3.2.1 Studies on flow past spheres

Explaining the helical nature of the cavity wake observed requires an understanding of the flow structures produced around and in the wake of a Leidenfrost sphere in contact line cases. However, given the complexity of the problem and the fact that no study has investigated this phenomenon so far, one can form an analogy with the much-studied flow structures produced by a fully submerged sphere in a homogeneous fluid stream for clarification and inspiration. The contact line and gaseous wake of a Leidenfrost sphere in this respect are relevant to the boundary layer separation at the trailing edge of a fully submerged sphere and the recirculation zone present in its wake respectively. We will therefore outline the many studies on this subject and how they relate directly to the observations made in this study.

Figure 8. (a) Images at 5 ms intervals of a helical cavity wake created by the impact of a $D_{0}=30$  mm steel sphere from $h_{r}=100$  cm. The first image is taken 138 ms after impact at a depth of $z=57$  cm. The ridges noted on the cavity surface outline and form three distinct interfacial regions which rotate synchronously in a clockwise direction. The yellow stars (at $z=59$  cm) mark the location of one specific ridge as it advects downstream in the sphere wake. (b) Red, blue and green colour maps superimposed individually on each of the three rotating interfacial sections. Here, $U_{0}=4.43~\text{m}~\text{s}^{-1}$ , $Fr=133$ , $St=0.27$ . See also supplementary movie 3.

Even though many studies have investigated the flow past a sphere in a homogeneous fluid, there still exists some disagreement on the existence of different instability modes with respect to increasing Reynolds number regimes. For $Re<24$ (Taneda 1956), a laminar flow is observed with no separation from the sphere surface, while for $Re=24$ – $200$ (Nakamura Reference Nakamura1976), the flow recirculates in a toroidal vortex behind the sphere in an axisymmetric and steady manner. When $Re=211{-}270$ (Johnson & Patel Reference Johnson and Patel1999), the flow remains steady but loses its axial symmetry as the toroidal vortex gets tilted, shifting the recirculating region to one side. The flow, however, does preserve a planar symmetry, and a pair of streamwise vortices are noted to extend downstream without shedding. For a higher regime of $Re=280$ –375, the flow assumes an unsteady planar-symmetric nature whereby the vortical structure is composed of hairpin vortices being shed in a periodic manner with a constant orientation.

Figure 9. Strouhal number versus Reynolds number for the shear-layer (high-frequency) and spiral (low-frequency) instabilities from past studies in comparison with the present results.

For $Re=400$ – $800$ , a distinct spiral-instability mode associated with large-scale vortex shedding is observed (Chomaz et al. Reference Chomaz, Bonneton and Hopfinger1993), which is related to the azimuthal rotation of the recirculation zone and hence of the separation line. The spiral mode is reported as being locked with the Kelvin–Helmholtz instability in this regime since the Strouhal number (a dimensionless frequency measure) of each instability varies identically with the Reynolds number. The wake structure corresponding to this phenomenon has been described as a spiral of vortex loops emitted regularly behind the sphere, whereby the vortex shedding occurs sinuously, resulting in a progressive wave-like motion of wavelength $\unicode[STIX]{x1D706}$ . For $Re>800$ , the two instabilities become unlocked while coexisting simultaneously (Möller Reference Möller1938; Cometta Reference Cometta1957; Kim & Durbin Reference Kim and Durbin1988; Sakamoto & Haniu Reference Sakamoto and Haniu1990) until the maximum investigated value of $Re=6\times 10^{4}$ . This is represented by a bifurcation of their frequencies (see figure 9) following $St=fD_{0}/U_{\infty }\sim Re^{1/2}$ for the small-scale Kelvin–Helmholtz instability (high-frequency mode, where $f$ is the instability frequency and $U_{\infty }$ is the undisturbed flow velocity) and $St=D_{0}/\unicode[STIX]{x1D706}\sim 0.2$ for the large-scale spiral instability (low-frequency mode).

