3.1. Centre detector

The signals of the centre detector for different experiments at a fixed height are shown in figure 2(a). On the vertical axis the change ${\rm\Delta}l$ of the equivalent path length (i.e. compared with the situation before impact) is plotted during the impact. When the ball passes through the measurement plane the signal of the central detectors drops due to the higher absorption coefficient for X-rays of metal compared with sand. This leads to an increase of ${\rm\Delta}l$ (I). Immediately after the ball passes, the signal ${\rm\Delta}l$ becomes negative (II), indicating that in the path of the ray there is less sand than would fit in the container, e.g. as would happen when an air cavity has formed in the wake of the ball. This implies that there is more air in the sand at this height, but it does not reveal how this air is distributed. The bed may have become very loose such that the air is evenly distributed, or the air may be concentrated in the centre as an air cavity. Some time later a negative peak again suggests the presence of air (III).

Figure 2.

(a) The measured signal of the centre detector as a function of time for four different realizations of an impact experiment using the same measurement height ( $z=8~\text{cm}$ below the surface, $\mathit{Fr}=92$ ). Two of the measurements are recorded with the upper detector bank and two measurements with the lower detector bank. When the ball passes through a ray the signal becomes higher, whereas an air cavity accounts for a lower signal. The first part of the signal, which corresponds to the passing of the ball (I), cavity creation, and cavity collapse (II), is very reproducible. The second part of the signal, corresponding to the rising air bubble (III), shows poor reproducibility. (b) The measured signals of all of the sensors of one detector array measured in a single experiment at one fixed height, $z$ . The number of detectors that see the ball (I) or the air cavities (II and III) provides an estimate of the size of the object. Similar-sized objects that are visible in the signal for a longer time move with a lower velocity through the measurement plane.

The different curves show four distinct realizations of the experiments, all recorded at the same measurement plane in the sand. Two of them are measured with the upper detector row and two of them with the lower array of detectors. The first part is very reproducible, which can be concluded from the fact that the equivalent path lengths and the durations of the peaks are equal. The second part, 200–400 ms after impact, is less reproducible. The measured values are similar, but the shape and timing of the peak are very different among different experiments. From all of the above we can deduce that it must be possible to accurately reconstruct the impact within the sand bed, up to a certain amount of time after the ball has impacted ( $t\leqslant 200~\text{ms}$ ). This time span must at least be sufficient to image the formation of the jet, judging from the time scale ( $t<100~\text{ms}$ ) on which the latter forms.

3.2. Cavity size and shape

Since data are available from an entire array of detectors, it is possible to obtain information about sizes and positions. The signals from the different detectors of one of the arrays are plotted above each other in figure 2(b). The number of sensors that detect the ball (the positive part of the signal) reflects the width of the ball. The negative signals are concentrated in the centre of the container, indicating that the additional air exists in the form of an air cavity rather than somehow dispersed through the sand bed. These negative signals are found immediately after the ball passes (II), showing that the air cavity is attached to the ball. Since a similar number of detectors ‘see’ both the cavity and the ball, the air cavity must have a width similar to that of the ball. The second air cavity (III), the signals of which arrive at the detectors after a considerable delay, is visible in more sensors than the ball, demonstrating a larger size of this second cavity, which can be interpreted as a detached air bubble rising through the sand bed. It should be noted that an object moving at a lower velocity will have a longer X-ray signal duration because it is in the field of view of the detector for longer. The fact that the signal duration of the air bubble is longer than that of the air cavity does not necessarily imply that the air bubble is larger; the magnitude of the signal, however, does give information about the size.

Figure 3.

