In Fig. 2, we present impacts of a room-temperature sphere (T_s = 30 °C) and a heated sphere (T_s = 200 °C) onto water (T_w = 30 °C) at impact velocity V₀ = 1.2 m/s. We observe that the cavity only exists in the heated sphere case.

Fig. 2.

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	Fig. 2. Comparison of impacts of spheres onto water at V0 = 1.2 m/s: (a) Ts = 30 °C, at time t = 5, 13.2, 30.2, 47.2, 64.4, 81.6 ms from impact; (b) Ts = 200 °C, at time t = 5, 19, 33, 51, 60.8, 81.6 ms from impact. In both cases, the water temperature is Tw = 30 °C.

For the room-temperature sphere, a cavity can be formed at larger impact velocities (V₀ > 4.2 m/s) as shown in Fig. 3. The features of the impacts are as follows: (a) small entrained air pocket; (b) small cavity; (c) partial cavity; and (d) full cavity. We consider the impact in Fig. 3(a) as the no cavity condition in the following description.

Fig. 3.

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	Fig. 3. Cavities at depth H = 7R0 formed during impacts of spheres with Ts = 30 °C, Tw = 30 °C, V0 = (a) 4.2 m/s, (b) 4.5 m/s, (c) 5 m/s, and (d) 5.6 m/s.

The cavity formation is related to the movement of the thin liquid film from the water (marked by the arrow in the first frame for both cases). In Fig. 2(a), the thin liquid film wraps around the sphere and adheres to the surface until closing at the apex. The sphere is surrounded by water and no cavity is formed. In Fig. 2(b), the thin liquid film is detached from the sphere surface after the impact, and a cavity is formed behind the sphere.

As shown by Duclaux et al.,^[10] the cavity can be formed only when the impact velocity is above a threshold value. We can conclude that the threshold velocity for the heated sphere (T_s = 200 °C) is far less than that for the room-temperature sphere.

We observe another cavity formation at low impact speed (V₀ = 1.2 m/s) as shown in Fig. 4. The water temperature is T_w = 95 °C, and the initial sphere temperature is T_s = 500 °C. The mist is formed above the water surface when the water temperature is close to the boiling point. The characteristics of cavity walls in Figs. 2(b) and 4 are different (coarse in Fig. 2(b) and smooth with regular ripples in Fig. 4).

Fig. 4.

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	Fig. 4. Cavities formed during the impact of a sphere at V0 = 1.2 m/s with Ts = 500 °C, Tw = 95 °C at time t = 24.6, 42.6, and 68.4 ms from impact.

The threshold velocity for room temperature spheres depends on the surface state and the liquid characteristic. The surface state may be changed after heating the sphere to a high temperature, and this might lead to the low threshold velocity for cases shown in Figs. 2(b) and 4. We heat a sphere to a temperature of 500 °C then cool the sphere to room temperature. We find that the cavity cannot be formed below the impact velocity of 4 m/s. The temperatures of our experimental spheres do not exceed the temperature of 500 °C. So the change of sphere surface states has little influence on the low threshold velocity.

As shown by Duez et al.,^[9] the threshold velocity is proportional to the capillary velocity, defined as σ/μ. The water–air surface tension σ and the water dynamic viscosity μ decrease as the water temperature increases, and μ decreases at a larger rate, which leads to the increase of the capillary velocity. Thus, the threshold velocity for room temperature spheres will be high for high temperature water. We carry out experiments on the impacts of room-temperature spheres onto 60 °C and 95 °C water. We find that the cavity cannot be formed below the impact velocity of 4 m/s. So, the changes of water characteristics are not associated with the low threshold velocity.

Since the low threshold velocity cannot be due to the change of surface state and water characteristic, we suspect that it is a consequence of vaporization of water on the surface of the spheres. We present the surface vaporization states: Fig. 5(b), a case shown by Fig. 2(b); Fig. 5(c), a case shown by Fig. 4. We find that the surface flows are dissimilar to those of the room-temperature sphere shown in Fig. 5(a).

Fig. 5.

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	Fig. 5. Close-up cavity images at depth H = R0, and (a) V0 = 5.6 m/s, Tw = 30 °C, and Ts = 30 °C; (b) V0 = 1.2 m/s, Tw = 30 °C, and Ts = 200 °C, a case shown by Fig. 2(b); (c) V0 = 1.2 m/s, Tw = 95 °C, and Ts = 500 °C, a case shown by Fig. 4.

When a sphere with moderate temperature is immersed into water, nucleate boiling will occur around the sphere.^[15] We also observe vapor bubbles generated by nucleate boiling on the surface, as shown in Fig. 5(b).

