3.1. Experimental observations
Experimental observations of crown dynamics and underwater cavity by free falling thick disks with different impact energies and aspect ratios are shown in this section. The effect of impact kinetic energy, Ei, was studied by variations of disk mass (i.e., a combination of disk’s density and volume) and release height. The images of crown and cavity formation of six cylindrical thick disks with a constant impact velocity induced from a release height of hr = 500 mm (i.e., hr/do = 6.56) and with different aspect ratios and densities are shown in Fig. 2. All images were recorded at their pinch-off time. Each column in Fig. 2 represents different disk densities, and each row shows disks with different aspect ratios. The geometrical characteristics of crown and cavity such as crown height, Hc, crown diameter, Dc, pinch-off depth, Dp, and frontal position, Df, of disks are shown in Fig. 2. As can be seen, the volume of the penetrated airflow increased with increasing impact velocity and air is drawn into water in the wake of the disk. Therefore, the underwater cavity in disks with relatively higher impact energy, as a result of higher disk mass, formed a greater cavity than the lighter disks. As can be seen in Fig. 2a, the disk with the lowest density, ρd/ρw = 1.080, and highest aspect ratio (i.e., χ = 3), formed smooth and partially sealed crown. As the impact energy of disks increased, the crown was sealed, and the crown surface became rougher as micro ripples were formed on the crown surface. Moreover, more water droplets fell inside the cavity and cavity distortion occurred as the impact energy of disks increased. The crown wall was almost vertical in Fig. 2a (i.e., Test No. 10, ρd/ρw = 1.080, χ = 3, Ei/Eo = 0.219), and it gradually curved as the disk impact energy increased (see Figs. 2b and 2c). Moreover, the thickness of disks increased the mass and accordingly increased the impact energy. It was observed that by increasing the thickness of disks, the slope of crown wall became more inclined towards the center of the crown due to high pressure drop inside the crown.
A careful observation of snapshot images indicated that pinch-off depth, Dp, was marginally changed by disk impact energy whereas the frontal position of disks increased considerably. Such difference in the frontal position of submerged disks was detected and can be seen in Figs. 2b and 2c. A conical air pocket was formed on the top of heaviest disk (see Fig. 2c) whereas such conical air pocket did not form in relatively lighter disks. Formation of a conical air pocket in disks with the highest density of ρd/ρw = 2.335 was independent of the disk aspect ratio as it is shown in Figs. 2c and 2f.
The images in the first and second rows of Fig. 2 show the effect of disk aspect ratio on crown and cavity formations. The mass of disks in the second-row images was double the first-row images and the additional mass (i.e., impact energy) dragged more air into the ambient water. The effect of disk density showed a marginal change in the location of pinch-off points; however, the additional mass by the second disk in tests with a lower aspect ratio increased the overall depth of the pinch-off. As can be seen in Figs. 2d-2f, for tests with χ = 1.5, the crown surface and underwater cavity became rougher than in tests with χ = 3. The crown surface showed stronger perturbations and micro-waves were observed due to higher turbulence and falling water droplets inside the crown in tests with higher impact energy (i.e., heavier disk, χ = 1.5). The edge of crown was curved inward due to gravity and sudden pressure reduction inside the crown. It is noteworthy to express that the conical cavity pocket attached to a disk with a density ratio of ρd/ρw = 2.335 is greater in χ = 1.5 than in disks with χ = 3. Those air pockets in Figs. 2c and 2f enclosed the disk at the pinch-off without being ruptured. Whereas, the presence of air bubbles around lighter disks indicated the initiation of air cavity rupture at the pinch-off.
The effects of impact energy on crown and cavity formation were studied by comparing the snapshot images of free-falling disks at the pinch-off. The impact energy of disks in different tests was controlled by varying the mass of disks and release height. Figure 3 shows the snapshot images of free-falling disks at the pinch-off indicating the effects of release height and aspect ratio in disks with the same density ratio of ρd/ρw = 1.384. Each column in Fig. 3 represents a particular release height, and each row shows a disk aspect ratio. As can be seen, the impact energy had significant effects on the crown shape and cavity evolution. In particular, the pinch-off depth, Dp, and frontal position, Df, varied with the impact energy. In disks with an aspect ratio of χ = 1.5, the crown distortion was noticeable due to creation of vacuum pressure inside the cavity. As can be seen, the crown was deformed and developed a rough and asymmetric surface.