(a–f) A series of plots at different times ((a) 40 ms, (b) 80 ms, (c) 100 ms, (d) 110 ms, (e) 250 ms, (f) 350 ms) after the ball has impacted onto the sand bed at $t=0$ for $\mathit{Fr}=92$ . The grey-scale value represents the normalized signal. For each plot, the horizontal axis displays the signals of the different detectors and the vertical axis repetitions of the experiment with cross-sections taken at different depths in the bed. We clearly see the cavity first being formed and subsequently closing (a,b), the resulting pinch-off (c), the formation of the jet and finally the entrapment of an air bubble in the sand (d). In the next plot (e), the air bubble detaches from the ball and slowly rises to the surface (f). (g–j) Tomographic reconstruction of a single horizontal cross-section through the bed, see the red dashed line in (a–f), at four different times for $\mathit{Fr}=89$ ((g) 40 ms, (h) 60 ms, (i) 100 ms, (j) 350 ms). From left to right: a measurement plane through the centre of the ball (g), through the air cavity immediately behind the ball (h), the air cavity close to the collapse (i) and a cross-section through the rising air bubble (j).

In figure 3(a–f) the change in equivalent path length is plotted as a grey-scale value (white for positive and black for negative ${\rm\Delta}l$ ) for different times during the experiment. The pixels in each row indicate the signals of the different detectors at a single height, whereas the different rows correspond to experiments carried out at different depths in the bed. This gives a first indication of what happens inside the sand bed. As the ball moves through the sand an air cavity behind the ball is generated (a). This air cavity grows while the ball moves and then starts to collapse under the influence of the lithostatic pressure (i.e. the ‘hydrostatic pressure’ due to the mass of grains above that point) in the bed (b,c). When the walls of the cavity touch (d), a jet shoots upward and an air bubble is entrained. The air bubble moves down with the ball and after it detaches it slowly rises to the surface (e,f). From this analysis it is clear that the events in (a–d) are highly reproducible, whereas the randomness and irregularity in the last two plots (e,f) reflects that the rising of the air bubble is not reproducible at all. The size of the latter varies considerably with height, indicating that in the different experiments to which it corresponds the bubble detaches from the ball at different points in time, leaving empty gaps in the reconstruction. The critical question that remains is to what extent the cavities and resulting jets that are created are axisymmetric, just as in water. To obtain more insight into this issue, we will now look at the full tomographic information that is available from the set-up.

3.3. Tomography

When the container is positioned in the centre of the X-ray set-up, a single horizontal cross-section can be imaged by three detector arrays spaced evenly around the container, allowing for tomographic measurements. Using tomography, we obtain the full 2D shape of the cross-section of the ball and the air cavities. In figure 3(g–j) the tomographic reconstruction at a single height, for four different times during the experiment, is shown. For the tomography the signals of all three detector banks are superpositioned on a square lattice of $140\times 140$ pixels. By applying a threshold intensity value the cavity and the ball shape can be extracted. In the first image (g) the ball is visible, of which the size and shape are known. Indeed, within the limits of the reconstruction the ball is found to be round, and also the size is correctly estimated. In figure 3(h) a reconstruction of the air cavity just after the ball has passed is provided. The air cavity is seen to have a similar degree of roundness to the ball, and has the same size as the ball, which is indeed what one would expect directly after the cavity is created. The second image of the air cavity (i) is taken just before the collapse, showing that the air cavity is still axisymmetric. The last reconstruction (j) is made at the time the bubble passes by. The bubble is not completely circular and is rising slightly off centre. This is one of the origins of the poor reproducibility of the air bubble.

3.4. A single array

By moving the set-up closer to one of the X-ray sources we are able to obtain a much higher spatial resolution. From the tomography analysis there are good indications that the cavity remains axisymmetric at least until the collapse. For such an axisymmetric cavity we can anticipate what the signals in the different detectors should look like, given the radius of the cross-section of the cavity. This is illustrated in the top half of figure 4(a). To quantify the cavity shape as a function of time the cavity radius for every measurement (each height) needs to be determined at each point in time.

Figure 4.

(a) The geometry of one horizontal cross-section of the bed. When the exact locations of the source and detectors with respect to the container are known it is possible to calculate the change of the equivalent path length ${\rm\Delta}l$ as a function of the angle ${\it\alpha}$ for both the case with a circular air cavity (upper half) and the situation in which a sand jet is present in the centre of the air cavity (lower half). In (b,c) the equivalent path length ${\rm\Delta}l$ is plotted as a function of the angle ${\it\alpha}$ for two situations: (b) an air cavity in the sand bed and (c) a jet within the air cavity. Both signals are fitted to the theoretical case of a circular cavity and a circular jet.