The spiral-instability mode was first speculated by Achenbach (Reference Achenbach1974), who recorded simultaneous signals from four hot-wire probes attached to the sphere surface. For $Re=6\times 10^{3}$ – $3\times 10^{5}$ , the signals showed a phase shift, which suggested that the vortex separation occurred at a point that rotates around the sphere periodically with the vortex shedding frequency ( $St=0.125$ – $0.2$ ), to possibly produce a helical wake configuration. This phenomenon was not found to occur beyond the upper critical Reynolds number $Re>3.7\times 10^{5}$ (Achenbach Reference Achenbach1972), i.e. in the supercritical regime, whereby the boundary layer transitions from laminar to turbulent flow, shifting downstream in the process to result in a sharp decrease in the drag coefficient. Moreover, no signal was recorded below $Re=6\times 10^{3}$ , which is much higher than the spiral-instability onset of $Re=400$ noted by Chomaz et al. (Reference Chomaz, Bonneton and Hopfinger1993). In contrast, Sakamoto & Haniu (Reference Sakamoto and Haniu1990), who conducted further experiments utilizing hot-wire measurements and flow visualization, showed that the shedding point of large-scale vortices begins to rotate slowly and irregularly about the streamwise axis of the sphere, starting at $Re=480$ . A similar observation was also reported in the work of Taneda (Reference Taneda1978) for $Re=10^{4}$ – $3.8\times 10^{5}$ with $St\approx 0.2$ in an investigation regime of $Re=10^{4}$ – $10^{6}$ . In comparison to these experimental studies where the spheres were kept fixed with respect to the moving fluid stream, Magarvey & Bishop (Reference Magarvey and Bishop1961) investigated wake structures behind falling spheres, as also used in this study. Their results for Strouhal numbers calculated from the vortex shedding frequencies in the range $Re=350$ – $500$ (see figure 9) are not found to be in good agreement with fixed sphere studies. This has been reasoned to be due to the ability of the falling spheres in reacting to the changing flow forces by displacement, which can result in different hydrodynamic forces and wake formation in comparison to fixed sphere cases.

A number of numerical studies have also investigated flow over spheres to further understand the vortex shedding mechanisms observed in experiments. Tomboulidesn, Orszag & Karniadakis (Reference Tomboulidesn, Orszag and Karniadakis1993) performed direct numerical simulations (DNS) and large eddy simulations (LES) to show that two distinct instability frequencies are indeed present at $Re=1000$ and $2\times 10^{4}$ . More recently, Constantinescu & Squires (Reference Constantinescu and Squires2004) investigated flows in both subcritical and supercritical regimes spanning $Re=10^{4}$ – $10^{6}$ , using detached-eddy simulations. For subcritical flows, the wake was shown to develop a helical-like vortical structure as the azimuthal angles at which the vortices were formed and shed varied strongly with time. In supercritical cases, the vortices were shed periodically with a single dominant frequency of $St=1.3$ while appearing to be locked at certain azimuthal locations for long intervals of time. This was in contrast to the experimental results of Taneda (Reference Taneda1978) and Achenbach (Reference Achenbach1974), who did not notice any periodic vortex shedding beyond the critical Reynolds number. A helical-like wake configuration was also obtained by Rodríguez et al. (Reference Rodríguez, Borell, Lehmkuhl, Perez Segarra and Oliva2011, Reference Rodríguez, Lehmkuhl, Borrell and Oliva2013) in the subcritical regime ( $Re=3700,10^{4}$ ) using DNS, which was shown to result from the shedding of large-scale vortices ( $St=0.215$ ) at random azimuthal locations in the shear layer. The vortices, however, moved downstream without any azimuthal circulation, indicating that the helical pattern formed primarily from the way in which the vortices were shed in time. The LES results of Yun, Kim & Choi (Reference Yun, Kim and Choi2006) also showed the vortical structures to follow nearly straight paths downstream. They suggested the observed helical structure to be closely related to changes in the azimuthal wall-pressure distribution of the sphere with time. The wall pressure was shown to convect azimuthally at an almost constant velocity to produce a tilted vortical structure in time and, consequently, a helical wake configuration.