A coarse sphere has a low threshold velocity compared with a smooth sphere.^[5] Both the coarseness of the sphere surface and the disturbance of the vapor bubbles can enhance the turbulence of the flow. Both coarse spheres and heated spheres with moderate temperatures can form cavities with rough walls. The rough cavities can also be formed at low velocities by impacting the spheres coated with viscous liquids due to the surface disturbance of the liquid layers reported by Bell^[26] and May.^[7] We suspect that the low threshold velocity shown in Fig. 2(a) is due to the disturbance of the vapor bubbles.

When a sphere with a high temperature (far above the boiling temperature) is immersed into water at low subcooling, film boiling occurs and a continuous vapor layer can be formed around the sphere.^[15] This vapor layer is also observed in Fig. 5(c), and the sphere inside the vapor layer looks smaller than the sphere shown in Fig. 5(a) due to the refraction of light through different media (water and vapor). The regular ripples shown in Fig. 4 are caused by the repeated compression of the vapor layer when the sphere moves downward in the water.

A drop of water hitting a solid surface may bounce, provided that the surface is highly hydrophobic. As a similar situation, when the drop hits a very hot solid, a thin layer of vapor will be generated between the drop and the solid and may make the drop rebound.^[27] We suspect that the vapor layer may have some similar properties to those of hydrophobic materials. From Fig. 1(a) taken by Quéré,^[28] we find that the contact angle between the small water drop and the 300 °C metallic plate is more than 160°.

A cavity can be formed at a low impact velocity for a hydrophobic sphere relative to a hydrophilic sphere.^[9,10] We compare the impact of a superhydrophobic sphere onto 30 °C water (Fig. 6(a)) with the impact of a heated sphere with an initial temperature of 400 °C onto 95 °C water (Fig. 6(b)), in which case a stable vapor layer can be generated around the sphere. The impact velocities in the above cases are both V₀ = 1.2 m/s.

Fig. 6.

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	Fig. 6. Comparison of impact between (a) a superhydrophobic sphere (Ts = 30 °C, Tw = 30 °C) and (b) a heated sphere (Ts = 400 °C, Tw = 95 °C) onto water at time t = 6, 23.6, 43, 53.6, 67.2, and 72.2 ms from impact, with an advance contact angle of about 170° and receding contact angle close to 160°. The impact velocities are both V0 = 1.2 m/s.

The superhydrophobic sphere is prepared by using the methods given by Vakarelski et al.^[16] with a commercially available coating agent (Glaco Mirror Coat Zero, Soft 99 Co.). The advance contact angle is about 170° and the receding contact angle is close to 160°.

The thickness of vapor layer increases as the temperatures of the sphere and water increase.^[29] The initial temperature of the sphere shown in Fig. 6(b) is 400 °C which is lower than 500 °C for the case shown in Fig. 4. So the vapor layer is thinner in Fig. 6(b) and cannot be easily seen by eye. As a result, we cannot find the regular ripples shown in Fig. 4. However, this is really in the film-boiling regime, and we will discuss the boiling regimes for the impacts by heated spheres onto 95 °C water in Subsection 3.2.

The cavity appearances for the two impact processes shown in Fig. 6 are very similar, and in both cases the cavity interfaces are smooth. The contact line emanates approximately from the equator of the superhydrophobic sphere as seen in Fig. 6(a), while the contact line is absent for the Leidenfrost impact shown in Fig. 6(b) due to the encapsulation of a thin vapor layer. We suspect that the vapor layer has a similar effect to the superhydrophobic sphere, which leads to the low threshold velocity for the cases shown in Figs. 4 and 6(b).

In Fig. 7, we present cavities at H = 6R₀ formed during impacts of spheres with different impact velocities and initial temperatures onto 95 °C water. As shown in Fig. 7(a), a cavity with rough interface is formed, which is similar to the cavity by the impact of a 200 °C sphere onto 30 °C water as shown in Fig. 2(b). In the nucleate-boiling regime, the heat transfer intensity is enhanced with the water and sphere temperature increasing. Large vapor bubbles can be seen around the sphere shown in Fig. 7(a).

Fig. 7.

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	Fig. 7. Cavities formed at depth H = 6R0 during impacts of spheres onto 95 °C water for (a) Ts = 250 °C and V0 = 1.2 m/s, (b) Ts = 300 °C and V0 = 1.2 m/s, (c) Ts = 350 °C and V0 = 1.2 m/s, (d) Ts = 350 °C and V0 = 3.4 m/s, (e) Ts = 400 °C and V0 = 5.6 m/s, and (f) Ts = 500 °C and V0 = 5.6 m/s.

