modes in both cases are almost the same frequency and strongly narrow band compared with the low modes. The high mode pressures are in-phase indicating axisymmetric shedding with the abundant case showing greater coherence than the deplete. The low mode, expected to be asymmetric, shows less coherence and ambiguous phase with apparent out-of-phase for the deplete and in-phase for the abundant. These observations are reflected in the XWT plots shown in figure 17. Here, the absolute value of the XWT is given, indicating when the two pressure sensors were simultaneously encountering pressure fluctuations of the same frequency. Qualitatively more low mode than high mode events are evident for the deplete case but conversely more high mode than low mode for the abundant case. These observations may be confirmed quantitatively from the durations over which each frequency has the greater amplitude to obtain an overall probability for each mode, as listed in figure 17. For the deplete case, the low mode has the dominant probability of 73 % and, for the abundant case, the high mode is dominant with a probability of 57.6 %. The unexpected in-phase CPSD result for the abundant case may be attributable to the lack of events due to the prevalence of the high mode combined with random circumference variation of the asymmetric shedding being unresolved from only two correlated pressure measurements.

Figure 17. Absolute value of the XWT of the pressure time series between the two sides. The two frequencies of interest for each regime are indicated by the horizontal lines. Rows (a), (c) and (e) are for the deplete configuration, while (b), (d) and (f) are for the abundant. Regime I is shown in figures (a) and (b) with cavitation numbers of 0.825 and 0.800, respectively. Regime II (c,d) is at a cavitation number of 0.750, and regime III (e,f) is 0.600.

These observations can be further explored using the high-speed imaging directly and via extracted space–time plots from a horizontal line through the centre of the sphere. Unfortunately, high-speed imaging was only taken for $\sigma =0.800$ and not at $\sigma =0.825$ so that the regime I deplete modes could be studied in isolation. Examples of the high mode have been identified in the imaging at $\sigma =0.800$ along with the regime II modes where the low mode is dominant. Space–time plots for the deplete and abundant cases at $\sigma =0.800$ are shown in figure 18 with coloured shaded periods indicating examples of several modes in regimes I and II as indicated. The frequencies for each modes were determined from the corresponding pressure time series using the wavelet transform. Movies for each mode as indicated are provided in the online material for $\sigma =0.800$ (movies 2 to 7). For the abundant case, there is a prevalence of highly coherent short period shedding events whereas, for the deplete case, there are fewer longer events. From the high-speed imaging it can be seen that cavity lengths just become long enough for coupled re-entrant jet and shock phenomena to form in both nucleation conditions. For $\sigma =0.800$, the shockwaves are triggered by re-entrant jet formation and can be seen to propagate upstream unlike at $\sigma =0.900$ where there were no re-entrant jets and the shock front is stalled not propagating upstream.

Figure 18. Spatio-temporal maps extracted from a horizontal line through the centre of the sphere of the cavity shedding for the deplete (a–c) and abundant (d–f) cases at $\sigma =0.8$. The duration of each row is 1 s. The flow is from bottom to top. Structures moving downstream have a positive gradient, and the upstream propagation of shockwaves have a negative gradient. Events at three distinct time scales are evident in the deplete case, while shorter modes are more dominant for the abundant flow. The segments highlighted in blue, orange and green are extracted in figure 24 and correspond to regime II low, regime II high and regime I high, respectively. For the abundant case, the segments highlighted correspond to regime I low, II high and I high, respectively. Videos of these segments are available online as supplementary material and movies 2–7.