Similarly to these studies, the azimuthal locations of ridges in the helical cavity wakes formed in our study are noted as being fixed with respect to the laboratory frame as they advect downstream (e.g. see the yellow marker in figure 8 a). This indicates that the helical cavity structure arises essentially from the rotation of ridges about the sphere surface. Using visual measurements to obtain the ridge rotation frequencies $f_{r}$ , the corresponding Strouhal numbers $St=f_{r}D_{0}/U_{i}$ (where $U_{i}$ is the instantaneous sphere velocity) are found to range between $St=0.22$ and $0.30$ for $Re\equiv Re_{i}=(\unicode[STIX]{x1D70C}U_{i}D_{0})/\unicode[STIX]{x1D707}=1.4\times 10^{5}$ – $2.4\times 10^{5}$ (see figure 9). Even though the spheres in this study are trailed by a gaseous cavity, where a recirculation zone is present otherwise for spheres in a homogeneous fluid, these Strouhal numbers are still close to $St\approx 0.2$ reported for the spiral-instability mode in homogeneous flow cases (Achenbach Reference Achenbach1974; Taneda Reference Taneda1978). Since the flow past spheres for the range of Reynolds numbers employed here has been shown by these studies to comprise a spiral-instability mode, it is possible for the corresponding large-scale flow structures produced around the sphere to affect the shape of the cavity wake and hence its shedding characteristics accordingly. Although one might expect the flow forces resulting from the spiral instability to bring about rotation in the free falling sphere itself, repeating experiments with marked spheres and front lighting showed no indication of spinning in the close-up videos recorded.

Figure 10. (a) Images with front lighting showing the early stages of penetration of a $D_{0}=30$ mm steel sphere in the formation of a helical cavity wake. (b) Close-up images of the right hemisphere for $t=4$ –7 ms from impact revealing Kelvin–Helmholtz (K–H) instabilities along the liquid–vapour interface. The K–H billows are further magnified in the insets and marked with black arrows in the image at $t=4$  ms. The contact line is indicated by grey arrows. Here, $U_{0}=4.2~\text{m}~\text{s}^{-1}$ , $Fr=120$ .

3.2.2 Analysis of initial sphere penetration

In order to further understand the origin and formation of the helical cavity wake, figure 10 presents a series of close-up images with front lighting showing the early stages of penetration of a $D_{0}=30$  mm steel sphere impacting at $4.2~\text{m}~\text{s}^{-1}$ . The images in figure 10(a) show the presence of a contact line around the sphere equator which results in nucleate boiling, similar to the observations of Marston et al. (Reference Marston, Vakarelski and Thoroddsen2012). However, instead of shedding the entire contact line backwards, the region of contact enlarges longitudinally at distinct circumferential locations along the sphere equator. One such location appears as an increasingly elongating grey area ( $t\geqslant$ 9 ms) above the bright spot on the sphere surface created by the reflection of lighting. The elongated contact area draws a stream of liquid into the cavity along the upper hemisphere surface of the sphere, which forms a column of liquid in its wake, and at the same time pulls the cavity wall inwards to form a ridge. This pulling effect also becomes apparent on the contact line itself as it assumes an inverted V-shape from localized backward movement, where the ridge originates. The formation of the liquid column, the ridge and the inverted V-shaped contact line feature hence stems essentially from the elongated contact area formed at the primary contact line, which is noted to begin a clockwise rotation shortly afterwards at $t=15$  ms.