With the increase of the sphere’s initial temperature, vapor bubbles begin to gather together, leading to the transition from nucleate boiling to film boiling. When the sphere’s initial temperature reaches to 300 °C, film boiling occurs and a vapor layer is formed on the surface of the sphere. The boiling intensity weakens due to the low thermal conductivity of vapor, and only a few small bubbles can be seen around the vapor layer. As the sphere temperature is not high enough, the vapor layer is unsteady. We cannot fully understand why the unsteady vapor layer leads to a cavity not fully developed as shown in Fig. 7(b). When the sphere’s initial temperature reaches to 350 °C, the vapor layer becomes stable, producing a very smooth cavity as shown in Fig. 7(c) similar to the case shown in Fig. 6(b). There are no obvious bubbles around the vapor layer due to the continuous decrease of heat transfer intensity.

A drop may rebound when hitting a heated surface if the vapor (generated by vaporization) pressure overcomes the drop’s dynamic pressure. The dynamic pressure of the drop is determined by We and the increase of the surface temperature can raise the vapor pressure. Thus the Leidenfrost temperature will increase with We increasing.^[28,30,31] So the Leidenfrost temperature rises with the velocity increase of the drop for a specific liquid. Vakarelski et al.^[29] also presented that the Leidenfrost temperature for a sphere falling through water is higher than that for a static sphere. They argued that the shear stresses induced on the vapor layer may cause the vapor layer to be unstable. In our experiments, we find a similar phenomenon: the destabilization of the vapor layer leads to an unstable cavity formed by the impact of a 350 °C sphere when increasing the impact velocity to 3.4 m/s, as shown in Fig. 7(d). We also find the disturbed cavity for a 400 °C sphere at an impact velocity of 5.6 m/s, as shown in Fig. 7(e). A stable vapor layer requires a high temperature of sphere at a high impact velocity. As shown in Fig. 7(f), when the initial temperature of the sphere is 500 °C, a stable vapor layer can be formed around the sphere, which leads to a smooth cavity with regular ripples at the impact velocity of 5.6 m/s.

In Fig. 8, we present the cooling progress of a 400 °C sphere held by the electromagnet still in 60 °C water. A stable vapor layer is formed on the surface after the sphere has been immersed into water. The vapor layer lasts about 30 s due to the low heat transfer intensity of film boiling. When the sphere temperature drops below the Leidenfrost temperature, vapor explosion occurs, then film boiling transfers to nucleate boiling with an explosive release of bubbles around the sphere.

Fig. 8.

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	Fig. 8. Cooling process of a 400 °C sphere in 60 °C water at time t = 1.84, 30.07, 30.08, and 30.35 s after the sphere has immersed into water.

In Fig. 9(a), the stable vapor layer cannot be formed by the impact of the 400 °C sphere onto water at V₀ = 1.2 m/s. A shell of vapor around the sphere grows and collapses radially as shown in the supplementary movie. There is only small entrained air, and no cavity is formed. In Fig. 9(b), when the initial temperature of the sphere is 500 °C, a vapor layer is formed in the initial stage of the impact. The appearance of the vapor layer is different from the stable vapor layer formed during the impact of a 500 °C sphere onto 95 °C water shown in Fig. 4. We can see the micro bubbles around the vapor layer in the first two frames of Fig. 9(b), and hardly any of the bubbles can be seen around the smooth and stable vapor layer shown in Fig. 4. The micro-bubble boiling has been considered as a particular film-boiling model for high subcooled water.^[18,20] The stability of the vapor layer cannot be maintained for a long time due to the high heat transfer intensity in this boiling regime. The vapor layer around the sphere grows and collapses after several milliseconds, which is similar to the case shown in Fig. 9(a). No cavity is formed in this impact process. In high subcooled water, stable film boiling cannot easily occur, and in our impact experiments the stable vapor layer is not observed for T_w = 30 °C and T_w = 60 °C.

Fig. 9.

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	Fig. 9. Impacts of heated spheres onto 60 °C water at V0 = 1.2 m/s for (a) Ts = 400 °C at time t = 6.6, 28.8, 50.6, and 81.6 ms after impact, and (b) Ts = 500 °C at time t = 6.6, 28.8, 50.6, and 81.6 ms after impact.

In our experiments, no cavities are formed by the impact of heated spheres with initial temperatures higher than 300 °C onto high subcooled water (T_w = 30 °C, T_w = 60 °C) at an impact velocity of 1.2 m/s. When increasing the impact velocity, in some cases, cavities are formed due to the influence of the velocity. However, none of the cavities can be fully developed and will break into fragments. In Fig. 10, we show some broken cavities for different impact velocities and initial sphere temperatures.