In figure 4(b), for a given time and height, the measured equivalent path length is plotted as a function of the angle between the detector and the centre detector when there is an air cavity present. These data can be fitted with the expectation of an axisymmetric air cavity in the centre of the container, as shown in the top half of figure 4(a). The equivalent path length of a circular cavity of radius $r$ as a function of the angle ${\it\alpha}$ is calculated to be

where $J$ is the known distance between the X-ray source and the centre of the container. This function is fitted to the obtained data to get the cavity radius $r$ , plotted as a red line in figure 4(b).

After the collapse a jet occurs inside the air cavity in some of the cross-sections, as illustrated in the bottom half of figure 4(a). This will change the signal as shown in figure 4(c), where we observe a shape similar to that of figure 4(b), but with a pronounced dimple in the centre. To calculate both the cavity radius and the jet radius, (3.1) is adapted such that the change in equivalent path length is calculated through two concentric circles. The larger circle is filled with air and the smaller one is filled with sand: ${\rm\Delta}l(r,{\it\alpha})-{\rm\Delta}l(R,{\it\alpha})$ , where $R$ is the radius of the jet. A fit for both $r$ and $R$ is plotted as a green line in figure 4(c).

This fitting procedure is repeated for every time step and every measurement height, such that we are able to reconstruct the full axisymmetric cavity and jet shape as a function of time. The result of this analysis is shown in figure 5. In blue, the cavity radius as a function of height is represented in the figure at several times after impact. The exact position of the ball could be extracted from the original data by looking at the maximum (see figure 2). The ball is plotted in figure 5 in red. Despite its small scale we were also able to reconstruct the jet that is created during the collapse. The last plot of figure 5 shows the jet in purple.

From the analogy to cavity collapse in a liquid, a secondary downward moving jet is expected to form from the pinch-off point as well (Lohse et al. Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004a ; Bergmann et al. Reference Bergmann, van der Meer, Gekle, van der Bos and Lohse2009). We find no clear evidence for this secondary jet, which may be connected to the fact that it is expected to be weaker than the upward one such that it is simply too thin to measure with the spatial accuracy of our experimental set-up.

Figure 5.

Several snapshots of the results from the analysis described in § 3.4. To obtain these images, the equivalent path lengths measured in the X-ray set-up are fitted to theoretical cavity shapes. Stitching together the experiments executed at different heights results in the blue air cavity. The ball position is also measured from the data and the ball is added to the images in red. For the last image the jet recreated from the data is visible in purple. The plots represent the situation 40, 60, 80, 100 and 120 ms after the impact for $\mathit{Fr}=92$ .

3.5. Cavity collapse

With the analysis described above the cavity radius is extracted as a function of time. It is now possible to take a closer look at the dynamics of the cavity collapse, both at and below the closure depth.

Collapsing air bubbles in incompressible liquids have been studied both theoretically and experimentally (Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006; Gordillo & Pérez-Saborid Reference Gordillo and Pérez-Saborid2006; Duclaux et al. Reference Duclaux, Caillé, Duez, Ybert, Bocquet and Clanet2007; Eggers et al. Reference Eggers, Fontelos, Leppinen and Snoeijer2007; Gekle et al. Reference Gekle, Gordillo, van der Meer and Lohse2009a ,Reference Gekle, Snoeijer, Lohse and van der Meer b ). Theory has shown that the time evolution of the cavity radius asymptotically and slowly converges to a power law with exponent $1/2$ . More specifically, the local slope ${\it\alpha}=\text{d}\ln r/\text{d}\ln t$ has been shown to satisfy ${\it\alpha}\approx 1/2+1/[4\sqrt{-\ln (t_{c}-t)}]$  (Gordillo & Pérez-Saborid Reference Gordillo and Pérez-Saborid2006; Duclaux et al. Reference Duclaux, Caillé, Duez, Ybert, Bocquet and Clanet2007; Eggers et al. Reference Eggers, Fontelos, Leppinen and Snoeijer2007; Gekle et al. Reference Gekle, Snoeijer, Lohse and van der Meer2009b ). In experiments and numerics this leads to data that over several decades appear to be extremely well fitted by a power law (Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006; Gekle et al. Reference Gekle, Snoeijer, Lohse and van der Meer2009b ), with a measured exponent that is slightly larger than $1/2$ .