The question then arises as to how these rotating elongated contact areas develop and also why a contact line is observed in the first place, even though the sphere is encapsulated by a vapour layer. From the close-up images shown in figure 10(b), a shear-layer K–H instability is readily observed at the liquid–vapour interface formed around the bottom hemisphere of the Leidenfrost sphere. The presence of a boundary beside the shear layer, as in this study, has been shown in theory (Hazel Reference Hazel1972) to enhance the destabilization of stratified shear flows, depending markedly on the proximity of the boundary to the shear layer. This suggests that the thickness of the vapour layer $\unicode[STIX]{x1D6FF}_{l}$ , in addition to velocity shear, plays a crucial role in the formation of the K–H instability observed in helical cavity cases. Although internal waves radiating energy from the shear-layer interface do not occur in two-layer stratified systems as in this study (Xuequan & Hopfinger Reference Xuequan and Hopfinger1986; Strang & Fernando Reference Strang and Fernando2001), it is plausible that the continuous influx of vapour at the liquid–vapour interface acts as a destabilizing factor in this regard. The instability manifests itself by overturning the interface to form several billows (Corcos & Sherman Reference Corcos and Sherman1976) along the boundary of the sphere (see the black arrows in the magnified inset at $t=4$  ms). These degenerate into turbulence further downstream at the sphere equator, mixing liquid into the vapour layer and hence enabling the formation of a contact line (see the grey arrows). Since the large-scale vortex shedding associated with the spiral instability appears to occur at a point that rotates around the sphere periodically (Achenbach Reference Achenbach1974, $St\approx 0.2$ at $Re\approx 2\times 10^{5}$ ), it is most likely that its presence aggravates this mixing at distinct azimuthal locations in a rotating manner. This can explain the formation of the elongated contact areas, and hence the ridges, which furthermore are noted to rotate at similar non-dimensional frequencies ( $St=0.22$ – $0.30$ , $Re\equiv Re_{i}=1.4\times 10^{5}$ – $2.4\times 10^{5}$ ; see figure 9).

Figure 11. (a) Images with front lighting showing the formation of a dual-cavity structure when a $D_{0}=25$  mm tungsten carbide sphere impacts at $U_{0}=4.2$ $\text{m}~\text{s}^{-1}$ . Here, $Fr=144$ . (b) Close-up images of the contact region for $t=5$ – $14$  ms from impact showing the contact line shedding event and the pacification of the K–H billows thereafter. (c) Edge enhancement of the cavity interface for $t=5$ – $26$  ms. The contact line is indicated by the black arrow. The dashed curve represents the surface of the sphere.

Figure 12. Unstable, stable and helical cavity wake regimes in the $Fr$ – $Re_{0}$ parameter space for (a) steel and (b) tungsten carbide spheres. Here, $D_{0}=10$ – $40$  mm,  $Re_{0}=2.7\times 10^{4}$ – $2.6\times 10^{5}$ , $Fr=33.3$ –400. The dual-coloured symbols represent a bistability where two different cavity wake types can coexist separately for the same impact parameters.

Similarly to figure 10, close-up images with front lighting showing the formation of a dual-cavity structure are presented in figure 11. The images in (a) have been magnified further near the contact region in (b) and, additionally, enhanced using an edge-detection MATLAB routine in (c). The contact line noted around the equator at $t=5$  ms in this case (see the black arrow in c) is shed backwards as the sphere descends further down with time, discontinuing the local nucleate boiling. Since the elongated contact areas associated with the primary contact line are also shed in the process, the formation of liquid columns also stops, and these later appear as falling streams inside the cavity (see figure 11(a), $t=14$  ms). The images in (b) and (c) reveal the K–H billows forming at the vapour layer interface to pacify with time until a smooth liquid–vapour interface is observed at $t=26$  ms. Since the contact line is shed much earlier at $t=6$  ms, when the billows are still very obvious, it appears that this phenomenon results due to the contact line shedding event, rather than vice versa.

3.2.3 Instability in stratified shear flows

Studies on the instability of stratified shear layers can be traced back to the works of Goldstein (Reference Goldstein1931) and Taylor (Reference Taylor1931), who analytically investigated unbounded parallel inviscid shear flows. Taylor (Reference Taylor1931) conjectured that a stable flow in such circumstances requires the gradient Richardson number $Ri_{g}=N_{L}^{2}/(\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z)^{2}$ (where $N_{L}$ is the Brunt–Väisälä frequency and $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ is the local shear) to exceed a value of 0.25 everywhere. This was proved in a theorem by Miles (Reference Miles1961), whereby the velocity and density were assumed to follow monotonic profiles. Howard (Reference Howard1961) simplified Miles’s theorem such that it did not require such hypotheses, which has come to be known as the classical Miles–Howard theorem. Shear flows in reality, however, are not only bounded but can also be influenced by viscous stresses, such that the minimum critical $Ri_{g}$ required for instability can vary significantly. A plethora of studies have investigated stratified shear flow in this regard, which can be found in the extensive reviews of Fernando (Reference Fernando1991) and Peltier & Caulfield (Reference Peltier and Caulfield2003).