Fig. 10.

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	Fig. 10. Broken cavities at depth H = 8R0 for (a) Tw = 30 °C, Ts = 350 °C, and V0 = 5.6 m/s, (b) Tw = 30 °C, Ts = 450 °C, and V0 = 3.4 m/s, (c) Tw = 60 °C, Ts = 400 °C and V0 = 3.4 m/s, and (d) Tw = 60 °C, Ts = 450 °C, and V0 = 5.6 m/s.

The nucleate-boiling and film-boiling regimes in the cooling progress are usually distinguished by the heat transfer intensity determined with the temperature of the sphere measured by an embedded thermal couple.^[16–18] The heat transfer intensity of nucleate boiling is usually an order of magnitude higher than that of film boiling. While high heat transfer intensity is also found for spheres whose temperature is far above the critical temperature of water.^[20] According to the classic theory of homogeneous nucleation, under these conditions the liquid–vapor phase transition occurs practically instantly, where the characteristic time is on the order of ns. The nucleate boiling cannot occur under such conditions reported by Yagov et al.^[18] and it is referred to as a particular mode of film boiling. The low heat transfer intensity exists only in stable film-boiling regime with a stable vapor layer around the sphere. There is no stable film-boiling regime for T_w = 30 °C and T_w = 60 °C by the impact of heated spheres in our experiments. When the temperature of the sphere is above 300 °C, vapor fragments or an unstable vapor layer may be formed on the surface. The broken cavities may be due to the high heat transfer intensity and the disturbance of the vapor layer or vapor fragments.

Figure 11 shows the dual cavities similar to those shown in Fig. 13 from Marston et al.^[23] In our experiments, the dual cavities can be formed for the cases of nucleate boiling (Figs. 11(a) and 11(b)) and film boiling (Fig. 11(c)). The ranges of the impact velocities and the initial sphere temperatures in which dual cavities can be formed for the three water temperatures are shown in Fig. 13.

Fig. 11.

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	Fig. 11. Dual cavities formed during the impacts for (a) Tw = 30 °C, Ts = 260 °C, and V0 = 1.2 m/s, (b) Tw = 30 °C, Ts = 280 °C, and V0 = 2.5 m/s, and (c) Tw = 95 °C, Ts = 350 °C, and V0 = 4.5 m/s.

In Fig. 12, we present the evolutionary process of a dual cavity for the nucleate boiling case. The water temperature is T_w = 30 °C, the initial sphere temperature is T_s = 280 °C, and the impact velocity is V₀ = 5.6 m/s. As shown in Fig. 12(a), before the formation of the dual cavity, a number of bubbles attach to the bottom hemisphere, the cavity-sphere contact line (marked by the arrow) is located at the top hemisphere surface. When the dual cavity is formed, as shown in Fig. 12(b), there are hardly any bubbles around the bottom hemisphere and the cavity-sphere contact line is located at the sphere equator. The bubbles around the bottom hemisphere are merged with the upper cavity due to the low pressure at the sphere equator, and the contact line moves downward, leading to a new cavity structure. Thereafter, the contact line is pinned at the sphere equator, as shown in Figs. 12(c) and 12(d). The top and bottom cavities are connected, air passes through the top cavity and enters into the bottom cavity till the surface seals.

Fig. 12.

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	Fig. 12. Evolutionary process of a dual cavity by the impact of a sphere for Tw = 30 °C, Ts = 280 °C, V = 5.6 m/s, at time t = 3.2, 7, 11.6, and 23.4 ms from impact.

Fig. 13.

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Fig. 13. (color online) Maps of V0–Ts parameter space, showing transitions between cavity features for different water temperatures: (a) Tw = 30 °C, (b) Tw = 60 °C, and (c) Tw = 95 °C. The vertical dashed line in panel (c) indicates temperature beyond which a stable vapor layer can be formed around the sphere at low impact velocities. The inclined dashed line indicates the transition from smooth cavity (bottom) to disturbed cavity (top). The impact features are classified as no cavity (◾), incomplete cavity (★), full cavity (▴), dual cavity (▾), and broken cavity (♦).

The dual cavity formed in the film-boiling regime is similar to that in the nucleate boiling regime. The vapor is disturbed by the impact with bubbles around it before the formation of the dual cavity, and the location of the contact line is also at the top hemisphere surface. Then the vapor layer becomes stable and a very smooth dual cavity can be formed. The bottom hemisphere is covered by the vapor layer, no contact line can be found for this case, as shown in Fig. 11(c). So, the formation of the dual cavity for both cases needs a large change of the flow states at the sphere surface, and the contact line must be located at the top hemisphere surface at the beginning of the impact.

