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Experimental study on the cavity dynamics of oblique impact of sphere on a viscous liquid floating on water
Ocean Engineering 194 (2019) 106597
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Experimental study on the cavity dynamics of oblique impact of sphere on a viscous liquid floating on water Tiezhi Sun a, b, Heng Wang a, Li Zou a, b, c, *, Zhi Zong a, b, c, Haitao Li a a
School of Naval Architecture, Dalian University of Technology, Dalian, 116024, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian, 116024, China c Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Oblique impact Two-layer liquid Splash formation Cavity dynamics Pinch-off
Oblique water entry of marine structure involves complex hydrodynamics phenomena including splash forma tion, asymmetrical cavity and pinch-off that have been observed in previous systematic experiments. However, very few studies focus on such phenomena induced by oblique impact of object in a two-layer liquid system. In this paper, we present an experimental investigation of oblique entry of a sphere into a deep pool of water within a layer of highly viscous dimethicone floating on the water. In the experiment, a high speed digital camera is established to obtain the instantaneous cavity and motion trajectories for both qualitative and quantitative analysis. The results show that the upper-layer viscous liquid introduces a significant change in the splash for mation mechanism and the special behaviors are characterized by stratified splash and corresponding asym metrical shrinkage structures. Meanwhile, it is observed that the difference in splash characteristics caused by different thicknesses of viscous fluid layer is obvious, and the pinch-off depth is independent of the thickness value which studied in our experiments. In addition, it is found that increasing the horizontal impact velocity will cause a delay in the splash closure and the vertical displacement of the sphere during the dimethicone-water enter process is independent of the horizontal impact velocity. Hydrodynamic force model is established to explain the trajectory of the sphere. Furthermore, the splash angle, pinch-off depth, pinch-off time and the water entry depth at pinch-off locations and the corresponding change tendency are obtained qualitatively and quantitatively. These results indicate that the dimethicone layer is critical in determining the cavity dynamics, especially splash formation and pinch-off characteristics.
1. Introduction Studies of objects impacting into the water has attracted the atten tion of researchers for over a century and is still a hot topic of current research (Worthington and Cole, 1897; Wang et al., 2019). The water entry problem is relevant to application of air-to-sea projectile (May 1952), ship slamming (Faltinsen, 1990), and stone skipping (Rosellini et al., 2005). The phenomenon of solid-liquid impact is accompanied by many modern cavity dynamics, such as splash inception, cavity forma tion, splash seal and pinch-off (Aristoff et al., 2010; Chen et al., 2019; Jiang et al., 2018; Mansoor et al., 2014; Sun et al., 2019). Particularly, one of the important problems has long been studying is the splash behavior and cavity dynamics during a solid object impacting liquid process. Water entry is a challenging problem, and associated with the
complex hydrodynamic feature. The pioneering experiments were con ducted by Worthington and Cole (1897, 1900) and Worthington (1908). They obtained the first high-speed photographs of the impact dynamics and cavity formation. Following Worthington’s works, May (1951; 1952) experimentally studied the effects of surface conditions on splashes formations and the types of cavity closures. Since then, surface characteristic of water entry objects has been studied extended to wettability (Duez et al., 2007), hydrophobicity (Korkmaz and Güzel, 2017), and temperature (Li et al., 2018). The effect of hydrophilic and hydrophobic surface of spheres on cavity formation was studied by Truscott and Techet (2009a), they found that transverse spin altered the spheres surface velocity distribution and cavity formation. In addition to these notable studies, other works on water entry problems involves the influence of object shape on cavity dynamics, such as spheres (May and Woodhull, 1948; Belden et al., 2016), rigid projectile (Lee et al., 1997),
* Corresponding author. School of Naval Architecture, Dalian University of Technology, Dalian, 116024, China. E-mail address: [email protected] (L. Zou). https://doi.org/10.1016/j.oceaneng.2019.106597 Received 29 July 2019; Received in revised form 15 September 2019; Accepted 13 October 2019 Available online 1 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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content and Froude number on cavity formation and dynamics mecha nism during water entry of objects are still not well understood. Hence, the aim of this experimental study is to provide new phenomenon of a thin layer of highly viscous liquid resting on a deep pool of water to alter splash characteristics and cavity dynamics of oblique impact spheres. Experiments with clean water using the same spheres are investigated to highlight and support the present findings. In Sec. II, we present our experimental setup, including the facility parameters, liquid properties and test cases. Section III first summarizes the effect of the highly viscous liquid layer on the cavity formation mechanism. Subsequently, the in fluence of the thickness of viscous liquid layer and the horizontal impact velocity on the cavity dynamics are analyzed. Moreover, we mainly analyze the characteristic of the splash angle, pinch-off time, and pinchoff depth. Finally, the conclusions are presented in Section IV.
slender axisymmetric bodies (Bodily et al., 2014) and wedges (Shams et al., 2015). Duclaux et al. (2007) performed an experimental study on the collapse of a transient cavity of air in water created by the impact of a sphere and analyzed the cavity evolution and pinch-off features. Speirs et al. (2019) used spheres to investigate the cavity shape, splash crown and scale pinch-off times. They found that there were three kinds of splash crowns, including thick rim forms, thin crown and classical cavities. Li et al. (2019) investigated the multiphase flow during water entry of spheres with different surface wettability at low Froude numbers. The results showed that the impact of the hydrophilic case creates a smaller splash and the hydrophilic and hydrophobic spheres experience different cavity regimes as the Froude number increases. Particular attention was paid to address the influence of liquid properties on cavity dynamics. The associated experiments involved multiple characteristic liquids, including glycerine-water mixture (Bell, 1924), viscoelastic liquid (Cheny and Walters, 1996), and ethanol (Grumstrup et al., 2007). Driscoll et al. (2010) performed an experiment to address the thin film formation during splashing of viscous liquids. They found that the time of sheet ejection depends on impact velocity, liquid viscosity, molecular weight and gas pressure. Le Goff et al. (2013) experimentally investigated the process of spheres impacted the viscous Newtonian liquids. They found that there was no cavity entrainment due to the spheres stopped before the cavity retracted. Marston et al. (2016) compared the splash characteristics produced by water and per fluorohexane. They found that surface tension played an important role in the crown closure. The interesting buckling phenomena occurred in both water and perfluorohexane crowns indicated much finer and more numerous rib structures for perfluorohexane. These novel observations and founding provided a new perspective for impact features of single-phase liquid system. More recently, Tan and Thomas (2018) experimentally studied the influence of an upper layer liquid on the phenomena and cavity formation. An unusual shallow seal of cavity pinching off close to the liquid surface had seen in their experiments. Moreover, according to the impact experiments of two-layer system of immiscible liquids conducted by Tan (2019), the influence of the vis cosity of an upper-layer liquid on cavity formation and overall dynamics was further highlighted. These works have contributed much to our understanding of water entry. Although the study on water entry has significantly increased during the past decades, little attention is paid to the cavity dynamics involving highly viscous liquid. Moreover, the influence of surface viscous liquid
Fig. 1 shows the schematic of the experimental setup. We conducted the impact using a 1.5 m � 0.8 m � 1.0 m (length � width � depth) glass tank. A standard billiard ball as the test model with a diameter D ¼ 57.2 mm, and mass of the sphere is ms ¼ 0.18 kg. The release mechanism consists of a pneumatic finger fixed on a car. The horizontal initial velocity U0 of the sphere is generated by a motor driving the car and the vertical entry velocity at the still free surface is generated by free pffiffiffiffiffiffiffiffi fall with V0 � 2gH, where g ¼ 9.81 m/s2 is the gravity acceleration and H is the release height. The water in the glass tank was covered by a layer of dimethicone with a density of ρ ¼ 941 kg/m3, surface tension σ ¼ 19.4 mN/m and dynamic viscosity μ ¼ 0.01 Pa s. The density of dimethicone is similar to that of water, but the surface tension and viscosity are significantly different from water. It is helpful to analyze the influence of upper layer highly viscous liquid on cavity dynamics. The thickness of the dimethicone layer h was 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, respectively. The spheres were released at the height of H ¼ 0.95 m to acquire the vertical impact velocity of V0 ¼ 4.32 m/s. And the horizontal impact velocity of the sphere is in a range of U0 ¼ 0.49–1.3 m/s due to the equipment limitations. The mo tion of the sphere and the cavity dynamics were recorded with a highspeed camera (Phantom V12.1) at a frame rate of 3000 frames per second with an image resolution of 1280 � 800 pixels. The value of exposure time used in the present experiments is 1/3000 s. Meanwhile, throughout the experiments the liquid temperature was kept constant at
Sphere
U0
Water tank Car
Pneumatic finger Dimethicone layer
U0 Water
2. Experimental setup
Sphere x
V0
o
h
y
(a)
(b)
Fig. 1. (a) Schematic illustration of the basic experimental setup. (Figure is not to scale.) (b) Definitions of the Cartesian coordinate system xoy and dimethicone layer thickness h. 2
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boundary of the upper layer splash is lower than the upper boundary of the lower layer splash. At the same time, the upper-right corner of the upper layer splash goes downwards, which leads the upper layer splash collapses. Therefore, the presence of the dimethicone layer modifies the splash evolution and seal patterns. In the early stage 4, the cavity below the still free surface appears pinch-off and it splits into two distinct cavities: a lower air cavity still fully attached to the sphere, and an upper air cavity connected to the free surface. It is worth noting that the pinch-off time of the two-layer liquid system is earlier than the water entry case. After pinch-off (t ¼ 110 ms) the cavity begins a rapid collapse below the free surface. Distinct jet of fluid ejects away from the point of pinch-off can be seen both in water entry case and water-dimethicone entry case. This jet phenomenon is also seen by other researchers (e.g. Worthington and Cole, 1897; Truscott and Techet, 2009b). In order to gain further insight into the effect of the upper-layer dimethicone on cavity formation of the oblique impact of a sphere, the comparison of cavity shape between water entry and upper-layer dimethicone entry is shown in Fig. 3. As shown in Fig. 3(a), it should be noted that the wall of the cavity formed under the free surface is smooth, which is consistent with the observations of single phase liquid system (May 1952; Mansoor et al., 2017; Hou et al., 2018). However, it is interesting to note that the cavity wall for water-dimethicone entry case is relatively rough and it is characterized by wave-like instability structure. Due to the difference in viscosity between water and dime thicone, the generation of the wave-like structures is driven by the shear-induced instability which occurs at the contact line between the surrounding water and the entrained dimethicone of the sphere. More over, note that ’ribbed’ structure is observed around the upper left part of the splash wall in water-dimethicone entry case, whereas the splash wall in the water still maintains good continuity. By comparing the pinch-off locations of the two conditions presented in Fig. 3(b), an interesting