The high-speed imaging (available online as movie 7) shows the abundant high mode shedding to be due to coupled re-entrant jet/shockwave phenomena with a high degree of axisymmetry driven or enhanced by the continuous supply of activated nuclei around the sphere periphery. The order of magnitude greater PSD amplitude of the abundant high mode compared with the deplete high mode is most likely attributable to this axisymmetric stabilisation with the abundant nuclei supply. Correspondingly this would also explain the greater probability of the abundant high mode and perhaps also a contributing factor to the unresolved CPSD phase for the suppressed low mode. The imaging also appears to show the low mode shedding to occur with only slight asymmetries in cavity length due to the nucleation promoting axisymmetric growth. Examination of the imaging and unsteady surface pressure for temporal segments at the low mode frequency do show oblique vortical structures in the sphere wake but in-phase pressures suggest on-body cavity growth is symmetric and instabilities develop after detachment. Again, these may be further contributing factors to the unresolved phase angles between the near and far unsteady pressures. The axisymmetric forcing of the abundant nucleation may also explain the regime I occurring at a slightly lower cavitation number than for the deplete case.

From the imaging (available online as movie 4), the deplete high mode appears due to local three-dimensional re-entrant jets and shockwaves that lead to early breakup of the growing cavities, resulting in much shorter cavity lengths and more frequent events compared with the cavity lengths that form during more coherent events that occur in regime II.

In summary, for both nucleation cases in regime I, cavity lengths become just long enough for coupled re-entrant jet and shockwave formation to drive weakly bi-modal shedding with the high-frequency, axisymmetric mode dominating. For the abundant case, the nuclei supply amplifies or forces axisymmetric cavity growth and shedding. The high-frequency modes in both nucleation cases have essentially equal frequencies higher than those for regime II due to similar cavity growth rates but shorter lengths at which destabilisation occurs.

Regime II shows strong bi-modal response in both nucleation cases (figure 14) with the deplete case having much greater PSD amplitudes in both modes than the abundant case. From figure 15 it can be seen that the high and low modes are highly coherent in- and out-of-phase, respectively. The low mode for the abundant case appears much weaker, as shown in figure 14, although a clearer comparison of the PSD at $\sigma =0.75$ is made in figure 19. The high modes in both nucleation cases are of greater amplitude than the low modes. The amplitudes of each mode for the deplete case are nearly an order of magnitude greater than the corresponding modes for the abundant case. These differences in amplitude correspond to those discussed above for the standard deviation across the attached regime attributed to differences in void fraction and coherence for each nucleation condition. It can also be seen from figure 19 that both modes for the abundant case occur at higher frequencies than the deplete modes. These differences can be explained from basic statistical analysis of the unsteady pressure waveforms. The pressure signals may be segregated into periods of high and low, corresponding to with and without a cavity present, as well as periods of rising and falling pressure, as shown schematically in figure 20 with the periods listed in table 2. These data show that the periods of pressure rise and fall, or cavity growth and shockwave/collapse phases are similar for each nucleation condition. The frequency differences are mostly attributable to periods where there is no cavity, or between a shedding event and re-nucleation, and the period for which a cavity is present. These observations are reflected, and may be seen qualitatively, in the space–time images for the deplete and abundant cases shown in figure 18. It is evident that there are far fewer shedding events in the deplete case than in the abundant and that the duration where the cavity is either present or not present are greater than those in the abundant case. Larger-scale segments of the space–time plots from figure 18 presented in figure 21 comparing equal periods of time for deplete and abundant cases show these differences more clearly. It can also be seen from figures 18 and 21 that most of the longer events in the deplete case are two successive or coupled events with a short duration of extinction between them. It will be shown that these events correspond with the regime II low mode. Similar observations can also be made from the XWT plots shown in figure 17 for $\sigma =0.75$. The probability or prevalence of the low mode is greater for the deplete case than the abundant. Differences in the duration statistics described for when there is and is not a cavity present and overall differences in the modal frequencies may then be attributable to the effects of nucleation. As argued for regime I, the continuous supply of nuclei for the abundant case not only shortens the periods between shed cavities but also drives the high axisymmetric mode preferentially over the low asymmetric mode.

Figure 19. Power spectral density of the pressure signals in regime II. Here $\sigma =0.75$ for the deplete (blue) and for the abundant (orange) cases.