A notable work among these studies by Strang & Fernando (Reference Strang and Fernando2001) investigated the entrainment and mixing in a stratified shear layer, separating an upper lighter layer (of thickness $D$ ) from a deep lower layer of homogeneous nature (two-layer case). For large Reynolds numbers ( $Re\sim 10^{4}$ ), the flow was shown to be governed primarily by the bulk Richardson number $Ri_{B}=\unicode[STIX]{x0394}bD/\unicode[STIX]{x0394}U^{2}$ (a parameter related to the average gradient Richardson number $\overline{Ri_{g}}$ , where $\unicode[STIX]{x0394}U$ is the velocity difference and $\unicode[STIX]{x0394}b=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}$ , with $\unicode[STIX]{x1D70C}_{0}$ as the density of the upper layer, is the buoyancy jump across the shearing interface). Kelvin–Helmholtz instabilities were noted to occur predominantly for $Ri_{B}<3.2$ ( $\overline{Ri_{g}}<0.39$ ) and subsided completely for $Ri_{B}\gtrsim 5$ ( $\overline{Ri_{g}}\gtrsim 1$ ). For $Ri_{B}<1.5$ ( $\overline{Ri_{g}}<0.09$ ), the entrainment rate was found to be independent of $Ri_{B}$ (or $\overline{Ri_{g}}$ ), indicating entrainment to occur as if stratification did not exist. Turbulent eddies in the shear layer were strong enough in these cases to scour the denser liquid against buoyancy forces. Furthermore, the turbulence properties in the mixed layer, such as the integral length scales and root mean square (r.m.s.) of velocity fluctuations, were found to be dependent on $\unicode[STIX]{x0394}U$ and $D$ , whereby the upper part of the layer was maintained in a turbulent state by wall-induced turbulence.

The stratified turbulent shear flow problem in this study is analogous to the two-layer case investigated by Strang & Fernando (Reference Strang and Fernando2001) in the sense that the vapour layer (of thickness $\unicode[STIX]{x1D6FF}_{l}$ ) and the PP1 fluid form the upper and lower parts of the mixed layer respectively in a rotated view frame about the sphere equator. Since the bulk Richardson number at the vapour–liquid interface in the initial stages of sphere descent ( $t=5$ – $6$  ms in figure 11, $Re_{i}\sim 8\times 10^{4}$ ) follows $Ri_{B}=\unicode[STIX]{x0394}b\unicode[STIX]{x1D6FF}_{l}/\unicode[STIX]{x0394}U^{2}\sim O(0.01)$ (where $\unicode[STIX]{x1D6FF}_{l}\approx 100$ – $200~\unicode[STIX]{x03BC}\text{m}$ at the sphere equator from Vakarelski et al. (Reference Vakarelski, Marston, Chan and Thoroddsen2011)), inadequate entrainment in the vapour layer itself is not expected to cause the contact line to shed backwards. It therefore appears that the thickness of the vapour layer $\unicode[STIX]{x1D6FF}_{l}$ formed after a time scale of one diameter penetration in the present case, i.e. $tU_{0}/D_{0}=1$ , becomes sufficiently large to prevent the entrained liquid in the mixed layer from maintaining contact with the sphere surface. In other words, $\unicode[STIX]{x1D6FF}_{l}<\unicode[STIX]{x1D6FF}_{m}$ at the equator for $t\leqslant 5$  ms, where $\unicode[STIX]{x1D6FF}_{m}$ corresponds to the mixed-layer thickness at the vapour–liquid interface. As $\unicode[STIX]{x1D6FF}_{l}$ is primarily dependent on the sphere temperature, velocity and inertial flow forces, it is understandable why the dual-cavity observations made by Marston et al. (Reference Marston, Vakarelski and Thoroddsen2012) were reported to be a sensitive function of $T_{s}$ and $U_{0}$ for a given sphere size.