observation is that the film wall at the rim of the lower splash breaks into a finger-like structure. The wave and rib structure features can be observed more clearly by the zoomed views in Fig. 3. It further highlights the upper-layer dimethicone plays an important role on splash formation and seal pattern. Surface tension is considered to play an important role in deter mining the splash evolution mechanism, and the corresponding impor tant parameter is the Weber number. (Mansoor et al., 2014). It is well known that the Weber number is the ratio between the inertial force and the surface tension force and it is expressed as We ¼ ρLV2/2σ , where L is the characteristic length, V is the velocity of fluid. In the splash, the relevant velocity and length scales must reflect the motion. Hence, the splash Weber number is defined as Wes ¼ ρδV2s /σ , where δ is the thick ness of the splash film, Vs is splash velocity. To analyze the splash stratification patterns, splash formation images at two typical moments and the corresponding schematic of the cavity wall profile and motion velocity at the stratified boundary are shown in Fig. 4. As shown in Fig. 4 (a), obvious splash stratification can be seen on the splash film of the experimental image. Based on the intersection line structure between the lower layer and the upper layer of the splash, we can conclude that the velocity distribution along the intersection line is linear. The instantaneous average velocity of the splash motion in Fig. 4(a) is approximately Vs ¼ 1.2 m/s. Meanwhile, the value of the film thickness is δ � 10 μm according to experimental findings by Zhang et al. (2012) and Marston et al. (2016). Hence, taking δ ¼ Οð10Þ μm and the splash velocity Vs ¼ Οð100 Þ m/s, we can obtain that Wes ¼ Οð10 1 Þ for water and Wes ¼ Οð100 Þ for dimethicone. This indicates surface tension does play a significant role in the splash formation. More importantly, the discontinuity of Weber number at the interface between the upper and lower layers is cause of the splash stratification. Moreover, as shown in Fig. 4(b), it is observed that the shrinkage near the intersection line on the left and right sides of the splash is asymmetrical. Actually, from the schematic illustration of Fig. 4(b), we can know that the splash motion
16 � C to minimize temperature-induced physical properties variation. Table 1 summarizes the experimental conditions in the present study. 3. Results and discussions 3.1. Cavity formation mechanism induced by a thin layer of highly viscous liquid floating on water Following the discussions of May (1975) and Truscott and Techet (2009b), we analyze the cavity formation problem in four distinct stages: (1) the flow forming stage, (2) the open cavity and splash growth stage, (3) the closed cavity and pinch-off stage and (4) the collapsing cavity stage. Fig. 2 presents sequence of images of the cavity formation of the sphere falls into the water-dimethicone and water. In stage 1 of the impact sequence (t ¼ 10 ms and t ¼ 20 ms), an initial horizontal splash of fluids forms as the sphere impact the still free surface and this splash extends radially outwards as the sphere descends into the liquids. At t ¼ 20 ms, the splash transitions from outwards to upwards growth. During stage 1, we can clearly see that the cavity formed by the impact of water-dimethicone are almost the same as the cavity formed by the impact of the water. In stage 2 of the impact sequence (t ¼ 30–60 ms), the entire sphere passes below the still free surface and an open air cavity begins to form in its wake. It is observed that the splash crown and the top of the subsurface air cavity bases, connected to the free surface, increase in diameter. Interestingly, left-right asymmetry in cavity below free sur face and splash above free surface can be observed in stage 2. It is well known that the initial stages of impact of object on the free surface are dominated by inertial effects. Hence, the subsurface air cavity both grows radially outwards and elongates vertically as the sphere descends. However, the splash of sphere entry onto the two-layer liquid system formed by a dimethicone layer floating on the water are quite different than onto only water. For the water-dimethicone entry case as shown in Fig. 2(a), the splash appears stratified and is characterized by upper and lower layers. This unique phenomenon has not been observed by other experiments (Worthington, 1908; Truscott and Techet, 2009b; Smolka and McLaughlin, 2019). Moreover, obvious buckling instability occurs around the upper-right part of the splash wall in the water-dimethicone entry case, which is marked by arrow at t ¼ 40 ms in Fig. 2(a). Whereas, the event of buckling instability is weak in water entry case. According to Marston et al. (2016), the collapse of the splash is mainly driven by surface tension and the buckling occurs when collapse rate is faster than the vertical velocity of the splash crown. Hence, we speculate that small surface tension is beneficial to accelerate the radial shrinkage and the corresponding buckling instability is easy to occur in water-dimethicone entry case. In stage 3 of the impact sequence (t ¼ 70–100 ms), the subsurface air cavity continues to stretch and curve as the sphere descends. The cavity evolution below the free surface in water entry case and waterdimethicone entry case is similar. Meanwhile, as shown in Fig. 2(b), the splash film is closed and no more air can flow into the cavity. However, for the two-layer liquid system presented in Fig. 2(a), the lower-left corner of the upper layer splash shrinks, then the bottom Table 1 Experimental conditions. Parameter
Symbol/Definition
Range/Values
Units
Diameter of sphere Mass of sphere Horizontal initial velocity Vertical initial velocity Dimethicone thickness Splash velocity Froude number Weber number Splash Weber number
D ms U0 V0 h Vs Fr ¼ U0/(gD/2)1/2 We ¼ ρV0D/2σ Wes ¼ ρVsδ/2σ
57.2 0.18 0.49–1.3 4.32 2–8 – 0.93–2.45 – –
mm kg m/s m/s mm m/s – – –
3
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Buckling instability 10 ms
20 ms 30 ms Collapse
70 ms
40 ms
Bottom boundary
Shrinkage
80 ms
Stratified splash
90 ms
50 ms
Jet
Pinch-off 100 ms
60 ms
105.33 ms
110 ms
(a) Cavity formation during the entry of a sphere into a two-layer liquid system that consists of a thin layer (thickness: 4 mm) of dimethicone floating on the surface of a deep water pool.
Single splash 10 ms
20 ms
30 ms
40 ms
50 ms
Pinch-off 70 ms
80 ms
90 ms
100 ms
107.67 ms
60 ms
Jet 110 ms
(b) Cavity formation during water entry of the sphere. Fig. 2. Comparison of the cavity formation of a sphere (D ¼ 57.2 mm) falls into the water-dimethicone and water. The initial moment t ¼ 0 is defined as the bottom of the sphere touching the still free surface. The initial impact velocity of V0 ¼ 4.32 m/s and U0 ¼ 1.25 m/s.
velocity Vsl on the left side is greater than the velocity Vsr on the right side. In other words, the difference in velocity will lead to the difference in Weber numbers. Therefore, the surface tension significantly changes the evolution mechanism of splash during the entry process in the two-layer liquid system. To further elucidate the splash stratification patterns, the schematic representation of instantaneous motion state and cavity profile in xoy plane is shown in Fig. 5. Gillbarg and Anderson (1948) stated that the cavity splash is subject to two chief forces: surface tension σ and a pressure drop ΔP caused by the air flow behind the sphere. It is conceivable that the presence of the pressure drop across the splash film can play a role in splash shrinkage. Visual inspection of the splash for mation just prior to shrinkage and seal as presented in Fig. 2. At these instantaneous moments of the upward motion of the cavity splash, the surface tension in the radial direction is not enough to overcome the inertial force. Therefore, no splash shrinkage is formed along the crown wall near the intersection line. However, the surface tension in the radial direction becomes prominent when the upper layer splashes slowly or even stops rising. Meanwhile, the cavity splash formed by water below the intersection line continues to expand due to inertial effects and from energy transfer by sphere at its instantaneous location. As a result, splash shrinkage appears at the junction of dimethicone and water of the splash.
3.2. Influence of the thickness of viscous liquid layer on cavity formation To analyze the cavity evolution and splash formation quantitatively, we introduce four parameters θ, dp, tp and dc. The definition of these parameters are presented in the following and shown in Fig. 6. The angle between the splash layered line and the horizontal direction is denoted as θ. The depth at which cavity pinch-off occurs is denoted dp and the corresponding pinch-off time is defined as tp. The total entry depth of the sphere dc is defined as the distance from the undisturbed free surface to the leading edge of the sphere at any moment during the water entry process. Fig. 7 shows the splash and cavity formation by the impact of spheres at different moments with the dimethicone layer thickness h of 0–8 mm. For t ¼ 70 ms as shown in Fig. 7(a), stratified splash can be observed when the dimethicone layer thickness varied from 2 mm to 8 mm, and the stratified boundary height decreases as the thickness of the dime thicone layer increases. On the one hand, it is indicated that the source of the liquid that forms the splash is more than 8 mm, and on the other hand, also indicates that the energy of the sphere during the water entry process is distributed to different regions of the splash. Note that for the 4 mm and 5 mm cases, the splash shrinkage appears on the left side of the stratified interface. Moreover, compared with the water entry case, a more pronounced ribbed structure can be seen in the splash seal region of the dimethicone-water entry cases. At the pinch-off time as shown in Fig. 7(b), it is observed that the obvious collapse of the upper layer splash occurs in the 3 mm and 4 mm 4
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4 mm. Therefore, the thickness of dimethicone is an important factor for the collapse phenomenon in the cavity splash, and we speculate that there may be a critical dimethicone thickness value between 2 mm and 5 mm in the present experiment. Meanwhile, by comparing the pinch-off depth dp (marked by solid line in Fig. 7(b), dimethicone-water entry cases are marked by black line and water entry case is marked by red line) and the entry depth dc (marked by dashed line in Fig. 7(b), dimethicone-water entry cases are marked by black line and water entry case is marked by red line), we found that the pinch-off positions and the entry depth of the different thickness of dimethicone-water entry cases are almost the same. How ever, the values of dp and dc in the dimethicone-water entry cases are smaller than that of the water entry case. This indicates that the high viscosity dimethicone floating above the water causes the pinch-off to occur earlier, but the thickness value range of the dimethicone we studied has almost no effect on the pinch-off position. For t ¼ 120 ms after pinch-off moment as shown in Fig. 7(c), it can be seen that the
Ribs
Smooth Water
Rough Dimethicone water
(a) t = 85 ms Fingers
Water
Vs
Dimethicone water
(b) Pinch-off time
Stratification line
Water film
Fig. 3. Cavity formation comparison between water entry and waterdimethicone entry at different times for the cases with entry velocity V0 ¼ 4.32 m/s and U0 ¼ 1.25 m/s. Compared to Fig. 2, the enlarged images in Fig. 3 highlights the stratified splash phenomenon, see text for the detailed explanation.
Dimethicone film Vs
No air flow
Air flow
No air flow a w
thickness of the dimethicone-water entry cases. However, the collapse intensity is weaker in the 2 mm and 5 mm dimethicone-water entry cases at this moment. As shown in Fig. 7, note that the water film in cavity splash of the 2 mm working condition is high enough to cause the splash seal to occur mainly on the upper side of the water film. Whereas, for the 5 mm test case, the rise height of the water film is limited due to the energy distribution of the falling sphere. Hence, it is observed that splash seal occurs on the upper side of the dimethicone layer, that is, away from the position of the splash stratification. Especially, the splash seal area is close to the stratified position, resulting in a more pronounced collapse and shrinkage when the thickness of dimethicone layer is 3 mm and
W0
D Fig. 5. Schematic representation of instantaneous motion state and cavity profile in xoy plane.
Vsl Dimethicone film
Splash stratification
Vs
Vsr
Water film Intersection line
(a) Splash stratification and the corresponding schematic illustration of the cavity wall profile and motion velocity. The entry time is t = 30 ms.