Figure 20. Schematic showing the average cavity shedding cycle for two cavitation numbers and each seeding configuration. The growth and shrinking phases are similar duration, but the time with and without the cavity present are shorter for the abundant cases.

Figure 21. Spatio-temporal map extracted from a horizontal line through the centre of the sphere of the cavitation extent for a cavitation number of 0.800. The (a) figure is deplete of nuclei, while the (b) is abundantly seeded. The flow is from bottom to top. Structures advecting downstream have a positive gradient, and the upstream propagation of shockwaves have a negative gradient.

Table 2. Average duration of each segment of the cavity shedding cycle for each nuclei concentration. The times are in milliseconds.

As noted above, the space–time plots shown in figures 18 and 21 for the deplete case show regular occurrence of paired shedding events of which an example is shown in figure 22. An initial cavity grows and a re-entrant jet and shockwave form detaching the first cloud that, as can be seen, is not advected downstream. A second cavity then develops followed by a re-entrant jet and shockwave but in this case the detached cloud may be seen to be advected downstream. Space–time segments of the synchronised dual high-speed photography (video is available online as movie 8) of the near and far side of the sphere are shown in figure 23. These images show a sequence of paired shedding events on each side qualitatively to be out-of-phase suggesting these events do correspond with the regime II low mode. These images are typical, as per much longer space–time sequences, in showing that the first event is not advected downstream whereas the second is and that these paired events tend to occur in clusters.

Figure 22. Spatio-temporal map extracted from a horizontal line through the centre of the sphere of the cavitation showing a single regime II, low-frequency shedding event, consisting of two cloud cavities. The first cavity grows before stalling near the end of the sphere. The second cavity grows and its shockwave extinguishes the cavity. The cavitation number is 0.800 and there is no additional seeding.

Figure 23. Spatio-temporal map extracted from a horizontal line through the centre of the sphere showing low-frequency regime II shedding from both sides of the sphere. The left column is the near side, while the right column is the far side. Flow is from right to left for the near-side, and left to right for the far-side. The long-duration shedding events are out-of-phase between the two sides. The cavitation number is 0.800 and the flow is deplete of nuclei. The corresponding video of this sequence is available as supplementary material movie 8.

To confirm this observation, space–time sequences and surface pressure time series may be locally analysed using the CWT to discern the features corresponding to each mode, as shown in figure 24 for the deplete case at $\sigma =0.8$. In this figure three sequences of equal duration (a–c) have been selected from figure 18, as indicated. Wavelet transform analysis of these sequences show the frequencies to correspond with those for the regime II low mode, regime II high mode and regime I high mode, respectively. The near- and far-side unsteady surface pressures acquired simultaneously with the high-speed imaging are shown in the second row of plots. The CWT and XWT amplitudes are shown in the third and forth row plots, respectively, and the horizontal lines show the frequencies of interest. The arrows in the CWT plots indicate the phase and the real component of the XWT is shown in the right-hand side plot. Positive values of the XWT real component indicate nominally in-phase and negative out-of-phase, and the magnitude the degree of coherence. The real component for the regime II low mode is negative indicating out-of-phase shedding with a relatively larger amplitude than the other modes. The real components for the regime I and II high modes are positive indicating in-phase shedding but with relatively less coherence compared with the regime II low mode. The greatest response for regime II in the deplete case occurs at about $\sigma =0.75$ for which a similar analysis as above has been carried out, as shown in figure 25. Selected temporal sequences with frequencies matching the low and high modes show corresponding out-of-phase and in-phase real components, respectively, with greater coherence than the $\sigma =0.8$ results.