4.3.1 Spline interpolation method

From the total hydrodynamic force data obtained in section (§ 4.2), the drag force $F_{d}$ resisting the motion of the sphere and the attached cavity wake can be obtained using

The effect (force) of added mass $m_{a}$ from accelerating the surrounding fluid can be expressed as $F_{a}(t)=m_{a}\ddot{z}(t)=C_{m}V_{c}\unicode[STIX]{x1D70C}\ddot{z}(t)$ , where $C_{m}$ is the added mass coefficient and $V_{c}$ is the combined volume of the sphere and cavity wake. Although the magnitude of $V_{c}$ varies with time due to cavity shedding, and $C_{m}$ in turn depends on the flow pattern around the sphere and trailing cavity (Berklite Reference Berklite1972; Sarpkaya Reference Sarpkaya1976), a constant mean value for the added mass obtained from $m_{a}=0.672\unicode[STIX]{x1D70C}[(1/6)\unicode[STIX]{x03C0}L_{c}D_{c}^{2}]$ for streamlined bodies (Dong Reference Dong1978) is assumed over the entire descent. The added mass for spheres falling without cavity attachments is simply given by $m_{a}=C_{m}V\unicode[STIX]{x1D70C}$ , where $C_{m}$ takes a constant value of $0.5$ .

Since the cavity adjoins the sphere at or close to its equator, the total buoyancy force $F_{b}$ due to hydrostatic pressure before cavity pinch-off (i.e. for an open sphere–cavity system exposed to the atmosphere) is given by $F_{b}(t)=\unicode[STIX]{x1D70C}(2/3)\unicode[STIX]{x03C0}R_{0}^{3}g+\unicode[STIX]{x1D70C}g[z(t)-R_{0}]\unicode[STIX]{x03C0}R_{0}^{2}$ . Following the pinch-off event, a closed sphere–cavity system completely surrounded by the fluid is formed whereby $F_{b}=\unicode[STIX]{x1D70C}V_{c}g$ . The combined volume of the sphere and the attached cavity wake $V_{c}$ is obtained by tracing their boundary around the vertical $Z$ -axis using spline interpolation (see figure 16 a–c), approximating it using a piecewise-linear curve to form a closed polygonal chain and, finally, revolving the polygonal chain about the $Z$ -axis of symmetry to give the volume of revolution. In mathematical terminology, if the sphere–cavity boundary curve closed around the $Z$ -axis is parameterized as $\langle R(s),Z(s)\rangle$ for $0\leqslant s\leqslant 1$ , then the volume of revolution about the $Z$ -axis can be expressed as (Wilson, Turcotte & Halpern Reference Wilson, Turcotte and Halpern2003)

The $V_{c}$ values corresponding to unstable, stable and helical cavity wake cases produced by the impact of $D_{0}=15$  mm steel and $D_{0}=25$  mm tungsten carbide spheres are computed using (4.4) at 20 ms time steps and plotted in figure 16(d,e); see the caption for details. Since the splines (in red) are interpolated between manually drawn boundary points (in yellow), the volume $V_{c}$ is subjective to how the cavity shape is perceived at each time step. For example, $V_{c}$ values obtained over repeated measurements for the stable–streamlined cavity of an apparently constant volume in figure 16(d) (see the bracketed area) are found to have a mean and mean absolute deviation (MAD) of $7.29~\text{cm}^{3}$ and $0.12~\text{cm}^{3}$ respectively. These results are found to be in close agreement with the true volume $V_{c}=7.31~\text{cm}^{3}$ (see the methodology of computation in § 4.3.2) and have a mean absolute percentage error (MAPE) of 1.7 %. The total error in volume measurement for unstable and helical wake cases, however, is higher due to cavity asymmetry and more difficult to pin down.