Shrinkage Intersection line
Dimethicone film
Vsl Dimethicone film Vs
Shrinkage Water film
Intersection line
Vsr
(b) Splash shrinkage and the corresponding schematic illustration of the cavity wall profile and motion velocity. The entry time is t = 60 ms. Fig. 4. Experimental splash images at two typical moments and the corresponding schematic illustration of the splash stratification and shrinkage. The initial impact velocity of V0 ¼ 4.32 m/s, U0 ¼ 1.25 m/s. The thickness of the dimethicone layer is h ¼ 4 mm. 5
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more in number and larger in scale for the dimethicone-water entry cases compared to the water entry. Therefore, the presence of the dimethicone layer leads to a more complicated splash seal mechanism. In addition, the shrinkage tendency at the interface of the upper and lower layers of the splash becomes weaker as the thickness of the dimethicone increases. Fig. 8(b) presents the splash pattern when the angle between the splash layered line and the horizontal direction θ � 12.5� . As can be seen from Fig. 8(a), the smaller the thickness of the dimethicone layer, the shorter the time required to reach a certain angle. In fact, the splash for these four cases is in different stages of evolution. For the h ¼ 2 mm case, it is the development stage of the splash crown. For the h ¼ 3 mm case, the upper layer of the splash begins to occur buckling instability. For the h ¼ 5 mm case, it is in the early stage of splash seal of the upper layer cavity. For the h ¼ 7 mm case, the splash is almost closed. Due to the sphere-liquid interaction, the kinetic energy of the sphere transfers to the surrounding fluids as the sphere decelerates. The water in splash will obtain less energy as the thickness of dimethicone increases. As a result, the angular velocity of the splash layered line of the thick dimethicone layer cases is smaller than that of the thin ones. Hence, it will take more time to reach the same angle of the splash layered line for the thicker dimethicone layer cases due to the energy distribution and the presence of the dimethicone layer.
dp
3.3. Influence of the horizontal impact velocity on cavity dynamics In order to analyze the effect of horizontal impact velocity on cavi tation formation, a comparison of the cavity morphology at four hori zontal impact velocities with a dimethicone thickness of 2 mm is given in Fig. 9. For t ¼ 70 ms before pinch-off moment as shown in Fig. 9(a), it is important to note that the size of the opening at the top of the splash is different and appears to increase with the increase of horizontal impact velocity. In other words, increasing the horizontal impact velocity causes a delay in the splash closure. Moreover, the angle θ increases as the speed increases, indicating that the large horizontal impact velocity causes the cavity asymmetry to be more pronounced. This observation is further confirmed at the pinch-off position (as shown in Fig. 9(b)), in particular, a complete splash seal is formed when U0 ¼ 0.51 m/s. As shown in Fig. 9(c), there is a line closure pattern observed (as marked with arrows in Fig. 9(c)) in the experiments due to the pressure differ ence between the inside and outside of the cavity under the still free surface. It is worth noting that there is no bubbles trace generated after pinch-off in the symmetrical water entry cases (Grumstrup et al., 2007; Aristoff and Bush, 2009; Tan and Thomas, 2018; Sun et al., 2019). Whereas, the bubbles trace was found in the asymmetrical water entry process of projectiles (Bodily et al., 2014). Due to the formation of the jet and the asymmetry of cavity in the oblique entry cases, the boundary of the cavity wall is easier to roll up into a ring vortex, which entrains air along the cavity wall and the subsequent shedding of gas bubbles at the tail of the attached cavity over the sphere (Spurk, 2002; Karn et al., 2016). Moreover, with the increase of horizontal impact velocity, the connected bubbles trace shifts from curve to approximate straight line due to the inertia effects. Curves of the horizontal and vertical displacement as a function of time at the above four different initial vertical impact velocities are shown in Fig. 10. The curves indicate that relation between displace ment and time follow a more non-linear relationship. Note that the horizontal displacement of the sphere rise faster with increasing impact velocity. However, as can be seen in Fig. 10(b), the values in the vertical displacement are almost the same for different initial vertical impact velocities. Therefore, we speculate that the vertical drag force during the dimethicone-water process is independent of the horizontal impact ve locity. To support this conclusion, we will analyze the force model of sphere during the entry case in the next part. In order to determine the sphere trajectory of the above analysis, Fig. 11 presents the instantaneous force model of sphere motion. The
dc
Fig. 6. Definitions of cavity parameters. The angle θ between the upper boundary of the lower layer splash and the horizontal direction, pinch-off depth dp and cavity depth dc.
water film breaks into a finger-like structure at the left rim of the lower crown in the 3 mm and 4 mm dimethicone-water entry cases. Interest ingly, a connected bubbles trace can be observed between the upper cavity near the free surface and the cavity attached to the tail of the sphere and the closure depth of the upper cavity under the free surface almost the same as the dimethicone thickness increases. The above re sults indicate that the thickness of dimethicone affects the splash for mation and evolution process. To further clarify the influence of the dimethicone layer and the thickness on the splash formation, enlarged views of the splash morphology above the free surface in water entry case and dimethiconewater entry cases with different thickness of dimethicone are shown in Fig. 8. Fig. 8(a) compares the results at the same water entry time of t ¼ 60 ms, it is noteworthy that the ribbed structures in the crown are 6
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Ribs Shrinkage
h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(a) Comparison of the splash and cavity formation at the same instantaneous moment before pinch-off. All images are taken at t = 70 ms.
Collapse
h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(b) Comparison of the splash and cavity formation at the pinch-off location. Fingers
Bubbles trace h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(c) Comparison of the splash and cavity formation at the same instant after pinch-off. All images are taken at t = 120 ms. Fig. 7. Splash and cavity formation by the impact of spheres at different moments with the dimethicone layer thickness h of 0–8 mm. The moment when the bottom of the sphere touches the still free surface is defined as the initial time t ¼ 0. The initial impact velocity of V0 ¼ 4.32 m/s and U0 � 0.75 m/s.
Fig. 8. Comparison of splash formation by the influence of the dimethicone layer thickness. The initial impact velocity of V0 ¼ 4.32 m/s and U0 � 1.0 m/s.
trajectory of the sphere at multiple times during the entry process is marked by red dashed line. It is worth noting that the sphere almost keeps linear motion within the time range of the present study. The coordinate directions have been defined in Fig. 1(b), and are also shown in Fig. 11(c). The force balance equation of oblique motion of the sphere is expressed as
(ms þ ma)as ¼ G þ FB þ FD þ FT
(3.1)
where ms ¼ ρs Vsphere ¼ 43 πρs R3s is the mass of the sphere, ρs is the density of the sphere, Vsphere is the volume of the sphere, and Rs is the radius of the sphere; The force required to accelerate the surrounding liquid is 7
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1.2
U0 = 0.51 m/s U0 = 0.74 m/s
U0 = 0.88 m/s U0 = 1.04 m/s
x/D
0.9
0.6
0.3
0.51 m/s
0.74 m/s
0.88 m/s
1.04 m/s
0.0 0.00
(a) t = 70 ms.
0.03
0.06
t (s)
0.09
0.12
(a) Time history curves of horizontal displacement of the sphere.
Splash seal
U0 = 0.51 m/s 6
U0 = 0.74 m/s U0 = 0.88 m/s
y/D
U0 = 1.04 m/s 4
2
0.51 m/s
0.74 m/s
0.88 m/s
1.04 m/s 0 0.00
(b) Pinch-off position.
0.03
0.06
t (s)
0.09
0.12
(b) Time history curves of vertical displacement of the sphere. Fig. 10. Comparison of horizontal displacement and vertical displacement of the sphere at different initial horizontal impact velocities of U0 ¼ 0.51 m/s, 0.74 m/s, 0.88 m/s, and 1.04 m/s. The corresponding initial vertical impact velocity V0 ¼ 4.32 m/s. The dimethicone layer thickness is h ¼ 2 mm.
Line
Curve
0.51 m/s
0.74 m/s
0.88 m/s
However, Cm is usually considered to be a constant of 0.5 (Newman, 1977; Truscott et al., 2012; Tan, 2019; Wang et al., 2019). Hence, we take it as a constant Cm ¼ 0.5 for analysis in the present study, and the added mass is given as ma ¼ 0:5ρw Vsphere . For the buoyancy force, ignoring the initial partial submerged process, it can be expressed as
FB ¼ -ρwgVsphere. The drag force is FD ¼ 0:5Cd ρw Ap jWj2 , where Cd is the drag coefficient, and it is typically taken to be constant for varying impact velocities and penetration depth in studies of subsonic water entry cases (Lee et al., 1997; Abraham et al., 2014); Ap is the projected wetted area of the submerged portion; W ¼ U þ V is the total velocity. Meanwhile, as shown in Fig. 11(b), the dynamic contact angles on the left and right sides of the sphere are defined as α and β, respectively. It is worth noting that both α and β are almost constant in the stable entry stage of the sphere. The projected wetted area can be expressed as Ap ¼
1.04 m/s
(c) t = 120 ms. Fig. 9. The image sequence depicting splash and cavity evolution caused by the sphere impacts still free surface with increasing initial horizontal impact ve locities of U0 ¼ 0.51 m/s, 0.74 m/s, 0.88 m/s, and 1.04 m/s. The corresponding initial vertical impact velocity V0 ¼ 4.32 m/s. The dimethicone layer thickness is h ¼ 2 mm.
πR2s in direction of movement of the sphere when the entry is more than
one diameter depth. However, the effects of initial impact conditions and gravity will cause the values of α and β to be different. Therefore, we introduce a correction coefficient Ko, which is a constant close to 1, and the corresponding projected wetted area becomes Ap ¼ Ko πR2s . More over, the surface tension force FT can be neglected as it is less than 1% of the gravitational force for the water entry of standard billiard balls (Truscott and Techet, 2009b). As mentioned in Section 2, the surface tension coefficient of dimethicone is much smaller than that of water, and Tan (2019) demonstrated that FT was on a much lower percentage of FD. Therefore, even if there is a dimethicone layer, the surface tension can be negligible in the force model. By making as the subject and simplifying Eq. (3.1), the equations in the x and y directions can be expressed as:
expressed as the added mass ma ¼ Cm ρw Vsphere , where Cm is the added mass coefficient, and ρw is the liquid density; as is the acceleration of the sphere; G ¼ mg is the gravity; FB is the buoyancy force; FD is the drag force; FT is the surface tension force. According to Aristoff et al. (2010), the added mass force maas should appear as an independent term at relatively small Froude numbers during the process of water entry. The maximum Froude number in this paper is no more than 10, therefore, the added mass force cannot be ignored. Meanwhile, it is noted that the added mass coefficient Cm is related to the submerged depth, and it could go from zero at first impact and up to 0.5 when the sphere is one radius deep (Shepard et al., 2019). 8
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Ocean Engineering 194 (2019) 106597
Cavity profile
20 ms
40 ms
FD
FB
60 ms
i
80 ms
j
100 ms
W=U+V
(a)
(b)
G
(c)
Fig. 11. (a) The trajectory of the sphere at multiple times during the entry process. (b) Instantaneous image of sphere motion, the red dashed line represents the trajectory of the sphere. (c) Schematic diagram of instantaneous force model of the sphere.
8 > > > < x€ ¼ asx ¼
Cd Ko ρw 2 U sinφ 4Rs ρs � � > 2 ρw Cd Ko ρw 2 > > 1 gþ V cosφ : y€ ¼ asy ¼ 3 ρs 4Rs ρs
(3.2)
where φ is the angle between FD and bj, as shown in Fig. 11(c). Based on the previous analysis, the value of φ is almost a constant due to the straight track of the sphere in the present study. In Eq. (3.2), it is indicated that the sphere will receive larger hori zontal acceleration at higher initial horizontal impact velocity. This can better explain the variation trend of the displacement curve at different initial horizontal impact velocities in Fig. 10(a). According to asy in Eq. (3.2), since the initial vertical impact velocities of the four conditions in Fig. 10(b) are the same, the vertical displacement versus time curves under different conditions present the same trend. The force model in Eq. (3.2) is mainly suitable for oblique impact of sphere into water for a relatively short time in the present study. We can speculate that the trajectory of the sphere will not remain straight at all times if the experimental recording time is long enough. As a result, the parameters Ko and φ in Eq. (3.2) will be time dependent. However, it should be noted that the measurement of these forces are also very difficult until now and the subtleties of these differences difficult to pinpoint. In order to better establish the force model, more measurement methods and flow field information are necessary in the future work.