The crown shape in tests with a lower aspect ratio of χ = 1.5 was fully sealed in most release heights except in Tests No. 13, 14, 16, and 17 having an impact energy less than 36% of the initial energy, Eo. The results indicated that the threshold value of impact energy to separate partial- from full-seal crowns is Ei/Eo= 0.36. Below the threshold impact energy ratio, the crown is partially sealed and above the threshold, it is fully sealed. Two cases of full-seal crown were observed for the aspect ratio of χ = 3, which were belong to Tests No. 9, and 12, having impact energies of Ei = 0.44Eo and 0.417Eo, respectively. It is worth noting that both cases had impact energies more than the threshold impact energy of Ei = 0.36Eo. Experimental observations show that pressure drop within the crown increased with increasing the release height and decreasing with the disk’s aspect ratio.
A sequence of snapshot images of free-falling thick disks entering into stagnant water is shown in Fig. 4. The selected time series images show Test No. 7 with an aspect ratio of χ = 3, density ratio of ρd/ρw = 1.080, normalized release height of hr/do = 5.25, and impact energy of Ei = 0.216Eo. The time in each frame is tagged underneath each image, and the pinch-off time for this test occurred at tp = 134.4 ms. The density of disk in Test No. 7 was close to the density of water and despite relatively high impact velocity, the crown was not fully sealed due to relatively small impact energy below the threshold impact energy of Ei = 0.36Eo. The consecutive images show a uniform and smooth crown surface with slight disturbances. The time reference (i.e., t = 0) in all experiments was defined when a disk hit the water surface and in Test No. 7 the entire disk submerged in water at t = 0.4 ms. The surface splash was formed, and the underwater cavity continuously expanded between t = 0.4 ms, and t = 80.4 ms. After the evolution phase, the edges of crown curved and turned inward to seal the top. The cavity volume continuously contracted near the top of the disk until the pinch-off occurred at tp = 134.4 ms.
Two jets were formed and moved in opposite directions at the pinch-off. The upward jet is called “the Worthington jet” and it penetrated in the water and the downward jet pushed the disk further downstream. The cavity detached from the top of the disk and remained a number of micro bubbles due to sudden formation of vapor as a result of negative pressure (140.4 ms ≤ t ≤ 440.4 ms). The Worthington jet contains enough energy to uplift the water around it. It then returned to the water and penetrated into the water and caused the second pinch-off at tp2 = 540.4 ms. The second pinch-off time was approximately four times of the initial pinch-off in this test (i.e., tp2/tp = 4.02). The second pinch-off cavity is characterized by a hollow spherical shape. After separation from the water surface, the second pinch-off cavity collided with the upcoming bubbles and agitated a region of ambient water between the top of the disk and water surface (i.e., see images for 460.4 ms ≤ t ≤ 960.4 ms). The final stage of evolution occurred for 1040.4 ms ≤ t ≤ 1240.4 ms (7.74 < t/tp < 9.23). At this stage, the disk descended freely while the remaining microbubbles approached the water surface.
It is interesting to study the effect of impact energy, by varying disk mass and release height, and aspect ratio on the water entry of thick disks. Our experimental observations indicated a completely different crown development and cavity formation for the tested aspect ratios. Figure 5 shows the time history of snapshot images of a cylindrical disk (i.e., Test No. 16, Ei = 0.179Eo) in stagnant water. The disk in Fig. 5 had the same density and release height (i.e., ρd/ρw = 1.080 and hr/do = 5.25) as Test No. 7, which was illustrated in Fig. 4, and the impact energy was doubled due to increasing the thickness of the disk. The time duration from the onset of impact was tagged below each image and the pinch-off time occurred at tp = 137.6 ms which was 3.2 ms longer than the same disk with half thickness (i.e., χ = 3).