Figure 16. Cavity interfaces traced around the vertical $Z$ -axis (purple line) using spline interpolation (in red) between manually drawn boundary points (in yellow) for (a) an unstable wake case, $D_{0}=15$  mm,  $\unicode[STIX]{x1D71A}=4.54$ ,  $U_{0}=2.21~\text{m}~\text{s}^{-1}$ ,  $Fr=67$ (b) a stable wake case, $D_{0}=15$  mm,  $\unicode[STIX]{x1D71A}=4.54$ ,  $U_{0}=2.97~\text{m}~\text{s}^{-1}$ ,  $Fr=120$ , and (c) a helical wake case, $D_{0}=25$  mm,  $\unicode[STIX]{x1D71A}=8.70$ ,  $U_{0}=4.20~\text{m}~\text{s}^{-1}$ ,  $Fr=144$ . (d) The total volume of the sphere and cavity $V_{c}$ versus time for the realizations in (a,b). (e) Plot of $V_{c}$ versus $t$ for unstable and stable cavity wakes produced by the impact of $D_{0}=25$ mm tungsten carbide spheres for $U_{0}=3.57~\text{m}~\text{s}^{-1}$ , $Fr=104$ and $U_{0}=4.20~\text{m}~\text{s}^{-1}$ , $Fr=144$ respectively and for the helical wake case in (c). The horizontal dashed lines correspond to times whereby the cavity has been completely shed off, i.e. $V_{c}=(4/3)\unicode[STIX]{x03C0}R_{0}^{3}$ .

From the plots shown in figure 16(d,e), the volume of air entrained by the helical wake upon pinch-off is computed to be significantly smaller (by ${\approx}60\,\%$ ) than the stable wake counterpart for the same impact parameters. The $V_{c}$ values for unstable wake cases generally decline at the steepest rate in a stepwise manner until the sphere volume $V$ (see dashed line) is reached upon completion of the shedding process. The buoyancy then becomes constant and can be obtained directly from $F_{b}=(4/3)\unicode[STIX]{x03C0}R_{0}^{3}\unicode[STIX]{x1D70C}g$ .

Figure 17. Drag coefficients for the unstable, stable and helical cavity wake cases produced by (a) $D_{0}=15$ mm steel and (b) $D_{0}=25$  mm tungsten carbide spheres in figure 16(d,e). The black filled, colour filled and unfilled symbols represent $C_{D}$ values before pinch-off, after pinch-off and after cavity detachment respectively.

Using equations (4.1) and (4.3) together with the relevant expressions for $F_{a}$ and $F_{b}$ applicable before pinch-off, after pinch-off and after cavity detachment discussed above, the general equation for the drag coefficient given by

can be expressed for each respective phase as

Drag coefficients corresponding to the unstable, stable and helical wake cases in figure 16 are computed using (4.6)–(4.8) for any such given phase observed (see the caption for details) and plotted versus time in figure 17. The plots reveal the $C_{D}$ values to follow a declining trend after cavity pinch-off in the former two cases. While the drag coefficients for stable wakes appear to approach a steady value with time (e.g. mean $\overline{C_{D}}=0.051$ in (a) corresponding to the bracket in figure 16 d), the coefficients for unstable wakes begin to increase rapidly at $t\approx 125$ ms until they reach a maximum. This dramatic increase stems from obtaining comparatively higher $C_{F}$ values (see figure 15) and primarily from significantly lower buoyancy force coefficients $C_{B}=F_{b}/(1/2)\unicode[STIX]{x1D70C}[{\dot{z}}(t)^{2}]\unicode[STIX]{x03C0}R_{0}^{2}$ computed for unstable wakes due to incessant cavity shedding. The effect of sharply declining $C_{F}$ values shortly before ( ${\approx}40$  ms) the shedding process is completed then becomes the dominating factor and accordingly causes the $C_{D}$ curves to undergo steep declines. Although the drag coefficients for the helical wake sphere increase steadily with time, the values obtained in the initial stages of descent (e.g. at $t=25$ ms, i.e. before pinch-off) are noted to be significantly smaller (by ${\approx}75\,\%$ ) in comparison with those obtained for its stable wake counterpart. Given that non-cavity-forming cases tend to have even higher force coefficients than cavity-forming cases during free-surface entry (Truscott et al. Reference Truscott, Epps and Techet2012), these results indicate helical cavity wake cases to produce an ideal configuration for the free-surface entry of spheres.