Fig. 12. The angle θ with different dimethicone layer thickness of h ¼ 3 mm, pffiffiffiffiffi 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of U0/ hg at the pinch-off locations. Solid curves are fitted data and markers are experimental results.
pffiffiffiffiffiffiffiffiffiffi plotted against Fr in Fig. 13, where Fr ¼ U0/ gD=2, U0 is the initial horizontal impact velocity at the still free surface, g is the gravitational acceleration and D is the diameter of the sphere). Interestingly, we observe roughly linear relationships between tp* and Fr. Based on the experimental data fitting, the measured dimensionless pinch-off time is tp* � 0.96 Fr, and tp* increases as the Froude number increases. Lee et al. (1997) rationalized the experiments of Gilbarg and Anderson (1948) regarding the apparent independence of the dimensionless pinch-off time on the vertical impact velocity. More recently, Truscott and Techet (2009b) provided further experimental evidence that the pinch-off time, tpV0/D ~ Fr, is roughly constant in water entry. Our experimental findings are generally similar to their measurements. However, our Fr is based on the horizontal impact velocity and is different from their vertical velocity definition. Moreover, the measured pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pinch-off time is tp� 1:92 D=ð2gÞ for all the water-dimethicone entry cases in the present study. Duclaux et al. (2007) found the prefactor k to be 2.06 for the impact of spheres and the corresponding relationship is tp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2:06 D=ð2gÞ. The difference of the prefactor k between water-dimethicone entry cases and water entry cases indicates the pinch-off time is effected by the dimethicone layer. Hence, we conclude that the presence of high viscosity dimethicone contributes to the cavity shrinkage and the pinch-off occurs earlier. Furthermore, based on the
3.4. Splash angle and typical pinch-off parameters characteristics In this section, we further quantitatively assess the effects of dime thicone and initial impact velocity on the cavity dynamics. Fig. 12 presents the angle θ mentioned above as a function of non-dimensional pffiffiffiffiffi velocity parameter U0* at the pinch-off locations, where U0* ¼ U0/ hg. From Fig. 7(b), it is observed that no obvious oil and water boundary line is formed in the splash for the dimethicone-water entry case of h ¼ 2 mm. Hence, the thickness of the dimethicone layer is in the range 3 mm � h � 8 mm in Fig. 12. In the process of extracting test results, conversion from pixels to angles yields an uncertainty of � 1� . It should be noted that the symbols denote experimental data and the corre sponding fitted curves are presented with second-order function. It in dicates that the value of θ increases as the decreases of the thickness of dimethicone layer, and it decreases as the increases of the original horizontal impact velocity. Pinch-off time is non-dimensionalized as tp* ¼ tpU0/D in Fig. 13. In the process of extracting test results, conversion from pixels to times yields an uncertainty of �0.66 ms in tp. It should be noted that deep seal is driven by hydrostatic pressure and thus pinch-off time scales with Froude number Fr (Speirs et al., 2018). The dimensionless time tp* is 9
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Ocean Engineering 194 (2019) 106597
h = 2 mm h = 3 mm h = 4 mm h = 5 mm
1.8 1.6
horizontal impact velocity. Fig. 15 shows the values of the water entry depth dc of the water entry cases and the dimethicone-water entry cases at the pinch-off mo ments as a function of the Froude number Fr. In the process of extracting test results, conversion from pixels to millimeters yields an uncertainty of �1.94 mm in dc. It is worth noting that the values of dc almost keep constant with the increase of Fr for the both types of water entry case and the dimethicone-water entry cases. In other words, here, in Fig. 15, the dc is independent of the horizontal impact velocity or, if a depen dence existed, it is very weak. Hence, combined with the experimental results in Fig. 7(b), it is further confirmed here that the values of dc in water are larger than that of dimethicone-water. It indicates that the presence of the dimethicone layer will affect the falling trajectory of the sphere. Meanwhile, an interesting finding is that the dc is independent of the thickness of the dimethicone layer under the experimental condi tions in the present study. Fig. 16 shows the values of the pinch-off depth as a function of the thickness of dimethicone layer. Based on the experimental data, we can get the value of the pinch-off depth is almost constant, where dp ¼ 0.161 m. For water entry cases, Duclaux et al. (2007) found the ratio of pinch-off depth to total entry depth is constant at dp/dc ¼ 0.5 and it was also experimentally confirmed by Speirs et al. (2018). However, the ratio of dp to dc is approximately 0.48 for the dimethicone-water entry cases in the present study. In other words, the pinch-off depth is less than half of the total depth. It indicates that the highly viscous dimethicone layer can promote the shrinkage of the cavity under the undisturbed free surface and subsequently leads to the early occurrence of pinch-off.
h = 6 mm h = 7 mm h = 8 mm
tpU0/D
1.4 1.2
0.96
1.0 0.8
1.0
1.2
1.4
1.6
1.8
Fr Fig. 13. Dimensionless pinch-off time with different dimethicone layer thick ness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number based on the initial horizontal impact (U0). Solid lines are fitted data and markers are experimental results. Present experimental data for the system with dimethicone above water, linear least-squares fit, tp* � 0.96 Fr.
pinch-off time tp, we can also draw that the cavity pinch-off time is in dependent of the thickness of the dimethicone layer and the horizontal impact velocity. Fig. 14 shows our experimental data for the dimensionless pinch-off depth dp* as a function of Fr 1, where dp* is expressed as dp/(U0tp). In the process of extracting test results, conversion from pixels to millimeters yields an uncertainty of �1.94 mm in dp. The scatter points with the same shape in Fig. 14 represent different thickness of dimethicone layer at the same horizontal impact velocity. We observe that a robust linear relationship between the dimensionless pinch-off depth and Fr 1. The slope k of dp* ~ Fr 1 for all the water-dimethicone entry cases is 6.15. Meanwhile, the pinch-off depth dp ¼ 5.9 D, it indicates that the pinch-off depth dp is independent of the thickness of the dimethicone layer and the
7
6
h = 2 mm h = 3 mm h = 4 mm h = 5 mm
4. Conclusions In the present study, the cavity dynamics of oblique entry of sphere through a viscous liquid floating on the water at low Froude number was investigated experimentally. High-speed camera system was used to captures the cavity formation and dynamics features for both qualitative and quantitative analysis. Meanwhile, water entry case was also per formed as a comparison to highlight the unique nature of cavity dy namics induced by the upper-layer of highly viscous liquid. From the experiments results, the main conclusions can be drawn as follows:
h = 6 mm h = 7 mm h = 8 mm
0.38
h = 0mm h = 2mm h = 3mm h = 4mm
h = 5mm h = 6mm h = 7mm h = 8mm
m/s
U0
m/s
6.15
5
dc = 0.345 m dc (m)
dp / (U0 tp)
0.36
0.34
dc = 0.335 m
4 0.32
3 0.5
0.6
0.7
0.8
0.9
1.0
1.1
-1
0.30
Fr
Fig. 14. Dimensionless pinch-off depth with different dimethicone layer thickness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number based on the initial horizontal impact (U0). Solid lines are fitted data and markers are experimental results. Present experimental data for system with dimethicone above water, linear least-squares fit, dp/ (U0tp) � 6.15 Fr 1.
1.0
1.2
1.4
1.6
1.8
Fr Fig. 15. Cavity length (dc) at pinch-off time with different dimethicone layer thickness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number (Fr) based on the initial horizontal impact (U0). Dashed lines are fitted data and markers are the experimental results. 10
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Ocean Engineering 194 (2019) 106597
U0 = 0.98
0.06 m/s
U0 = 0.86
0.06 m/s
although we have explained the splash stratification, it still needs to be strengthened in the future. Especially, setting up liquid dynamics model based on mathematical method with viscosity and density as changing parameters at the junction of dimethicone and water will help us improve understanding of the mechanism of splash stratification.
U0 = 0.75
0.03 m/s
U0 = 0.53
0.03 m/s
Acknowledgments
0.19
dp (m)
0.18
0.17
This work was supported by the National Natural Science Foundation of China (51709042, 51679037, 51639003), the China Postdoctoral Science Foundation (2019T120211, 2018M631791), the Natural Sci ence Foundation of Liaoning Province of China (20180550619), and the Fundamental Research Funds for the Central Universities (DUT18RC(4) 018, DUT2017TB05).
dp=0.161 m 0.16
References
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2
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h (mm)
6
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8
Fig. 16. Pinch-off depth (dp) as function of the thickness of dimethicone layer (h).
(1) The highly viscous liquid floating on the water substantially modifies the cavity formation mechanism. Compared to contin uous splash in water, the splash appears stratified and is char acterized by upper and lower layers for the water-dimethicone entry case. The discontinuity of Weber number at the interface of water and dimethicone due to the difference in surface tension is the most likely reason of stratified splash. (2) The thickness of dimethicone layer affects the cavity evolution patterns. Compared with the water entry case, a more pro nounced ribbed structure can be seen in the splash seal region of the dimethicone-water entry cases. The high viscosity dimethi cone floating on the water causes the pinch-off to occur earlier, but the thickness value range of the dimethicone we studied has almost no effect on the pinch-off position. Moreover, we found that the smaller the thickness of the dimethicone layer, the faster the intersection line in the splash reaches the same angle due to difference in the energy distribution of the dimethicone layer. (3) Through the analysis of the cavity formation, trajectory and force models of the sphere at different horizontal impact velocity, it is found that increasing the horizontal impact velocity will causes a delay in the splash closure. Meanwhile, it is observed that the vertical displacement during the dimethicone-water process is independent of the horizontal impact velocity. The hydrody namic force model of the sphere is established and subsequent explanation for the oblique water entry problem with a high viscosity dimethicone floating on the water. (4) Splash angle, pinch-off depth, pinch-off time and the water entry depth at pinch-off locations are obtained qualitatively and quantitatively by processing the experimental images. It is observed that roughly linear relationships between the dimen sionless pinch-off time tpU0/D and the Froude number, and the expression is tpU0/D � 0.96 Fr. Meanwhile, the pinch-off depth dp is independent of the thickness of the dimethicone layer and the horizontal impact velocity. Moreover, the entry depth dc almost keep constant at the pinch-off time as the increase of Fr for the both types of water entry case and the dimethicone-water entry cases within thickness range in the present study. The present work has provided experimental results to advance the understanding of the effects of highly viscous liquid layer on the cavity dynamics of oblique impact of sphere. As for the future research, a group of experiments with large thickness value of dimethicone layer would beneficial to expand some conclusions in this paper. In addition, 11
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Rosellini, L., Hersen, F., Clanet, C., Bocquet, L., 2005. Skipping stones. J. Fluid Mech. 543, 137–146. Shepard, T.G., Kane, S., Wielgos, S., Eshmawy, A., 2019. Trajectory model for vertical sphere water-entry in presence of deep-seal cavity. Appl. Ocean Res. 82, 478–488. Shams, A., Jalalisendi, M., Porfiri, M., 2015. Experiments on the water entry of asymmetric wedges using particle image velocimetry. Phys. Fluids 27 (2), 027103. Smolka, L.B., McLaughlin, C.K., 2019. Sphere entry through an oil lens floating on water. Phys. Rev. Fluids 4 (4), 044001. Speirs, N.B., Pan, Z., Belden, J., Truscott, T.T., 2018. The water entry of multi-droplet streams and jets. J. Fluid Mech. 844, 1084–1111. Speirs, N.B., Mansoor, M.M., Belden, J., Truscott, T.T., 2019. Water entry of spheres with various contact angles. J. Fluid Mech. 862. Spurk, J.H., 2002. On the gas loss from ventilated supercavities. Acta Mech. 155, 125–135. Sun, T., Wang, H., Zong, Z., Zhang, G., Wang, A., Xu, C., 2019. Splash formation and cavity dynamics of sphere entry through a viscous liquid resting on the water. AIP Adv. 9 (7), 075211. Tan, B.C.W., Thomas, P.J., 2018. Influence of an upper layer liquid on the phenomena and cavity formation associated with the entry of solid spheres into a stratified twolayer system of immiscible liquids. Phys. Fluids 30 (6), 064104.