Figure 24. The spatio-temporal map from a horizontal line through the centre of the sphere of cloud cavity position (a–c) for the nuclei deplete case at $\sigma = {0.800}$. These time series were selected to exemplify when each of the three frequencies, $f_{dL}^{\rm II}$ (a,d,g,j), $f_{dH}^{\rm II}$ (b,e,h,k) and $f_{dH}^{\rm I}$ (c,f,i,l) were dominant. The spatio-temporal map has a horizontal blue line indicating the downstream location of the pressure tap. Sample time series (d,e,f) of the pressure coefficient for the near (blue) and far (orange) pressure sensors. The filled contours (g,h,i) are the real value of the wavelet transform of the near pressure signal and in (j), (k) and (l) are the XWT between the two sides of the sphere. The arrows indicate the phase difference with an arrow to the right indicating zero phase while an arrow to the left is ${\rm \pi}$ out-of-phase. The horizontal lines indicate the frequency of interest and figure $m$ is the time average of the real component of the XWT for each of the three frequencies. The time-axis scale is constant across the samples and the duration is 0.2 s for each. These segments are indicated in figure 18.

Figure 25. The spatio-temporal map from a horizontal line through the centre of the sphere of cloud cavity position (a,b) for the nuclei deplete case at $\sigma = {0.750}$. These time series were selected to exemplify when each of the two frequencies, $f_{dL}^{\rm II}$ (a,c,e,g) and $f_{dH}^{\rm II}$ (b,d,f,h), were dominant. The spatio-temporal map has a horizontal blue line indicating the downstream location of the pressure tap. Sample time series (c,d) of the pressure coefficient for the near (blue) and far (orange) pressure sensors. The filled contours (e,f) are the real value of the wavelet transform of the near pressure signal and in (g), (h) are the XWT between the two sides of the sphere. The arrows indicate the phase difference with an arrow to the right indicating zero phase while an arrow to the left is ${\rm \pi}$ out-of-phase. The horizontal lines indicate the frequency of interest. The time integral of the real value of the XWT is given in $i$. The time-axis scale is constant across the samples and the duration is 0.2 s for each.

The segments of high-speed imaging identified from the wavelet transform analyses to have frequencies corresponding to the regime II low mode may be analysed to gain insights into the topology of this asymmetric mode. High-speed imaging of the regime II low mode for the space–time segments shown in figures 21 and 23 are provided in the online material as movies 2 and 8, respectively, and annotated in movie 9. In both these movies a sequence of oblique vortices shedding alternatively from nominally diametrically opposing sides of the sphere can be seen. That is, as can also be seen, the planar symmetry has no preferred azimuthal orientation and may temporally vary between events or clusters of events. The sequence of shedding events in movie 2 has shedding with a nominally horizontal plane of symmetry whereas movie 8 has an oblique plane of symmetry. In both cases cavity growth and subsequent re-entrant jet and shockwave formation are asymmetric resulting in oblique detachment of the shed cavity. The shed vortical rings are distorted or stretched with downstream advection forming classical hairpin shapes with the heads of successive events on diametrically opposing sides of the sphere. The connectivity between each hairpin head and the legs of the previously shed hairpin may also be seen.

From these observations the paired events evident in the space–time plots can be explained. As noted above, the space–time segments in figures 21 and 23 show paired events characterised by the first shedding event not showing downstream advection whereas the second does. The paired shedding is also characterised with an interval of cavity extinction between them typically shorter than those separating the pairs. It can be shown from the high-speed imaging that the first shedding event of the pair corresponds to the cavity on the near side receding or rotating about the sphere to the far side while simultaneously on the far side the second cavity of the pair on that side is released. That is, the first event contributes to the junction between each shed hairpin on the far side and the second event shows the downstream advection of the hairpin head shed on the near side.