Is the behaviour similar for a granular pinch-off, and what is the underlying mechanism? In figure 6(a) the cavity radius is plotted as a function of time on a double logarithmic scale, with the time given relative to the closing of the void at $t_{c}$ , i.e.  $t_{c}-t$ .

The behaviour is clearly consistent with a power law during the collapse of the cavity. The best fit exponent (green line) gives $r\sim (t_{c}-t)^{0.66}$ , but due to limited resolution close to pinch-off we are not able to quantitatively compare with values found by Gekle et al. (Reference Gekle, Snoeijer, Lohse and van der Meer2009b ) for the void collapse in water. However, values for the exponent as high as $0.66$ have also been found in liquid impact experiments (Bergmann et al. Reference Bergmann, van der Meer, Stijnman, Sandtke, Prosperetti and Lohse2006), which suggests that the mechanism of cavity collapse in a granular bed is quite similar to that in liquids. More specifically, our findings are consistent with the model of radial cavity collapse initiated by a hydrostatic (or lithostatic) driving pressure and continued by the inertia of the medium, as was already suggested in Lohse et al. (Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004a ).

Figure 6.

(a) Double logarithmic plot of the cavity radius as a function of time at $z=8.0~\text{cm}$ below the surface, which is the closure depth for $\mathit{Fr}=92$ . The cavity is created immediately after the ball has passed and in the beginning has roughly the same radius as the ball. As the cavity is collapsing, the signal decreases. The inset shows the same data in a linear plot. (b) The closing velocity $|{\dot{r}}|$ as a function of depth $z$ , (i) estimated by dividing the distance and the duration of the collapse (starting from the maximum cavity radius, squares) and (ii) calculated from a power-law fit as in (a), evaluated at a radial distance of $r=20~\text{mm}$ from the symmetry axis (triangles). The vertical dashed line marks the pinch-off point.

The final question we address in this section is that of how the cavity collapses away from the point of first closure. To answer this question, in figure 6(b) we plot the absolute value of the radial closing velocity $|{\dot{r}}|$ as a function of the depth $z$ , estimated in two ways. The first method we use is a global one in which we divide the maximum cavity radius by the time interval from the time this maximum is reached to cavity closure (squares in figure 6 b). The second one is local and makes use of power-law fits of the type $r(t)=a(t_{c}-t)^{b}$ for every depth $z$ , from which the velocity ${\dot{r}}$ at an (otherwise arbitrary) fixed distance $r_{0}=20~\text{mm}$ is calculated as ${\dot{r}}=a(b-1)$ $(t_{c}-t)^{b-1}=a(b\!-\!1)[r_{0}/a]^{(b-1)/b}$ (triangles). It should be noted that in this second method, $t_{c}$ , $a$ and $b$ are fitted separately for each value of $z$ . The two methods are consistent in the sense that they give the same trend with depth, but as expected the second methods provides larger values for $|{\dot{r}}|$ since the velocity diverges towards $r=0$ . In any case, clearly, the results of both methods show that $|{\dot{r}}|$ increases slowly with depth.

In the literature there has been some discussion about secondary lower pinch-offs that may be responsible for the thickening of the lower part of the jet (Royer et al. Reference Royer, Corwin, Flior, Cordero, Rivers, Eng and Jaeger2005, Reference Royer, Corwin, Conyers, Flior, Rivers, Eng and Jaeger2008; von Kann et al. Reference von Kann, Joubaud, Caballero-Robledo, Lohse and van der Meer2010). In the context of this discussion our present result indicates that a second – deeper – pinch-off is capable of creating a jet that could be almost as strong and fast as the first pinch-off. This supports the view put forward in von Kann et al. (Reference von Kann, Joubaud, Caballero-Robledo, Lohse and van der Meer2010) that the thick–thin structure first reported in Royer et al. (Reference Royer, Corwin, Flior, Cordero, Rivers, Eng and Jaeger2005) is caused by a secondary jet catching up with the first. By bursting through the primary pinch-off region it is then assumed to create the thick part of the visible jet.