Tan, B.C.W., 2019. Cavity emergence and the increase in drag following the entry of solid spheres into a stratified, two-layer system of immiscible liquids. Phys. Fluids 31 (2), 022104. Truscott, T.T., Techet, A.H., 2009. A spin on cavity formation during water entry of hydrophobic and hydrophilic spheres. Phys. Fluids 21 (12), 121703. Truscott, T.T., Techet, A.H., 2009. Water entry of spinning spheres. J. Fluid Mech. 625, 135–165. Truscott, T.T., Epps, B.P., Techet, A.H., 2012. Unsteady forces on spheres during freesurface water entry. J. Fluid Mech. 704, 173–210. Wang, J., Faltinsen, O.M., Lugni, C., 2019. Unsteady hydrodynamic forces of solid objects vertically entering the water surface. Phys. Fluids 31 (2), 027101. Worthington, A.M., 1908. A Study of Splashes. Longmans, Green, and Company. Worthington, A.M., Cole, R.S., 1897. V. Impact with a liquid surface, studied by the aid of instantaneous photography. Philos. Trans. R. Soc. Lond. - Ser. A Contain. Pap. a Math. or Phys. Character 189, 137–148. Worthington, A.M., Cole, R.S., 1900. IV. Impact with a liquid surface studied by the aid of instantaneous photography. Paper II. Philos. Trans. R. Soc. Lond. - Ser. A Contain. Pap. a Math. or Phys. Character 194 (252–261), 175–199. Zhang, L.V., Toole, J., Fezzaa, K., Deegan, R.D., 2012. Evolution of the ejecta sheet from the impact of a drop with a deep pool. J. Fluid Mech. 690, 5–15.
12
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Experimental study on the cavity dynamics of oblique impact of sphere on a viscous liquid floating on water Tiezhi Sun a, b, Heng Wang a, Li Zou a, b, c, *, Zhi Zong a, b, c, Haitao Li a a
School of Naval Architecture, Dalian University of Technology, Dalian, 116024, China State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian, 116024, China c Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, 200240, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: Oblique impact Two-layer liquid Splash formation Cavity dynamics Pinch-off
Oblique water entry of marine structure involves complex hydrodynamics phenomena including splash forma tion, asymmetrical cavity and pinch-off that have been observed in previous systematic experiments. However, very few studies focus on such phenomena induced by oblique impact of object in a two-layer liquid system. In this paper, we present an experimental investigation of oblique entry of a sphere into a deep pool of water within a layer of highly viscous dimethicone floating on the water. In the experiment, a high speed digital camera is established to obtain the instantaneous cavity and motion trajectories for both qualitative and quantitative analysis. The results show that the upper-layer viscous liquid introduces a significant change in the splash for mation mechanism and the special behaviors are characterized by stratified splash and corresponding asym metrical shrinkage structures. Meanwhile, it is observed that the difference in splash characteristics caused by different thicknesses of viscous fluid layer is obvious, and the pinch-off depth is independent of the thickness value which studied in our experiments. In addition, it is found that increasing the horizontal impact velocity will cause a delay in the splash closure and the vertical displacement of the sphere during the dimethicone-water enter process is independent of the horizontal impact velocity. Hydrodynamic force model is established to explain the trajectory of the sphere. Furthermore, the splash angle, pinch-off depth, pinch-off time and the water entry depth at pinch-off locations and the corresponding change tendency are obtained qualitatively and quantitatively. These results indicate that the dimethicone layer is critical in determining the cavity dynamics, especially splash formation and pinch-off characteristics.
1. Introduction Studies of objects impacting into the water has attracted the atten tion of researchers for over a century and is still a hot topic of current research (Worthington and Cole, 1897; Wang et al., 2019). The water entry problem is relevant to application of air-to-sea projectile (May 1952), ship slamming (Faltinsen, 1990), and stone skipping (Rosellini et al., 2005). The phenomenon of solid-liquid impact is accompanied by many modern cavity dynamics, such as splash inception, cavity forma tion, splash seal and pinch-off (Aristoff et al., 2010; Chen et al., 2019; Jiang et al., 2018; Mansoor et al., 2014; Sun et al., 2019). Particularly, one of the important problems has long been studying is the splash behavior and cavity dynamics during a solid object impacting liquid process. Water entry is a challenging problem, and associated with the
complex hydrodynamic feature. The pioneering experiments were con ducted by Worthington and Cole (1897, 1900) and Worthington (1908). They obtained the first high-speed photographs of the impact dynamics and cavity formation. Following Worthington’s works, May (1951; 1952) experimentally studied the effects of surface conditions on splashes formations and the types of cavity closures. Since then, surface characteristic of water entry objects has been studied extended to wettability (Duez et al., 2007), hydrophobicity (Korkmaz and Güzel, 2017), and temperature (Li et al., 2018). The effect of hydrophilic and hydrophobic surface of spheres on cavity formation was studied by Truscott and Techet (2009a), they found that transverse spin altered the spheres surface velocity distribution and cavity formation. In addition to these notable studies, other works on water entry problems involves the influence of object shape on cavity dynamics, such as spheres (May and Woodhull, 1948; Belden et al., 2016), rigid projectile (Lee et al., 1997),
* Corresponding author. School of Naval Architecture, Dalian University of Technology, Dalian, 116024, China. E-mail address: [email protected] (L. Zou). https://doi.org/10.1016/j.oceaneng.2019.106597 Received 29 July 2019; Received in revised form 15 September 2019; Accepted 13 October 2019 Available online 1 November 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.
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content and Froude number on cavity formation and dynamics mecha nism during water entry of objects are still not well understood. Hence, the aim of this experimental study is to provide new phenomenon of a thin layer of highly viscous liquid resting on a deep pool of water to alter splash characteristics and cavity dynamics of oblique impact spheres. Experiments with clean water using the same spheres are investigated to highlight and support the present findings. In Sec. II, we present our experimental setup, including the facility parameters, liquid properties and test cases. Section III first summarizes the effect of the highly viscous liquid layer on the cavity formation mechanism. Subsequently, the in fluence of the thickness of viscous liquid layer and the horizontal impact velocity on the cavity dynamics are analyzed. Moreover, we mainly analyze the characteristic of the splash angle, pinch-off time, and pinchoff depth. Finally, the conclusions are presented in Section IV.
slender axisymmetric bodies (Bodily et al., 2014) and wedges (Shams et al., 2015). Duclaux et al. (2007) performed an experimental study on the collapse of a transient cavity of air in water created by the impact of a sphere and analyzed the cavity evolution and pinch-off features. Speirs et al. (2019) used spheres to investigate the cavity shape, splash crown and scale pinch-off times. They found that there were three kinds of splash crowns, including thick rim forms, thin crown and classical cavities. Li et al. (2019) investigated the multiphase flow during water entry of spheres with different surface wettability at low Froude numbers. The results showed that the impact of the hydrophilic case creates a smaller splash and the hydrophilic and hydrophobic spheres experience different cavity regimes as the Froude number increases. Particular attention was paid to address the influence of liquid properties on cavity dynamics. The associated experiments involved multiple characteristic liquids, including glycerine-water mixture (Bell, 1924), viscoelastic liquid (Cheny and Walters, 1996), and ethanol (Grumstrup et al., 2007). Driscoll et al. (2010) performed an experiment to address the thin film formation during splashing of viscous liquids. They found that the time of sheet ejection depends on impact velocity, liquid viscosity, molecular weight and gas pressure. Le Goff et al. (2013) experimentally investigated the process of spheres impacted the viscous Newtonian liquids. They found that there was no cavity entrainment due to the spheres stopped before the cavity retracted. Marston et al. (2016) compared the splash characteristics produced by water and per fluorohexane. They found that surface tension played an important role in the crown closure. The interesting buckling phenomena occurred in both water and perfluorohexane crowns indicated much finer and more numerous rib structures for perfluorohexane. These novel observations and founding provided a new perspective for impact features of single-phase liquid system. More recently, Tan and Thomas (2018) experimentally studied the influence of an upper layer liquid on the phenomena and cavity formation. An unusual shallow seal of cavity pinching off close to the liquid surface had seen in their experiments. Moreover, according to the impact experiments of two-layer system of immiscible liquids conducted by Tan (2019), the influence of the vis cosity of an upper-layer liquid on cavity formation and overall dynamics was further highlighted. These works have contributed much to our understanding of water entry. Although the study on water entry has significantly increased during the past decades, little attention is paid to the cavity dynamics involving highly viscous liquid. Moreover, the influence of surface viscous liquid
Fig. 1 shows the schematic of the experimental setup. We conducted the impact using a 1.5 m � 0.8 m � 1.0 m (length � width � depth) glass tank. A standard billiard ball as the test model with a diameter D ¼ 57.2 mm, and mass of the sphere is ms ¼ 0.18 kg. The release mechanism consists of a pneumatic finger fixed on a car. The horizontal initial velocity U0 of the sphere is generated by a motor driving the car and the vertical entry velocity at the still free surface is generated by free pffiffiffiffiffiffiffiffi fall with V0 � 2gH, where g ¼ 9.81 m/s2 is the gravity acceleration and H is the release height. The water in the glass tank was covered by a layer of dimethicone with a density of ρ ¼ 941 kg/m3, surface tension σ ¼ 19.4 mN/m and dynamic viscosity μ ¼ 0.01 Pa s. The density of dimethicone is similar to that of water, but the surface tension and viscosity are significantly different from water. It is helpful to analyze the influence of upper layer highly viscous liquid on cavity dynamics. The thickness of the dimethicone layer h was 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, 8 mm, respectively. The spheres were released at the height of H ¼ 0.95 m to acquire the vertical impact velocity of V0 ¼ 4.32 m/s. And the horizontal impact velocity of the sphere is in a range of U0 ¼ 0.49–1.3 m/s due to the equipment limitations. The mo tion of the sphere and the cavity dynamics were recorded with a highspeed camera (Phantom V12.1) at a frame rate of 3000 frames per second with an image resolution of 1280 � 800 pixels. The value of exposure time used in the present experiments is 1/3000 s. Meanwhile, throughout the experiments the liquid temperature was kept constant at
Sphere
U0
Water tank Car
Pneumatic finger Dimethicone layer
U0 Water
2. Experimental setup
Sphere x
V0
o
h
y
(a)
(b)
Fig. 1. (a) Schematic illustration of the basic experimental setup. (Figure is not to scale.) (b) Definitions of the Cartesian coordinate system xoy and dimethicone layer thickness h. 2