The transitions between different cavity features shown in the previous sections are summarized in Fig. 13. We also consider the small entrained air pocket shown in Fig. 3(a) as the no cavity case. The impacts shown in Figs. 3(b), 3(c), and 7(b), in which the cavities cannot be fully developed, are considered as the incomplete cavity cases.

The distribution characteristics of impact features for 95 °C water are different from those for 30 °C and 60 °C water due to the different heat transfer and boiling characteristics for high subcooled and low subcooled water. The distribution characteristics of cavity properties are similar for 30 °C and 60 °C water, and from Fig. 13(a) to Fig. 13(b) the data points seem to move left.

Like cold spheres, when the vaporization effect is not obvious in the film-boiling regime, cavities can only be formed at high impact velocities (see the data points in the first columns for these three different water temperatures). The minimum initial temperature of the sphere for a full cavity at an impact velocity of 1.2 m/s decreases with the increase of water temperature. As described in Subsection 3.1, the cavities formed by the impacts of spheres with moderate initial temperatures are due to the disturbance of the vaporized bubbles around the spheres. In high temperature water, more bubbles can be formed, which leads to the disturbance being enhanced (e.g., Fig. 7(a)).

Dual cavities can only be formed in a small temperature range for all water temperatures. For T_w = 30°C and 60 °C, the temperature range is between 250 °C and 300 °C in the nucleate boiling regimes. The dual cavities are found to be formed at low impact velocities for low sphere temperatures and at high impact velocities for high sphere temperatures. For T_w = 95 °C, the dual cavity is formed under the condition of film boiling when the initial temperatures of the spheres are above 300 °C.

For 95 °C water, the full cavities are formed in the whole range of initial sphere temperatures. The cavity formed at an impact velocity of 1.2 m/s for T_s = 300 °C is incomplete, and the heat transfer is in the transition stage between nucleate boiling and film boiling. Smooth cavities can be formed when the initial sphere temperatures are beyond 300 °C at low impact velocities (e.g., Figs. 4, 6(b), and 7(c)). The vapor layers formed around the spheres can be disturbed by high impact velocities, thus resulting in disturbed cavities (see the data points beyond the inclined dashed line in Fig. 13(c)) (e.g., Figs. 7(d) and 7(e)). By increasing the initial sphere temperature, the smooth cavities can be formed at higher impact velocities (e.g., Fig. 7(f)). Stable vapor layers are not easy to form at high subcooled water, so we only find broken cavities beyond the temperature of 300 °C for 30 °C and 60 °C water due to the high heat transfer intensity.

Pinch-off (deep closure) is an important characteristic for cavity dynamics. The high-speed camera is set to have a small field of vision to obtain the highresolution snapshots. So, we can obtain none of the positions of the spheres when pinch-off occurs at high impact velocity. Here, we just investigate the pinch-off time, τ, when pinch-off occurs at one point, hence we can obtain the precise values. In accordance with Duclaux et al.,^[10] the pinch-off time is nearly independent of the impact velocity for our experiments, so we use the same symbols for different impact velocities with the same temperature of spheres and water.

The pitch-off time is nearly independent of both the initial sphere temperature and the water temperature, as shown in Fig. 14. Our best fit to this time yields τ ≈ 67 ms. Duclaux et al.^[10] found the pitch-off time with a prefactor k = 2.06 for a hydrophobic sphere. In our experiments we find the prefactor k = 2.1. Marston et al.^[23] found k = 2.09 for heated spheres impacting onto a fluorinated liquid. There are very small differences between these three prefactor values, so the pitch-off characteristics are similar for these experiments.

Fig. 14.

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	Fig. 14. Pinch-off time versus initial sphere temperature for Tw = 30 °C (□), 60 °C (☆), and 95 °C (+). The dashed line indicates the mean value from the data sets.

The initial radius of cavity expansion is equal to the sphere radius when the contact line is located at the sphere equator. For most cases, the contact line is pinned at the sphere equator after the impact, as in most cases in our experiments. So the sphere radius R₀ is usually used to represent the initial cavity radius in the expression, . The prefactor value k is related to the initial expansion speed of the cavity, as described by Duclaux et al.^[10] The contact lines for low initial sphere temperatures for T_w = 30 °C and 60 °C are slightly above the equator of the sphere due to the low disturbance intensity, as shown in Fig. 2(b), thus the initial cavity radius is smaller than R₀, which also leads to the low initial expansion speed of the cavity, so the pinch-off time has relatively low values for these cases (see the data points inside the dashed ellipse).