As can be seen in Fig. 5, the crown was fully sealed at the pinch-off and the crown surface was asymmetric and distorted. The detached cavity volume after the pinch-off was larger in the thicker cylinder and it deformed from a spherical shape to a hyperbolic/paraboloid surface near the water surface. A comparison of both cases right after the impact at t = 20.4 ms, indicated that the rising splash curtain was proportional to the thickness of each disk; however, the splash curtain turned inward in the thicker and heavier disk (i.e., χ = 1.5). The crown was fully sealed in the thicker and heavier disk (see Fig. 5) and the underwater cavity stretched further downward due to higher pressure drop inside the crown. Therefore, the pinch-off depth became larger in thicker disks (i.e., χ = 1.5) in comparison to the case with a larger aspect ratio of χ = 3.
As it was shown in Fig. 4, the crown was not fully sealed, and the inside surface of the cavity was exposed to the atmospheric pressure. Therefore, the pinch-off occurred earlier and the pinch-off depth became smaller than those cases with a full seal crown. As can be seen in Fig. 5 (160.4 ms ≤ t ≤ 220.4 ms; 1.165 ≤ t/tp ≤ 1.6), the upward Worthington jet destabilized the crown, ruptured the crown surface, and caused significant disturbances at the water surface. Considering the same disk with a higher aspect ratio and evolving at the same period, the Worthington jet did not interact with the crown as the crown in Fig. 4 was not fully sealed. As a result, the partially closed crown was intact, and the crown surface remained smooth and without any micro ripples (see Fig. 4). The higher momentum in disk with χ = 1.5 dragged more air into water and more bubbles were detached from the disk after the pinch-off. The normalized settling time in both tests with aspect ratios of χ = 3 and 1.5 were ts/tp = 9.2 and 6.4, respectively. This indicated that the overall settling time of the disk decreased by approximately 44% as the impact energy of the disk doubled.
It is important to correlate the impact velocity of cylindrical disks with their physical characteristics for further analysis of crown development and estimation of pinch-off. The impact velocity is used to calculate impact Froude number (Fro = Vo/(gr)1/2) and dimensionless moment of inertia (I* = πρde/64ρfdo) which representing the physical characteristics of the disk. The correlation between dimensionless moment of inertia and impact Froude number can be used for prediction of impact velocity and other characteristic time and length scales associated with the falling cylindrical objects. The impact energy in each disk can be also calculated by knowing the mass and impact velocity of the disks.
Figure 6a shows the correlations between I* and Fro for all tests in this study. As can be seen, the impact Froude number significantly increased as the release height went beyond four times of the disk diameter (i.e., hr = 4do). As a result, two prediction models were proposed for relatively near and far release heights. The correlations between I* and Fro indicated that the impact Froude number was correlated with the disks’ mass. Two linear equations with the coefficients of determination of R2 = 0.88 and 0.72 were proposed for near and far release heights as:
Fr o= (11.7χ − 2.7) I* + (0.7 − 0.25χ) for hr/do ≤ 4 (1a)
Fr o= (16.5χ − 6.6) I* + (3.05 − 0.5χ) for hr/do > 4 (1b)
In the field, the impact Froude number can be predicted using the geometry and density of disks without measuring the impact velocity. Figure 6b shows the relationship between the impact Froude number and normalized release height. For a constant release height, the densest disk with χ = 1.5 (i.e., the heaviest disk) had the maximum value of Froude number, and the lightest disk with χ = 3 had the lowest value of Froude number. Multi-regression analysis was developed to correlate the initial parameters with impact Froude number as:
Fr o = (0.051ρd/ρw + 0.29) hr/do + (0.38ρd/ρw − 0.23) for χ = 3 (2a)
Fr o = (0.056ρd/ρw + 0.38) hr/do + (0.28ρd/ρw − 0.1) for χ = 1.5 (2b)
In Fig. 6b, the solid lines represent disks with χ = 3 and dashed lines represent disks with χ = 1.5. The coefficients of determination for Eq. (2a) and (2b) are R2 = 0.76 and 0.98, respectively.