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boundary of the upper layer splash is lower than the upper boundary of the lower layer splash. At the same time, the upper-right corner of the upper layer splash goes downwards, which leads the upper layer splash collapses. Therefore, the presence of the dimethicone layer modifies the splash evolution and seal patterns. In the early stage 4, the cavity below the still free surface appears pinch-off and it splits into two distinct cavities: a lower air cavity still fully attached to the sphere, and an upper air cavity connected to the free surface. It is worth noting that the pinch-off time of the two-layer liquid system is earlier than the water entry case. After pinch-off (t ¼ 110 ms) the cavity begins a rapid collapse below the free surface. Distinct jet of fluid ejects away from the point of pinch-off can be seen both in water entry case and water-dimethicone entry case. This jet phenomenon is also seen by other researchers (e.g. Worthington and Cole, 1897; Truscott and Techet, 2009b). In order to gain further insight into the effect of the upper-layer dimethicone on cavity formation of the oblique impact of a sphere, the comparison of cavity shape between water entry and upper-layer dimethicone entry is shown in Fig. 3. As shown in Fig. 3(a), it should be noted that the wall of the cavity formed under the free surface is smooth, which is consistent with the observations of single phase liquid system (May 1952; Mansoor et al., 2017; Hou et al., 2018). However, it is interesting to note that the cavity wall for water-dimethicone entry case is relatively rough and it is characterized by wave-like instability structure. Due to the difference in viscosity between water and dime thicone, the generation of the wave-like structures is driven by the shear-induced instability which occurs at the contact line between the surrounding water and the entrained dimethicone of the sphere. More over, note that ’ribbed’ structure is observed around the upper left part of the splash wall in water-dimethicone entry case, whereas the splash wall in the water still maintains good continuity. By comparing the pinch-off locations of the two conditions presented in Fig. 3(b), an interesting observation is that the film wall at the rim of the lower splash breaks into a finger-like structure. The wave and rib structure features can be observed more clearly by the zoomed views in Fig. 3. It further highlights the upper-layer dimethicone plays an important role on splash formation and seal pattern. Surface tension is considered to play an important role in deter mining the splash evolution mechanism, and the corresponding impor tant parameter is the Weber number. (Mansoor et al., 2014). It is well known that the Weber number is the ratio between the inertial force and the surface tension force and it is expressed as We ¼ ρLV2/2σ , where L is the characteristic length, V is the velocity of fluid. In the splash, the relevant velocity and length scales must reflect the motion. Hence, the splash Weber number is defined as Wes ¼ ρδV2s /σ , where δ is the thick ness of the splash film, Vs is splash velocity. To analyze the splash stratification patterns, splash formation images at two typical moments and the corresponding schematic of the cavity wall profile and motion velocity at the stratified boundary are shown in Fig. 4. As shown in Fig. 4 (a), obvious splash stratification can be seen on the splash film of the experimental image. Based on the intersection line structure between the lower layer and the upper layer of the splash, we can conclude that the velocity distribution along the intersection line is linear. The instantaneous average velocity of the splash motion in Fig. 4(a) is approximately Vs ¼ 1.2 m/s. Meanwhile, the value of the film thickness is δ � 10 μm according to experimental findings by Zhang et al. (2012) and Marston et al. (2016). Hence, taking δ ¼ Οð10Þ μm and the splash velocity Vs ¼ Οð100 Þ m/s, we can obtain that Wes ¼ Οð10 1 Þ for water and Wes ¼ Οð100 Þ for dimethicone. This indicates surface tension does play a significant role in the splash formation. More importantly, the discontinuity of Weber number at the interface between the upper and lower layers is cause of the splash stratification. Moreover, as shown in Fig. 4(b), it is observed that the shrinkage near the intersection line on the left and right sides of the splash is asymmetrical. Actually, from the schematic illustration of Fig. 4(b), we can know that the splash motion
16 � C to minimize temperature-induced physical properties variation. Table 1 summarizes the experimental conditions in the present study. 3. Results and discussions 3.1. Cavity formation mechanism induced by a thin layer of highly viscous liquid floating on water Following the discussions of May (1975) and Truscott and Techet (2009b), we analyze the cavity formation problem in four distinct stages: (1) the flow forming stage, (2) the open cavity and splash growth stage, (3) the closed cavity and pinch-off stage and (4) the collapsing cavity stage. Fig. 2 presents sequence of images of the cavity formation of the sphere falls into the water-dimethicone and water. In stage 1 of the impact sequence (t ¼ 10 ms and t ¼ 20 ms), an initial horizontal splash of fluids forms as the sphere impact the still free surface and this splash extends radially outwards as the sphere descends into the liquids. At t ¼ 20 ms, the splash transitions from outwards to upwards growth. During stage 1, we can clearly see that the cavity formed by the impact of water-dimethicone are almost the same as the cavity formed by the impact of the water. In stage 2 of the impact sequence (t ¼ 30–60 ms), the entire sphere passes below the still free surface and an open air cavity begins to form in its wake. It is observed that the splash crown and the top of the subsurface air cavity bases, connected to the free surface, increase in diameter. Interestingly, left-right asymmetry in cavity below free sur face and splash above free surface can be observed in stage 2. It is well known that the initial stages of impact of object on the free surface are dominated by inertial effects. Hence, the subsurface air cavity both grows radially outwards and elongates vertically as the sphere descends. However, the splash of sphere entry onto the two-layer liquid system formed by a dimethicone layer floating on the water are quite different than onto only water. For the water-dimethicone entry case as shown in Fig. 2(a), the splash appears stratified and is characterized by upper and lower layers. This unique phenomenon has not been observed by other experiments (Worthington, 1908; Truscott and Techet, 2009b; Smolka and McLaughlin, 2019). Moreover, obvious buckling instability occurs around the upper-right part of the splash wall in the water-dimethicone entry case, which is marked by arrow at t ¼ 40 ms in Fig. 2(a). Whereas, the event of buckling instability is weak in water entry case. According to Marston et al. (2016), the collapse of the splash is mainly driven by surface tension and the buckling occurs when collapse rate is faster than the vertical velocity of the splash crown. Hence, we speculate that small surface tension is beneficial to accelerate the radial shrinkage and the corresponding buckling instability is easy to occur in water-dimethicone entry case. In stage 3 of the impact sequence (t ¼ 70–100 ms), the subsurface air cavity continues to stretch and curve as the sphere descends. The cavity evolution below the free surface in water entry case and waterdimethicone entry case is similar. Meanwhile, as shown in Fig. 2(b), the splash film is closed and no more air can flow into the cavity. However, for the two-layer liquid system presented in Fig. 2(a), the lower-left corner of the upper layer splash shrinks, then the bottom Table 1 Experimental conditions. Parameter
Symbol/Definition
Range/Values
Units
Diameter of sphere Mass of sphere Horizontal initial velocity Vertical initial velocity Dimethicone thickness Splash velocity Froude number Weber number Splash Weber number
D ms U0 V0 h Vs Fr ¼ U0/(gD/2)1/2 We ¼ ρV0D/2σ Wes ¼ ρVsδ/2σ
57.2 0.18 0.49–1.3 4.32 2–8 – 0.93–2.45 – –
mm kg m/s m/s mm m/s – – –
3
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Buckling instability 10 ms
20 ms 30 ms Collapse
70 ms
40 ms
Bottom boundary
Shrinkage
80 ms
Stratified splash
90 ms
50 ms
Jet
Pinch-off 100 ms
60 ms
105.33 ms
110 ms
(a) Cavity formation during the entry of a sphere into a two-layer liquid system that consists of a thin layer (thickness: 4 mm) of dimethicone floating on the surface of a deep water pool.
Single splash 10 ms
20 ms
30 ms
40 ms
50 ms
Pinch-off 70 ms
80 ms
90 ms
100 ms
107.67 ms
60 ms
Jet 110 ms
(b) Cavity formation during water entry of the sphere. Fig. 2. Comparison of the cavity formation of a sphere (D ¼ 57.2 mm) falls into the water-dimethicone and water. The initial moment t ¼ 0 is defined as the bottom of the sphere touching the still free surface. The initial impact velocity of V0 ¼ 4.32 m/s and U0 ¼ 1.25 m/s.
velocity Vsl on the left side is greater than the velocity Vsr on the right side. In other words, the difference in velocity will lead to the difference in Weber numbers. Therefore, the surface tension significantly changes the evolution mechanism of splash during the entry process in the two-layer liquid system. To further elucidate the splash stratification patterns, the schematic representation of instantaneous motion state and cavity profile in xoy plane is shown in Fig. 5. Gillbarg and Anderson (1948) stated that the cavity splash is subject to two chief forces: surface tension σ and a pressure drop ΔP caused by the air flow behind the sphere. It is conceivable that the presence of the pressure drop across the splash film can play a role in splash shrinkage. Visual inspection of the splash for mation just prior to shrinkage and seal as presented in Fig. 2. At these instantaneous moments of the upward motion of the cavity splash, the surface tension in the radial direction is not enough to overcome the inertial force. Therefore, no splash shrinkage is formed along the crown wall near the intersection line. However, the surface tension in the radial direction becomes prominent when the upper layer splashes slowly or even stops rising. Meanwhile, the cavity splash formed by water below the intersection line continues to expand due to inertial effects and from energy transfer by sphere at its instantaneous location. As a result, splash shrinkage appears at the junction of dimethicone and water of the splash.
3.2. Influence of the thickness of viscous liquid layer on cavity formation To analyze the cavity evolution and splash formation quantitatively, we introduce four parameters θ, dp, tp and dc. The definition of these parameters are presented in the following and shown in Fig. 6. The angle between the splash layered line and the horizontal direction is denoted as θ. The depth at which cavity pinch-off occurs is denoted dp and the corresponding pinch-off time is defined as tp. The total entry depth of the sphere dc is defined as the distance from the undisturbed free surface to the leading edge of the sphere at any moment during the water entry process. Fig. 7 shows the splash and cavity formation by the impact of spheres at different moments with the dimethicone layer thickness h of 0–8 mm. For t ¼ 70 ms as shown in Fig. 7(a), stratified splash can be observed when the dimethicone layer thickness varied from 2 mm to 8 mm, and the stratified boundary height decreases as the thickness of the dime thicone layer increases. On the one hand, it is indicated that the source of the liquid that forms the splash is more than 8 mm, and on the other hand, also indicates that the energy of the sphere during the water entry process is distributed to different regions of the splash. Note that for the 4 mm and 5 mm cases, the splash shrinkage appears on the left side of the stratified interface. Moreover, compared with the water entry case, a more pronounced ribbed structure can be seen in the splash seal region of the dimethicone-water entry cases. At the pinch-off time as shown in Fig. 7(b), it is observed that the obvious collapse of the upper layer splash occurs in the 3 mm and 4 mm 4
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4 mm. Therefore, the thickness of dimethicone is an important factor for the collapse phenomenon in the cavity splash, and we speculate that there may be a critical dimethicone thickness value between 2 mm and 5 mm in the present experiment. Meanwhile, by comparing the pinch-off depth dp (marked by solid line in Fig. 7(b), dimethicone-water entry cases are marked by black line and water entry case is marked by red line) and the entry depth dc (marked by dashed line in Fig. 7(b), dimethicone-water entry cases are marked by black line and water entry case is marked by red line), we found that the pinch-off positions and the entry depth of the different thickness of dimethicone-water entry cases are almost the same. How ever, the values of dp and dc in the dimethicone-water entry cases are smaller than that of the water entry case. This indicates that the high viscosity dimethicone floating above the water causes the pinch-off to occur earlier, but the thickness value range of the dimethicone we studied has almost no effect on the pinch-off position. For t ¼ 120 ms after pinch-off moment as shown in Fig. 7(c), it can be seen that the
Ribs
Smooth Water
Rough Dimethicone water
(a) t = 85 ms Fingers
Water
Vs
Dimethicone water
(b) Pinch-off time
Stratification line
Water film
Fig. 3. Cavity formation comparison between water entry and waterdimethicone entry at different times for the cases with entry velocity V0 ¼ 4.32 m/s and U0 ¼ 1.25 m/s. Compared to Fig. 2, the enlarged images in Fig. 3 highlights the stratified splash phenomenon, see text for the detailed explanation.
Dimethicone film Vs
No air flow
Air flow
No air flow a w
thickness of the dimethicone-water entry cases. However, the collapse intensity is weaker in the 2 mm and 5 mm dimethicone-water entry cases at this moment. As shown in Fig. 7, note that the water film in cavity splash of the 2 mm working condition is high enough to cause the splash seal to occur mainly on the upper side of the water film. Whereas, for the 5 mm test case, the rise height of the water film is limited due to the energy distribution of the falling sphere. Hence, it is observed that splash seal occurs on the upper side of the dimethicone layer, that is, away from the position of the splash stratification. Especially, the splash seal area is close to the stratified position, resulting in a more pronounced collapse and shrinkage when the thickness of dimethicone layer is 3 mm and
W0
D Fig. 5. Schematic representation of instantaneous motion state and cavity profile in xoy plane.