Figure 18. Magnified image of a stable–streamlined cavity produced by the impact of a $D_{0}=10$  mm steel sphere at $U_{0}=4.2$ $\text{m}~\text{s}^{-1}$ , $Fr=360$ . The inflow section is described by an ellipse and the outflow section by a parabola which are superimposed on the image over lengths $L_{e}$ and $L_{r}$ respectively.

4.3.2 Algebraic curve fitting method

Since the stable–streamlined cavity wakes formed in this study have well-defined shapes which remain consistent during sphere descent, more accurate drag coefficients for these cases can be obtained by utilizing high-magnification imaging and computing the corresponding $V_{c}$ volumes by characterizing their profiles using algebraic curves (to eliminate the element of manual input in boundary tracing). The sphere–cavity configuration in this regard can be modelled as a streamlined body of revolution consisting of an ellipse forebody and a parabola afterbody (Chen et al. Reference Chen, Xia, Liu and Wang2010). The ellipse inflow section is expressed as

and the parabola outflow section is given by

where $L_{e}$ is the inflow section length, $L_{r}$ is the outflow section length (see figure 18) and $L_{c}=L_{e}+L_{r}$ is the total cavity length. By performing disk integration on each section over its respective length, the total volume of the stable–streamlined configuration can be given as

Given that the spheres approach a terminal velocity $U_{t}$ in stable cavity wake cases, the steady-state drag coefficients can finally be obtained using

These are presented along with the corresponding Reynolds numbers $Re=(\unicode[STIX]{x1D70C}U_{t}D_{c})/\unicode[STIX]{x1D707}$ in table 2 for different sizes of steel and tungsten carbide spheres.

The results obtained for stable cavity wake cases produced by steel spheres range between $C_{D}=0.038$ and $0.043$ for $Re=1.9\times 10^{4}$ – $7.1\times 10^{4}$ and those produced by tungsten carbide spheres range between $C_{D}=0.021$ and $0.024$ for $Re=3.1\times 10^{4}$ – $8.3\times 10^{4}$ . If the sphere cross-sectional area is used to non-dimensionalize the drag coefficient for comparison purposes with results in § 4.3.1, a $C_{D}$ value of 0.049 is obtained for the $D_{0}=15$  mm steel sphere stable cavity configuration. This is in agreement with the range of values and mean computed in figure 17(a) in the corresponding steady-state regime ( $C_{D}=0.037$ – $0.065$ , $\overline{C_{D}}=0.051$ ), which shows the $V_{c}$ calculation procedure used therein to provide a good estimate for $C_{D}$ values.

Comparing results with past studies, the drag coefficients obtained for the streamlined cavity configuration are an order of magnitude smaller than those reported for a solid room-temperature sphere, $0.43<C_{D}<0.5$ (Achenbach Reference Achenbach1972), and up to ${\approx}80\,\%$ smaller than the low limiting value for a simple Leidenfrost vapour layer sphere, $C_{D}=0.1$ (Vakarelski et al. Reference Vakarelski, Marston, Chan and Thoroddsen2011), for the same range of pre-crisis Reynolds numbers as investigated in this study. This diminution in $C_{D}$ is attributed to the combined effect of a reduced form drag (or pressure drag), resulting from the streamlined shape of the sphere–cavity configuration, and a lowered skin-friction drag (or viscous drag) due to the presence of a partial-slip boundary condition at the vapour layer next to the solid sphere (Vakarelski et al. Reference Vakarelski, Berry, Chan and Thoroddsen2016). While form drag is the dominant source of drag for spheres, or more generally bluff bodies at high Reynolds numbers $Re=10^{4}$ – $10^{6}$ , both form and skin-friction drag are expected to have a notable contribution towards the total drag experienced by streamlined bodies. Determining the precise contribution of the streamlined sphere–cavity shape and the partial-slip boundary condition in this regard towards lowering the effective drag coefficient, however, is beyond the scope of this work and will require further investigative studies in the future.