This topology has been reported on in relation to low Reynolds number shedding about spheres and other axisymmetric bluff bodies in single-phase flows. In liquid–liquid systems (Magarvey & Bishop Reference Magarvey and Bishop1961) using flow visualisation observed alternate vortex shedding from diametrically opposing sides of a descending drop at a Reynolds number of 380. Goldburg & Florsheim (Reference Goldburg and Florsheim1966) repeated the measurements of Magarvey & Bishop (Reference Magarvey and Bishop1961) using solid spheres and other bluff objects and observed regimes with similar modes of alternate hairpin shedding, as described above. Sakamoto & Haniu (Reference Sakamoto and Haniu1995) report similar results for captive models and provide a summary of observed regimes and modes. Such modes have also been predicted by various computations (e.g. by Johnson & Patel Reference Johnson and Patel1999; Sheard, Thompson & Hourigan Reference Sheard, Thompson and Hourigan2004). Indeed the interesting effect of cavitation to create large-scale von Kármán type shedding at transcritical Reynolds numbers about spheres similar to that in single-phase flows at low Reynolds numbers was noted by Brandner et al. (Reference Brandner, Walker, Niekamp and Anderson2010).

As discussed above, for the abundant case in regime II, the supply of activated nuclei increases the frequency of both modes and increases the probability of the high axisymmetric mode compared with the deplete case. The modal topology, however, appears essentially unchanged, although, as the high-speed imaging and data show, shed vortical cavities are not as well defined and of shorter wavelengths than the deplete case due to the effects of discrete bubble activation. The CWT and XWT have been applied to short sequences of the simultaneous surface pressure and high-speed imaging data that correspond to the abundant regime II high and low mode frequencies at $\sigma =0.75$, as shown in figure 26. These data show similar results as for the deplete case (figure 25) in terms of the low mode being out-of-phase and the high mode in-phase. Although the high mode shows greater coherence and the low mode less in the abundant case than for the deplete case, presumably due to the nucleation effects enhancing the high mode, a large number of shedding cycles would need to be analysed to confirm this observation.

Figure 26. The spatio-temporal map from a horizontal line through the centre of the sphere of cloud cavity position (a,b) for the nuclei abundant case at $\sigma = {0.750}$. These time series were selected to exemplify when each of the two frequencies, $f_{aL}^{\rm II}$ (a,c,e,g) and $f_{aH}^{\rm II}$ (b,d,f,h), were dominant. The spatio-temporal map has a horizontal blue line indicating the downstream location of the pressure tap. Sample time series (c,d) of the pressure coefficient for the near (blue) and far (orange) pressure sensors. The filled contours (e,f) are the real value of the wavelet transform of the near pressure signal and in (g), (h) are the XWT between the two sides of the sphere. The arrows indicate the phase difference with an arrow to the right indicating zero phase while an arrow to the left is ${\rm \pi}$ out-of-phase. The horizontal lines indicate the frequency of interest. The time integral of the real value of the XWT is given in $i$. The time-axis scale is constant across the samples and the duration is 0.2 s for each.

In regime III, bi-modal shedding continues but with reduced amplitude and an initial step increase in frequency associated with the transition from attached to detached shedding, as can be seen in figure 14. de Graaf et al. (Reference de Graaf, Brandner and Pearce2017) showed this initial step increase in frequency to be the result of greater cavity growth rates once detached despite greater cavity lengths. The PSD spectrograms (figure 14) and XWT (figure 17) show the bi-modal behaviour but at reduced amplitudes compared with regimes I and II. The CPSD shows inconsistent phase depending on both the cavitation number and the nucleation condition possibly attributable to both low amplitudes and insufficient data. A direct comparison of the PSD for each nucleation case at $\sigma =0.6$ is shown in figure 27. Few differences are apparent with low and high modes having similar frequencies and a slight reduction in amplitude for both modes between deplete and abundant cases. This confirms the earlier observation from the standard deviation of the surface pressures that the cavity volumes are sufficiently large that differences in nucleation have little effect on the shedding behaviour. Observation of the high-speed imaging also confirms this result. The low-frequency mode cavity topology is uncertain due to cavity lengths being greater than the camera field of view.

Figure 27. Power spectral density of the pressure signals in regime III ($\sigma =0.600$) for the deplete (blue) and for the abundant (orange) cases.