Vsl Dimethicone film
Splash stratification
Vs
Vsr
Water film Intersection line
(a) Splash stratification and the corresponding schematic illustration of the cavity wall profile and motion velocity. The entry time is t = 30 ms.
Shrinkage Intersection line
Dimethicone film
Vsl Dimethicone film Vs
Shrinkage Water film
Intersection line
Vsr
(b) Splash shrinkage and the corresponding schematic illustration of the cavity wall profile and motion velocity. The entry time is t = 60 ms. Fig. 4. Experimental splash images at two typical moments and the corresponding schematic illustration of the splash stratification and shrinkage. The initial impact velocity of V0 ¼ 4.32 m/s, U0 ¼ 1.25 m/s. The thickness of the dimethicone layer is h ¼ 4 mm. 5
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more in number and larger in scale for the dimethicone-water entry cases compared to the water entry. Therefore, the presence of the dimethicone layer leads to a more complicated splash seal mechanism. In addition, the shrinkage tendency at the interface of the upper and lower layers of the splash becomes weaker as the thickness of the dimethicone increases. Fig. 8(b) presents the splash pattern when the angle between the splash layered line and the horizontal direction θ � 12.5� . As can be seen from Fig. 8(a), the smaller the thickness of the dimethicone layer, the shorter the time required to reach a certain angle. In fact, the splash for these four cases is in different stages of evolution. For the h ¼ 2 mm case, it is the development stage of the splash crown. For the h ¼ 3 mm case, the upper layer of the splash begins to occur buckling instability. For the h ¼ 5 mm case, it is in the early stage of splash seal of the upper layer cavity. For the h ¼ 7 mm case, the splash is almost closed. Due to the sphere-liquid interaction, the kinetic energy of the sphere transfers to the surrounding fluids as the sphere decelerates. The water in splash will obtain less energy as the thickness of dimethicone increases. As a result, the angular velocity of the splash layered line of the thick dimethicone layer cases is smaller than that of the thin ones. Hence, it will take more time to reach the same angle of the splash layered line for the thicker dimethicone layer cases due to the energy distribution and the presence of the dimethicone layer.
dp
3.3. Influence of the horizontal impact velocity on cavity dynamics In order to analyze the effect of horizontal impact velocity on cavi tation formation, a comparison of the cavity morphology at four hori zontal impact velocities with a dimethicone thickness of 2 mm is given in Fig. 9. For t ¼ 70 ms before pinch-off moment as shown in Fig. 9(a), it is important to note that the size of the opening at the top of the splash is different and appears to increase with the increase of horizontal impact velocity. In other words, increasing the horizontal impact velocity causes a delay in the splash closure. Moreover, the angle θ increases as the speed increases, indicating that the large horizontal impact velocity causes the cavity asymmetry to be more pronounced. This observation is further confirmed at the pinch-off position (as shown in Fig. 9(b)), in particular, a complete splash seal is formed when U0 ¼ 0.51 m/s. As shown in Fig. 9(c), there is a line closure pattern observed (as marked with arrows in Fig. 9(c)) in the experiments due to the pressure differ ence between the inside and outside of the cavity under the still free surface. It is worth noting that there is no bubbles trace generated after pinch-off in the symmetrical water entry cases (Grumstrup et al., 2007; Aristoff and Bush, 2009; Tan and Thomas, 2018; Sun et al., 2019). Whereas, the bubbles trace was found in the asymmetrical water entry process of projectiles (Bodily et al., 2014). Due to the formation of the jet and the asymmetry of cavity in the oblique entry cases, the boundary of the cavity wall is easier to roll up into a ring vortex, which entrains air along the cavity wall and the subsequent shedding of gas bubbles at the tail of the attached cavity over the sphere (Spurk, 2002; Karn et al., 2016). Moreover, with the increase of horizontal impact velocity, the connected bubbles trace shifts from curve to approximate straight line due to the inertia effects. Curves of the horizontal and vertical displacement as a function of time at the above four different initial vertical impact velocities are shown in Fig. 10. The curves indicate that relation between displace ment and time follow a more non-linear relationship. Note that the horizontal displacement of the sphere rise faster with increasing impact velocity. However, as can be seen in Fig. 10(b), the values in the vertical displacement are almost the same for different initial vertical impact velocities. Therefore, we speculate that the vertical drag force during the dimethicone-water process is independent of the horizontal impact ve locity. To support this conclusion, we will analyze the force model of sphere during the entry case in the next part. In order to determine the sphere trajectory of the above analysis, Fig. 11 presents the instantaneous force model of sphere motion. The
dc
Fig. 6. Definitions of cavity parameters. The angle θ between the upper boundary of the lower layer splash and the horizontal direction, pinch-off depth dp and cavity depth dc.
water film breaks into a finger-like structure at the left rim of the lower crown in the 3 mm and 4 mm dimethicone-water entry cases. Interest ingly, a connected bubbles trace can be observed between the upper cavity near the free surface and the cavity attached to the tail of the sphere and the closure depth of the upper cavity under the free surface almost the same as the dimethicone thickness increases. The above re sults indicate that the thickness of dimethicone affects the splash for mation and evolution process. To further clarify the influence of the dimethicone layer and the thickness on the splash formation, enlarged views of the splash morphology above the free surface in water entry case and dimethiconewater entry cases with different thickness of dimethicone are shown in Fig. 8. Fig. 8(a) compares the results at the same water entry time of t ¼ 60 ms, it is noteworthy that the ribbed structures in the crown are 6
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Ribs Shrinkage
h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(a) Comparison of the splash and cavity formation at the same instantaneous moment before pinch-off. All images are taken at t = 70 ms.
Collapse
h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(b) Comparison of the splash and cavity formation at the pinch-off location. Fingers
Bubbles trace h = 0 mm
h = 2 mm
h = 3 mm
h = 4 mm
h = 5 mm
h = 6 mm
h = 7 mm
h = 8 mm
(c) Comparison of the splash and cavity formation at the same instant after pinch-off. All images are taken at t = 120 ms. Fig. 7. Splash and cavity formation by the impact of spheres at different moments with the dimethicone layer thickness h of 0–8 mm. The moment when the bottom of the sphere touches the still free surface is defined as the initial time t ¼ 0. The initial impact velocity of V0 ¼ 4.32 m/s and U0 � 0.75 m/s.
Fig. 8. Comparison of splash formation by the influence of the dimethicone layer thickness. The initial impact velocity of V0 ¼ 4.32 m/s and U0 � 1.0 m/s.
trajectory of the sphere at multiple times during the entry process is marked by red dashed line. It is worth noting that the sphere almost keeps linear motion within the time range of the present study. The coordinate directions have been defined in Fig. 1(b), and are also shown in Fig. 11(c). The force balance equation of oblique motion of the sphere is expressed as
(ms þ ma)as ¼ G þ FB þ FD þ FT
(3.1)
where ms ¼ ρs Vsphere ¼ 43 πρs R3s is the mass of the sphere, ρs is the density of the sphere, Vsphere is the volume of the sphere, and Rs is the radius of the sphere; The force required to accelerate the surrounding liquid is 7
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1.2
U0 = 0.51 m/s U0 = 0.74 m/s
U0 = 0.88 m/s U0 = 1.04 m/s
x/D
0.9
0.6
0.3
0.51 m/s
0.74 m/s
0.88 m/s
1.04 m/s
0.0 0.00
(a) t = 70 ms.
0.03
0.06
t (s)
0.09
0.12
(a) Time history curves of horizontal displacement of the sphere.
Splash seal
U0 = 0.51 m/s 6
U0 = 0.74 m/s U0 = 0.88 m/s
y/D
U0 = 1.04 m/s 4
2
0.51 m/s
0.74 m/s
0.88 m/s
1.04 m/s 0 0.00
(b) Pinch-off position.
0.03
0.06
t (s)
0.09
0.12
(b) Time history curves of vertical displacement of the sphere. Fig. 10. Comparison of horizontal displacement and vertical displacement of the sphere at different initial horizontal impact velocities of U0 ¼ 0.51 m/s, 0.74 m/s, 0.88 m/s, and 1.04 m/s. The corresponding initial vertical impact velocity V0 ¼ 4.32 m/s. The dimethicone layer thickness is h ¼ 2 mm.
Line
Curve
0.51 m/s
0.74 m/s
0.88 m/s
However, Cm is usually considered to be a constant of 0.5 (Newman, 1977; Truscott et al., 2012; Tan, 2019; Wang et al., 2019). Hence, we take it as a constant Cm ¼ 0.5 for analysis in the present study, and the added mass is given as ma ¼ 0:5ρw Vsphere . For the buoyancy force, ignoring the initial partial submerged process, it can be expressed as
FB ¼ -ρwgVsphere. The drag force is FD ¼ 0:5Cd ρw Ap jWj2 , where Cd is the drag coefficient, and it is typically taken to be constant for varying impact velocities and penetration depth in studies of subsonic water entry cases (Lee et al., 1997; Abraham et al., 2014); Ap is the projected wetted area of the submerged portion; W ¼ U þ V is the total velocity. Meanwhile, as shown in Fig. 11(b), the dynamic contact angles on the left and right sides of the sphere are defined as α and β, respectively. It is worth noting that both α and β are almost constant in the stable entry stage of the sphere. The projected wetted area can be expressed as Ap ¼
1.04 m/s
(c) t = 120 ms. Fig. 9. The image sequence depicting splash and cavity evolution caused by the sphere impacts still free surface with increasing initial horizontal impact ve locities of U0 ¼ 0.51 m/s, 0.74 m/s, 0.88 m/s, and 1.04 m/s. The corresponding initial vertical impact velocity V0 ¼ 4.32 m/s. The dimethicone layer thickness is h ¼ 2 mm.
πR2s in direction of movement of the sphere when the entry is more than
one diameter depth. However, the effects of initial impact conditions and gravity will cause the values of α and β to be different. Therefore, we introduce a correction coefficient Ko, which is a constant close to 1, and the corresponding projected wetted area becomes Ap ¼ Ko πR2s . More over, the surface tension force FT can be neglected as it is less than 1% of the gravitational force for the water entry of standard billiard balls (Truscott and Techet, 2009b). As mentioned in Section 2, the surface tension coefficient of dimethicone is much smaller than that of water, and Tan (2019) demonstrated that FT was on a much lower percentage of FD. Therefore, even if there is a dimethicone layer, the surface tension can be negligible in the force model. By making as the subject and simplifying Eq. (3.1), the equations in the x and y directions can be expressed as:
expressed as the added mass ma ¼ Cm ρw Vsphere , where Cm is the added mass coefficient, and ρw is the liquid density; as is the acceleration of the sphere; G ¼ mg is the gravity; FB is the buoyancy force; FD is the drag force; FT is the surface tension force. According to Aristoff et al. (2010), the added mass force maas should appear as an independent term at relatively small Froude numbers during the process of water entry. The maximum Froude number in this paper is no more than 10, therefore, the added mass force cannot be ignored. Meanwhile, it is noted that the added mass coefficient Cm is related to the submerged depth, and it could go from zero at first impact and up to 0.5 when the sphere is one radius deep (Shepard et al., 2019). 8
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Cavity profile
20 ms
40 ms
FD
FB
60 ms
i
80 ms
j
100 ms
W=U+V
(a)
(b)
G
(c)
Fig. 11. (a) The trajectory of the sphere at multiple times during the entry process. (b) Instantaneous image of sphere motion, the red dashed line represents the trajectory of the sphere. (c) Schematic diagram of instantaneous force model of the sphere.