To conclude this section, for the Froude numbers studied in our set-up ( $\mathit{Fr}=9{-}92$ ) we have unambiguously shown that the jet at atmospheric pressures originates from the primary pinch-off point of the cavity, and not from the pressurized air bubble as was suggested in Royer et al. (Reference Royer, Corwin, Flior, Cordero, Rivers, Eng and Jaeger2005) based upon X-ray experiments in a much smaller set-up. In fact, the entrained air bubble is observed to move down with the sphere (see figures 3 d and 5) until it detaches and starts rising through the sand bed. This latter process will be discussed in the next section.

This points to a cavity creation and collapse mechanism that is very similar to that in a liquid (Bergmann et al. Reference Bergmann, van der Meer, Gekle, van der Bos and Lohse2009), for which the cavity is created from the inertia of the impacting object, after which it collapses due to the hydrostatic pressure in the liquid. This results in a trade-off where the upper part of the cavity is created early but the time needed for the collapse is long – due to the small value of the hydrostatic pressure – whereas the lower part of the cavity is created at a much later time but the collapse is fast. This leads to a pinch-off somewhere in the middle of the cavity, where the final collapse is governed by inertia only, leading to a time evolution of the cavity radius that slowly converges to a power law with exponent $1/2$ (Eggers et al. Reference Eggers, Fontelos, Leppinen and Snoeijer2007; Gekle et al. Reference Gekle, Snoeijer, Lohse and van der Meer2009b ). The large pressure created upon collapse on the symmetry axis finally leads to the formation of the jet (Gekle et al. Reference Gekle, Gordillo, van der Meer and Lohse2009a ).

The experimental technique used in this paper opens the possibility of directly observing the formation of the thick–thin structure reported in Royer et al. (Reference Royer, Corwin, Flior, Cordero, Rivers, Eng and Jaeger2005) and von Kann et al. (Reference von Kann, Joubaud, Caballero-Robledo, Lohse and van der Meer2010) at reduced ambient pressure. However, from high-speed imaging experiments in the current set-up it turned out not to be possible to observe these structures at atmospheric pressures, such that additional research at reduced ambient pressures is needed for this purpose.

5.1. Packing fraction after the experiment

The local packing fraction after the experiment in the entire sand bed, calculated with (5.1), is shown in figure 8 a. These measurements were taken several seconds after the ball had come to a halt, which assures that there are no air cavities or bubbles left, and that only packing fraction variations are detected. It should be noted that the packing fraction before the experiment was equal to 0.41, uniformly throughout the container. This means that the bed is compactified during the experiment.

We see a clear compacted region (pink area) next to where the ball has stopped. The packing fraction above the ball (in the centre of the plot) is relatively low, and is lowest just on top of the ball. The packing fraction below the ball slowly decreases with depth back to a value of 0.41. Where do these packing fraction variations originate from, and what do they teach us about the events below the surface?

Figure 8.

(a) A representation of the packing fraction of the bed after an experiment at $\mathit{Fr}=52$ . The data are taken several seconds after the impact events have terminated, assuring that there are no air pockets left in the sand. The colour indicates the packing fraction ( ${\it\phi}$ ). Even though the change in packing fraction is small, a compactified area next to the ball can be observed, while the vertical strip above the ball is relatively loose. The compaction below the ball decreases with depth. (b) The averaged packing fraction plotted as a function of time. To obtain the curves, the signals of 20 different experiments are averaged. The three different curves give the signals from three different detectors, i.e. for three different values of ${\it\alpha}$ . The transparent area around the curves indicates the statistical error. All three signals show a clear increase in packing fraction just before the ball the ball blocks out most X-rays.

Figure 9.