3.2. Surface seal and crown characteristics at the pinch-off
The crown geometry is affected by the impact energy of disk. The impact energy varies by variations in disk density, aspect ratio, and release height. In this study, the crown diameter at the pinch-off, Dc, was measured by the in-house MATLAB code. The crown diameter was normalized by the disk diameter and the variations of normalized crown diameter with impact Froude number are shown in Fig. 7. The correlation between normalized crown diameter and impact Froude number was linear in both near and far release conditions and can be described as:
D c /do = 0.28Fro+(0.16hr/do+0.78) (3)
The dashed lines in Fig. 7 show ± 5% variations from the proposed model. As can be seen in Fig. 7, the normalized crown diameter increased dramatically in tests with far release conditions (i.e., hr/do > 4). Therefore, release height is a determinative parameter in variations of crown diameter in comparison to density and aspect ratio of disks. The shape and diameter of the crown are completely altered at the threshold release height of hr/do = 4. In order to understand the shape effect, the correlation of crown diameter with impact Froude number for solid spheres in water were extracted from the study of Sun et el. (2019) and included in Fig. 7. It was noticed that all release height ratios were greater than the threshold limit (i.e., hr/do > 4) in the study of Sun et el. (2019), and impact Froude number ranged from 6.15 to 8.15. As can be seen in Fig. 7, the correlation of normalized crown diameter with Fro in solid spheres was similar to disks released from the normalized height of hr/do > 4, which ranges from 2.25 < Dc/do < 3. Moreover, the linear correlation between impact Froude number and normalized crown diameter was similar in both sphere and disk. However, the normalized crown diameters in spheres with the same impact Froude number were significantly lower than the corresponding disk which indicates the shape effect.
The height of surface crown at the pinch-off was measured from the water surface and the results were normalized by the square root of release height and disks’ diameter in form of Hc/(dohr)1/2. Figure 8 shows a linear correlation between normalized crown height and impact Froude number with a coefficient of determinations of R2 = 0.81 as:
H c /(dohr)1/2 = 0.12Fro + 0.2 (4)
The dashed lines in Fig. 8 show ± 10% variations from the proposed model. The total duration of crown formation and its collapse is defined by the crown time, tc, and it was measured in all experiments. The results were normalized by the square root of disk diameter and gravitational acceleration as tc/(do/g)1/2. Figure 9 shows the variations of normalized crown time versus impact Froude number and a power-law model showed the best fit in correlation of crown time with Fro as:
t c /(do/g)1/2 = 2.16Fro0.3 (5)
The coefficient of determinations of the above equation is R2 = 0.92 and dashed curves in Fig. 9 show ± 15% variations from the proposed model. As can be seen in Fig. 9, the crown time increased non-linearly with impact Froude number indicating that the crown time is also correlated with impact energy. The normalized crown time in this study varied between 2 and 3.5 times of characteristic time scale and it increased by increasing the release height as well.
3.3. Underwater cavity characteristics at pinch-off
The pinch-off is a prominent phenomenon during water entry of solid disks. Different characteristics of underwater cavity were measured at the pinch-off and the results are presented in this section. The pinch-off depth, Dp, in all experiments was measured and the results were normalized with the disk’s diameter. Figure 10a shows the correlation between normalized pinch-off depth and impact Froude number for all experiments. The results showed that the location of pinch-off was directly affected by release height. A power-law model was proposed to predict the pinch-off depth based on impact Froude number as:
D p /do = 0.45Fro0.9 (6)
The power of impact Froude number in Eq. (6) indicates that the correlation between normalized pinch-off depth and impact Froude number is almost linear. The dashed curves in Fig. 10 show ± 10% variations from the proposed model. A comparison between the present results (i.e., gravity-driven) and the proposed equation of Glasheen and McMahon (1996) for prediction of pinch-off depth in disks driven by force transducer indicated that the addition of impact force increased the pinch-off depth (see Fig. 10b). The results showed that dimensionless pinch-off depths in forced-driven disks had a linear relationship with the disk Froude number, Fr = urms/(gr)1/2; however, the rate of change of normalized pinch-off depth was 55% smaller, in gravity-driven disks than the forced-driven disks.