8 > > > < x€ ¼ asx ¼
Cd Ko ρw 2 U sinφ 4Rs ρs � � > 2 ρw Cd Ko ρw 2 > > 1 gþ V cosφ : y€ ¼ asy ¼ 3 ρs 4Rs ρs
(3.2)
where φ is the angle between FD and bj, as shown in Fig. 11(c). Based on the previous analysis, the value of φ is almost a constant due to the straight track of the sphere in the present study. In Eq. (3.2), it is indicated that the sphere will receive larger hori zontal acceleration at higher initial horizontal impact velocity. This can better explain the variation trend of the displacement curve at different initial horizontal impact velocities in Fig. 10(a). According to asy in Eq. (3.2), since the initial vertical impact velocities of the four conditions in Fig. 10(b) are the same, the vertical displacement versus time curves under different conditions present the same trend. The force model in Eq. (3.2) is mainly suitable for oblique impact of sphere into water for a relatively short time in the present study. We can speculate that the trajectory of the sphere will not remain straight at all times if the experimental recording time is long enough. As a result, the parameters Ko and φ in Eq. (3.2) will be time dependent. However, it should be noted that the measurement of these forces are also very difficult until now and the subtleties of these differences difficult to pinpoint. In order to better establish the force model, more measurement methods and flow field information are necessary in the future work.
Fig. 12. The angle θ with different dimethicone layer thickness of h ¼ 3 mm, pffiffiffiffiffi 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of U0/ hg at the pinch-off locations. Solid curves are fitted data and markers are experimental results.
pffiffiffiffiffiffiffiffiffiffi plotted against Fr in Fig. 13, where Fr ¼ U0/ gD=2, U0 is the initial horizontal impact velocity at the still free surface, g is the gravitational acceleration and D is the diameter of the sphere). Interestingly, we observe roughly linear relationships between tp* and Fr. Based on the experimental data fitting, the measured dimensionless pinch-off time is tp* � 0.96 Fr, and tp* increases as the Froude number increases. Lee et al. (1997) rationalized the experiments of Gilbarg and Anderson (1948) regarding the apparent independence of the dimensionless pinch-off time on the vertical impact velocity. More recently, Truscott and Techet (2009b) provided further experimental evidence that the pinch-off time, tpV0/D ~ Fr, is roughly constant in water entry. Our experimental findings are generally similar to their measurements. However, our Fr is based on the horizontal impact velocity and is different from their vertical velocity definition. Moreover, the measured pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pinch-off time is tp� 1:92 D=ð2gÞ for all the water-dimethicone entry cases in the present study. Duclaux et al. (2007) found the prefactor k to be 2.06 for the impact of spheres and the corresponding relationship is tp pffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 2:06 D=ð2gÞ. The difference of the prefactor k between water-dimethicone entry cases and water entry cases indicates the pinch-off time is effected by the dimethicone layer. Hence, we conclude that the presence of high viscosity dimethicone contributes to the cavity shrinkage and the pinch-off occurs earlier. Furthermore, based on the
3.4. Splash angle and typical pinch-off parameters characteristics In this section, we further quantitatively assess the effects of dime thicone and initial impact velocity on the cavity dynamics. Fig. 12 presents the angle θ mentioned above as a function of non-dimensional pffiffiffiffiffi velocity parameter U0* at the pinch-off locations, where U0* ¼ U0/ hg. From Fig. 7(b), it is observed that no obvious oil and water boundary line is formed in the splash for the dimethicone-water entry case of h ¼ 2 mm. Hence, the thickness of the dimethicone layer is in the range 3 mm � h � 8 mm in Fig. 12. In the process of extracting test results, conversion from pixels to angles yields an uncertainty of � 1� . It should be noted that the symbols denote experimental data and the corre sponding fitted curves are presented with second-order function. It in dicates that the value of θ increases as the decreases of the thickness of dimethicone layer, and it decreases as the increases of the original horizontal impact velocity. Pinch-off time is non-dimensionalized as tp* ¼ tpU0/D in Fig. 13. In the process of extracting test results, conversion from pixels to times yields an uncertainty of �0.66 ms in tp. It should be noted that deep seal is driven by hydrostatic pressure and thus pinch-off time scales with Froude number Fr (Speirs et al., 2018). The dimensionless time tp* is 9
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Ocean Engineering 194 (2019) 106597
h = 2 mm h = 3 mm h = 4 mm h = 5 mm
1.8 1.6
horizontal impact velocity. Fig. 15 shows the values of the water entry depth dc of the water entry cases and the dimethicone-water entry cases at the pinch-off mo ments as a function of the Froude number Fr. In the process of extracting test results, conversion from pixels to millimeters yields an uncertainty of �1.94 mm in dc. It is worth noting that the values of dc almost keep constant with the increase of Fr for the both types of water entry case and the dimethicone-water entry cases. In other words, here, in Fig. 15, the dc is independent of the horizontal impact velocity or, if a depen dence existed, it is very weak. Hence, combined with the experimental results in Fig. 7(b), it is further confirmed here that the values of dc in water are larger than that of dimethicone-water. It indicates that the presence of the dimethicone layer will affect the falling trajectory of the sphere. Meanwhile, an interesting finding is that the dc is independent of the thickness of the dimethicone layer under the experimental condi tions in the present study. Fig. 16 shows the values of the pinch-off depth as a function of the thickness of dimethicone layer. Based on the experimental data, we can get the value of the pinch-off depth is almost constant, where dp ¼ 0.161 m. For water entry cases, Duclaux et al. (2007) found the ratio of pinch-off depth to total entry depth is constant at dp/dc ¼ 0.5 and it was also experimentally confirmed by Speirs et al. (2018). However, the ratio of dp to dc is approximately 0.48 for the dimethicone-water entry cases in the present study. In other words, the pinch-off depth is less than half of the total depth. It indicates that the highly viscous dimethicone layer can promote the shrinkage of the cavity under the undisturbed free surface and subsequently leads to the early occurrence of pinch-off.
h = 6 mm h = 7 mm h = 8 mm
tpU0/D
1.4 1.2
0.96
1.0 0.8
1.0
1.2
1.4
1.6
1.8
Fr Fig. 13. Dimensionless pinch-off time with different dimethicone layer thick ness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number based on the initial horizontal impact (U0). Solid lines are fitted data and markers are experimental results. Present experimental data for the system with dimethicone above water, linear least-squares fit, tp* � 0.96 Fr.
pinch-off time tp, we can also draw that the cavity pinch-off time is in dependent of the thickness of the dimethicone layer and the horizontal impact velocity. Fig. 14 shows our experimental data for the dimensionless pinch-off depth dp* as a function of Fr 1, where dp* is expressed as dp/(U0tp). In the process of extracting test results, conversion from pixels to millimeters yields an uncertainty of �1.94 mm in dp. The scatter points with the same shape in Fig. 14 represent different thickness of dimethicone layer at the same horizontal impact velocity. We observe that a robust linear relationship between the dimensionless pinch-off depth and Fr 1. The slope k of dp* ~ Fr 1 for all the water-dimethicone entry cases is 6.15. Meanwhile, the pinch-off depth dp ¼ 5.9 D, it indicates that the pinch-off depth dp is independent of the thickness of the dimethicone layer and the
7
6
h = 2 mm h = 3 mm h = 4 mm h = 5 mm
4. Conclusions In the present study, the cavity dynamics of oblique entry of sphere through a viscous liquid floating on the water at low Froude number was investigated experimentally. High-speed camera system was used to captures the cavity formation and dynamics features for both qualitative and quantitative analysis. Meanwhile, water entry case was also per formed as a comparison to highlight the unique nature of cavity dy namics induced by the upper-layer of highly viscous liquid. From the experiments results, the main conclusions can be drawn as follows:
h = 6 mm h = 7 mm h = 8 mm
0.38
h = 0mm h = 2mm h = 3mm h = 4mm
h = 5mm h = 6mm h = 7mm h = 8mm
m/s
U0
m/s
6.15
5
dc = 0.345 m dc (m)
dp / (U0 tp)
0.36
0.34
dc = 0.335 m
4 0.32
3 0.5
0.6
0.7
0.8
0.9
1.0
1.1
-1
0.30
Fr
Fig. 14. Dimensionless pinch-off depth with different dimethicone layer thickness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number based on the initial horizontal impact (U0). Solid lines are fitted data and markers are experimental results. Present experimental data for system with dimethicone above water, linear least-squares fit, dp/ (U0tp) � 6.15 Fr 1.
1.0
1.2
1.4
1.6
1.8
Fr Fig. 15. Cavity length (dc) at pinch-off time with different dimethicone layer thickness of h ¼ 2 mm, 3 mm, 4 mm, 5 mm, 6 mm, 7 mm, and 8 mm as function of Froude number (Fr) based on the initial horizontal impact (U0). Dashed lines are fitted data and markers are the experimental results. 10
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Ocean Engineering 194 (2019) 106597
U0 = 0.98
0.06 m/s
U0 = 0.86
0.06 m/s
although we have explained the splash stratification, it still needs to be strengthened in the future. Especially, setting up liquid dynamics model based on mathematical method with viscosity and density as changing parameters at the junction of dimethicone and water will help us improve understanding of the mechanism of splash stratification.
U0 = 0.75
0.03 m/s
U0 = 0.53
0.03 m/s
Acknowledgments
0.19
dp (m)
0.18
0.17
This work was supported by the National Natural Science Foundation of China (51709042, 51679037, 51639003), the China Postdoctoral Science Foundation (2019T120211, 2018M631791), the Natural Sci ence Foundation of Liaoning Province of China (20180550619), and the Fundamental Research Funds for the Central Universities (DUT18RC(4) 018, DUT2017TB05).
dp=0.161 m 0.16
References
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2
4
h (mm)
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Fig. 16. Pinch-off depth (dp) as function of the thickness of dimethicone layer (h).
(1) The highly viscous liquid floating on the water substantially modifies the cavity formation mechanism. Compared to contin uous splash in water, the splash appears stratified and is char acterized by upper and lower layers for the water-dimethicone entry case. The discontinuity of Weber number at the interface of water and dimethicone due to the difference in surface tension is the most likely reason of stratified splash. (2) The thickness of dimethicone layer affects the cavity evolution patterns. Compared with the water entry case, a more pro nounced ribbed structure can be seen in the splash seal region of the dimethicone-water entry cases. The high viscosity dimethi cone floating on the water causes the pinch-off to occur earlier, but the thickness value range of the dimethicone we studied has almost no effect on the pinch-off position. Moreover, we found that the smaller the thickness of the dimethicone layer, the faster the intersection line in the splash reaches the same angle due to difference in the energy distribution of the dimethicone layer. (3) Through the analysis of the cavity formation, trajectory and force models of the sphere at different horizontal impact velocity, it is found that increasing the horizontal impact velocity will causes a delay in the splash closure. Meanwhile, it is observed that the vertical displacement during the dimethicone-water process is independent of the horizontal impact velocity. The hydrody namic force model of the sphere is established and subsequent explanation for the oblique water entry problem with a high viscosity dimethicone floating on the water. (4) Splash angle, pinch-off depth, pinch-off time and the water entry depth at pinch-off locations are obtained qualitatively and quantitatively by processing the experimental images. It is observed that roughly linear relationships between the dimen sionless pinch-off time tpU0/D and the Froude number, and the expression is tpU0/D � 0.96 Fr. Meanwhile, the pinch-off depth dp is independent of the thickness of the dimethicone layer and the horizontal impact velocity. Moreover, the entry depth dc almost keep constant at the pinch-off time as the increase of Fr for the both types of water entry case and the dimethicone-water entry cases within thickness range in the present study. The present work has provided experimental results to advance the understanding of the effects of highly viscous liquid layer on the cavity dynamics of oblique impact of sphere. As for the future research, a group of experiments with large thickness value of dimethicone layer would beneficial to expand some conclusions in this paper. In addition, 11
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