The packing fraction around the ball at several times during impact with a Froude number of 52 ((a)  $-300$  ms, (b) 67 ms, (c) 83 ms, (d) 99 ms, (e) 115 ms, (f) 131 ms, (g) 147 ms). To obtain these images we move along with the ball and average the signals around it. Panel (a) shows the bed before impact (average packing fraction of 0.41). In the other panels the ball moves from 8.5 cm (67 ms after impact) below the surface to 16 cm (147 ms after impact) below the surface. During this movement we observe a compacted region to the sides and in front of the ball (red), a growing compactified region (yellow) and a strong compression above the ball where the air cavity pinches off. It should be noted that the white area does not exactly represent the measured ball shape but simply the area in which no reliable data for density changes in the sand could be calculated due to the presence of the ball.

5.2. Time-resolved packing fraction

To understand the packing fraction of the sand around the ball after the experiment we need to look at the local compaction while the ball is moving through the sand. To determine what happens with the sand just in front of the ball we zoom in on the signal before the ball passes by at a given height. This gives the bed density underneath the moving ball. To obtain sufficient data the signals of 20 different experiments at 10 different heights are averaged. The moment the ball passes by is used to synchronize the signals in time. To smoothen the signal a central scheme is used where the trend of the 10 previous points is extrapolated beyond the central point. In figure 8(b) the result of this analysis is shown. The blue curve ( ${\it\alpha}=0^{\circ }$ ) passes through the centre of the ball and therefore detects the ball first. The other two curves show the signal through the side of the ball ( ${\it\alpha}=3.1^{\circ }$ ) and completely beside the ball ( ${\it\alpha}=6.3^{\circ }$ ). All three curves show a clear increase of the signal before the front of the ball passes. This shows that there is a compaction of the sand just before the ball arrives, or there is a compacted region being pushed in front of the ball. From the red curve ( ${\it\alpha}=6.3^{\circ }$ ) we deduce that there is also compaction next to the ball. This compacted region is still present after the experiment is finished, as can be seen in figure 8(a).

5.3. Packing fraction during penetration

The density differences of the sand during the experiment are very small, but it is possible to detect them if the signal is averaged over a sufficiently large time window (see figure 8 b). To image the packing fraction variations during the penetration around the ball we switch to the frame of reference of the ball such that we are able to perform a time averaging. The result of this procedure is shown in figure 9. The red area below and next to the ball indicates a compacted region, just as we saw in the previous section. In time we see that the compacted region below the ball (yellow area) grows downwards relative to the ball with an average speed of $0.8~\text{m}~\text{s}^{-1}$ and obtains a size that is several times that of the sphere. It is in fact likely that the speed of the compaction front is proportional to that of the sphere as it reflects the compression of the sand in front of the ball. Finally, it should be noted that the measurements presented in this section correspond to a depth-averaged value of the packing fraction since the packing fraction is integrated over the path of the X-rays.

When comparing these findings with the data obtained by Royer et al. (Reference Royer, Corwin, Eng and Jaeger2007, Reference Royer, Conyers, Corwin, Eng and Jaeger2011), who reported a much smaller compacted region in front of the ball at atmospheric pressures for comparable Froude numbers, we may attribute the difference to the higher sensitivity of our set-up to packing fraction changes.

In addition to what happens below the ball, we can also investigate the compaction above the ball. First, the air cavity is visible (blue), and when the cavity collapses a growing red area indicates a compacted region next to the pinch-off. The data in figure 8(a) (taken after the experiment was carried out) show a relatively uncompacted area in the centre above the ball. This must be connected to the rising bubble rearranging the sand particles in its path. It suggests that the sand at the bottom of the bubble is deposited loosely, pointing to a slow and unpressurized mechanism.

Physically, this all points to the following series of events. During the impact the sand is compacted in front of the ball (figure 9, as expected from earlier work (Royer et al. Reference Royer, Corwin, Eng and Jaeger2007, Reference Royer, Conyers, Corwin, Eng and Jaeger2011)), whereas the expansion and contraction of the cavity does not induce significant packing fraction changes (figure 9 b–d). This stands to reason since during the contraction the sand is essentially in free fall (i.e. not supported by the cavity walls). As soon as the cavity closes, however, there is a large pressure buildup on the symmetry axis which causes the sand to compact (figure 9 e–g). This compaction is partially undone by the surfacing air bubble, which in its wake creates a loosely packed region around the symmetry axis (figure 8 a).