The results for a gravity-driven sphere with various impact Froude numbers and a diameter of do = 57.2 mm from the study of Sun et al. (2019) were also included in Fig. 10b for comparison. The density ratio of solid sphere was ρ/ρw = 1.84, which is close to the density ratio of disks used in this study. Also, the diameter of spheres (do = 57.2 mm) is comparable to the diameter of disks used in the study of Sun et al. (2019) (do = 76.2 mm). In addition, the data related to spheres with higher density ratio of ρ/ρw = 7.80, from the study of Mansoor et al. (2014), was added in Fig. 10b. In comparison to Sun et al. (2019), spheres with a higher density ratio had greater pinch-off depth. A comparison between spheres and disks indicated that for a constant impact Froude number, the pinch-off depth in spheres is smaller than disks with the same Froude number which may be due to formation of boundary layer in solid spheres and flow separation in thick disks.
The pinch-off time in each test was measured from the impact time till the time that the underwater cavity pinches off. The results were normalized with the disk’s diameter and gravitational acceleration in form of tp/(do/g)1/2, and were plotted with the impact Froude number, Fro. Figure 11 shows that the normalized pinch-off time was classified based on the threshold release heights. It is inferred from the results that it took longer for disks with higher density to pinch-off than the lighter ones. Linear equations were proposed for prediction of pinch-off time for near and far release conditions as:
t p /(do/g)1/2 = 0.13Fro + 1.36 for hr/do ≤ 4 (7a)
t p /(do/g)1/2 = 0.13Fro + 1.2 for hr/do > 4 (7b)
The coefficients of determination for the near and far release conditions were R2 = 0.73 and 0.90, respectively. As can be seen, the pinch-off time increased with the disks’ density below and above the threshold release height. Moreover, by increasing the disks’ thickness the pinch-off time increased as well. The results for heavy solid spheres (i.e., ρ/ρw = 7.80) with different radius and impact Froude numbers from the study of Mansoor et al. (2014) were also added to Fig. 11. As can be seen, the pinch-off time of spheres increased by increasing the relative sphere density which is compatible with the results of the disks’ pinch-off time. As shown in Fig. 11, the size of sphere was not a determinative factor in variations of pinch-off time. The dimensionless pinch-off time of spheres with radiuses of r = 7.5 mm, 10 mm, 12.5 mm was constant with approximately tp/(do/g)1/2 = 1.45 for all cases in the study of Mansoor et al. (2014). The dimensionless pinch-off time of light spheres (i.e., ρ/ρw = 1.84) with a diameter of do = 57.2 mm, and impact Froude numbers ranging between Fro = 6.15 and 8.15 from the study of Sun et al. (2019) were also added to Fig. 11 for comparison. The diameter of sphere in the study of Sun et al. (2019) was higher than Mansoor et al. (2014) while the density of sphere was smaller. This resulted in a relatively smaller pinch-off time in the lighter sphere despite having a larger diameter. As illustrated in Fig. 11, the pinch-off time for disks and spheres with the same geometry and density remained almost the same. For example, the normalized pinch-off time in disks with a density ratio of ρd/ρw = 2.335 and aspect ratio of χ = 1.5, is approximately tp/(do/g)1/2 = 1.68 whereas the normalized pinch-off time for spheres with ρd/ρw = 7.8 and do = 25 mm was tp/(do/g)1/2 = 1.46.
3.4. Motion of thick disk in water
A large amount of energy is dissipated by the impact once a solid object impacts the water surface. The energy losses due to the impact were calculated for all experiments with various initial energies, Eo. The normalized energy losses, ΔE/Eo, are plotted versus the impact Froude number, Fro, Fig. 12 shows that the normalized energy losses for each release height followed the same trend. As can be seen in Fig. 12, disks with the smallest density (i.e., ρd/ρw = 1.080) and χ = 3, had the maximum energy losses at the impact. Moreover, less energy was dissipated by the impact as release height and density of disk increased. At each release height, the normalized energy losses decreased by increasing impact Froude number. The minimum value of energy losses in all experiments and release heights was in the test with the highest mass (i.e., ρd/ρw = 2.335 and χ = 1.5).
Figure 13 shows the variations of disk’s frontal position versus non-dimensional time of the falling disks, t/T, where T = (do/g)1/2 is the characteristic time scale. The subplots in Fig. 13 show the frontal position of disks with an aspect ratio of χ = 1.5, and 3, and with different release heights. As can be seen in Fig. 13a, the variations of normalized frontal position with normalized time for hr/do = 2.62, and 3.94 were compatible with each other. At the moment of impact until the depth equal to the disk diameter (i.e., Zf/do = 1), all disks had the same frontal positions. However, after the position of Zf = do, the effect of disk’s mass on variations of frontal position with time became effective. It was found that disks with higher densities, fell faster than disks with lower densities. Figure 13a shows the trajectories of disks with near release height (i.e., hr/do ≤ 4). As can be seen, disks with a density ratio of ρd/ρw = 1.080, 1.384, and 2.335 reached the position of Zf/do = 4 at different normalized times of t/T = 17.5, 10, 5, respectively. Figure 13b shows the trajectories of disks with far release conditions (i.e., hr/do > 4) and for disks with an aspect ratio of χ = 3. A comparison between trajectories of disks in near and far release conditions indicated that far release condition reduced the duration of disks’ descending time by approximately 70% of the total falling time. For example, considering a constant point of Zf/do = 4, disks with a density ratio of ρd/ρw = 1.080, 1.384, and 2.335 reached Zf/do = 4, at t/T = 12.25, 7, 3.5, respectively. Figures 13c and 13d show the trajectories of thick disks in both near and far release conditions, respectively. It can be seen that reducing the aspect ratio of disks by half (i.e., from χ = 3 to 1.5), decreased the falling duration of disks by 50%.
The frontal velocity of gravity-driven falling disks was measured from the impact till the settling stage and the results were normalized by the disks’ diameter and gravitational acceleration in form of (gdo)1/2. Figure 14 shows the variations of normalized frontal velocity of disks with normalized time, t/T, for different aspect ratios and release conditions. As can be seen in Fig. 14, frontal velocities decayed with time and they were significantly altered by the near and far release conditions. As expected, disks with the highest density had the highest frontal velocities. The impact velocities in tests started from a maximum velocity during the water entry and decreased non-linearly until disks reached the bottom of water tank. As can be seen in Fig. 14a, both normalized release heights of hr/do = 2.62, 3.94 with an aspect ratio of χ = 3, had similar trends, and the trajectory of velocities was classified based on the density of disks. Normalized impact velocities ranged between 1 < V/(gdo)1/2 < 1.75 and at the final settling phase the velocity of disks reduced to a new range of 0.25 < V/(gdo)1/2 < 0.75.
Figure 14b shows the trajectories of frontal velocity of disks with far release conditions (i.e., hr/do = 5.25, 6.56) and an aspect ratio of χ = 3. In these cases, the impact velocities were higher (i.e., 1.75 < V/(gdo)1/2 < 2.5) than those cases with near field release (hr/do ≤ 4). Settling velocities were approximately 40% more than settling velocities with the near field release. Figures 14c and 14d show the normalized velocity results for χ = 1.5 and in both near (hr/do ≤ 4) and far field (hr/do > 4) release conditions, respectively. In Figs. 14c and 14d with χ = 1.5, impact and settling velocities were greater than those disks with χ = 3 and this may be due to higher mass of disk. A prediction model based on multi-regression analysis was proposed for estimation of disk velocity as:
V/(gdo)1/2 = 0.3(t/T)0.62 for χ = 3 & hr/do ≤ 4 (8a)
V/(gdo)1/2 = 0.56(t/T)0.95 for χ = 3 & hr/do > 4 (8b)
V/(gdo)1/2 = 0.36(t/T)0.24 for χ = 1.5 & hr/do ≤ 4 (8c)
V/(gdo)1/2 = 0.48(t/T)0.51 for χ = 1.5 & hr/do > 4 (8d)
The average coefficient of determination was R2 = 0.98.