Numerical analysis of the bubble jet impact on a rigid wall

Applied Ocean Research 50 (2015) 227–236

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Numerical analysis of the bubble jet impact on a rigid wall Li Shuai, Li Yun-bo, Zhang A-man ∗ School of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, PR China

a r t i c l e

i n f o

Article history: Received 17 September 2014 Received in revised form 9 December 2014 Accepted 9 February 2015 Available online 6 March 2015 Keywords: Bubble Jet impact Boundary integral method

a b s t r a c t The main characteristic of the bubble dynamics near a rigid wall is the development of a high speed liquid jet, generating highly localized pressure on the wall. In present study, the bubble dynamic behaviors and the pressure impulses are investigated through experimental and numerical methods. In the experiment, the dynamics of a spark-generated bubble near a steel plate are captured by a high-speed camera with up to 650,000 frames per second. Numerical studies are conducted using a boundary integral method with incompressible assumption, and the vortex ring model is introduced to handle the discontinued potential of the toroidal bubble. Meanwhile, the pressure on the rigid wall is calculated by an auxiliary function. Calculated results with two different stand-off parameters show excellent agreement with experimental observations. A double-peaked or multiple-peaked structure occurs in the pressure profile during the collapse and rebounding phase. Generally, the pressure at the wall center reaches the first peak soon after the jet impact, and the second peak is caused by the rapid migration of the bubble toward the wall, and the subsequent peaks may be caused by the splashing effect and the rebounding of the toroidal bubble. At last, both agreements and differences are found in the comparison between the present model and a hybrid incompressible–compressible method in Hsiao et al. (2014). The differences show that the compressibility of the flow is another influence factor of the jet impact. However, the main features of the jet impact could be simulated using the present model. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Bubble dynamics near a rigid wall has many applications in ocean engineering because of the possible damage threat that can be caused by the impulsive pressure pulses during the collapse phase. The growth and collapse of micron-sized bubbles near propeller blades holds the key to understanding the deleterious effects of cavitation [1,2]. The interaction between underwater explosion (UNDEX) bubble and warship has important naval applications [3–6]. Early in 1917, Rayleigh [7] developed a spherical bubble model (Rayleigh–Plesset equation) based on the incompressible velocity potential theory, which could be used to explain the very high pressure near the bubble surface during the contraction stage of a bubble. However, the bubble cannot always keep a sphere shape during its whole life, especially near boundaries [8,9] or affected by gravity [10]. The development of a high speed liquid jet is the main feature of a non-spherical bubble. Lauterborn [9] found the velocity of the boundary-induced bubble jet is approximately 120 m/s, which would cause severe damage to the structures. The collapse of the bubble onto a cylinder was found to be the most severe structure load, generating a peak velocity almost twice that caused by

∗ Corresponding author. Tel.: +86 45182518443; fax: +86 45182518296. E-mail address: [email protected] (A.-m. Zhang). 0141-1187/$ – see front matter © 2015 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apor.2015.02.003

the shock wave [11]. The load caused by bubble collapse is complex and not yet fully understood. Experiment is the most direct method to study the bubble dynamic behavior and the load caused by a collapsing bubble, including UNDEX bubble [3], laser-induced cavitation bubble [12–16], and spark-generated bubble [17,18]. However, the real UNDEX experiment is hard to conduct for high costs and high risk. Besides, the gas inside the bubble is not transparent, so the development of the jet is invisible. The size of the laser-induced bubble and the spark-generated bubble is limited to several micrometers, thus the measurement of the impulsive forces is difficult because the size of pressure gauge is often larger than the bubble size [13]. In Tong’s work [13], the transducer is approximately the same size of the bubble diameter. Therefore, the experimental signals just indicate the force in a relatively large area. To sum up, it is still very difficult to measure the pressure caused by the bubble jet impact. Boundary integral method (BIM) is widely used to study the dynamic behavior of a non-spherical violently oscillating gas bubble for close to four decades [3]. The simulated results trace the main features of the bubble motion, such as expansion, collapse, jet and rebound. In addition, the velocity and pressure in the fluid domain could be calculated to analyze the mechanism of these phenomena. The transition of the bubble from a singly-connected to a doubly-connected form induces circulation in the flow around the toroidal bubble, which becomes a barrier to the numerical simulation of the subsequent bubble motion. Therefore, only a few works

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on the dynamic of toroidal bubble have been published before. Best [19] introduced a cut in the flow domain when jet impact occurs, rendering the flow domain remain singly-connected. Wang [20] came up with a simpler model by placing a vortex ring inside the bubble. The strength of the vortex ring is chosen to be equal to the potential difference of the jet tip and the opposite bubble surface just before the jet impacts. These two models are extensively and successfully applied to handle the toroidal bubble, and the second one is used in this paper. Valuable studies on jet impact have been performed using BIM. The “splashing” effect [12] occurs after the liquid jet impact onto the boundary with the dimensionless stand-off distances between 0.8 and 1.2, which is a source of very high pressures on a circular ring around the collapsing bubble. Brujan [14] studied the final stage of the collapse of a cavitation bubble close to a rigid boundary. It was found that two high pressure regions are located on the axis of symmetry above and below the bubble, and the high pressure region on the rigid boundary covers only a small area localized around the jet tip. In Wang’s work [21], there appears to be an optimal initiation distance for which the liquid jet thus formed is most damaging. The optimum stand-off is found to be around 1.3 in the absence of buoyancy. However, a quantitative relationship between the impulses and the bubble collapse is still incomplete due to the extremely rapid and complicated unsteady flow phenomena, and there is little information about the rebounding toroidal bubble. In this work, the motion of a bubble near a rigid wall is studied both experimentally and numerically. Two cases are selected with the stand-off distances being 1.51 and 1.05 times the maximum radius of the bubble. The numerical results of the cases indicate different pressure characteristics due to the direct and indirect jet impacts on the wall. There is always some numerical instability when calculating the pressure load on the rigid wall [21]. In the present study, ∂˚/∂t is evaluated with the auxiliary function method [22] to avoid making finite difference of the velocity potential, thus obtaining a better result of the pressure. At last, a comparison is made between the present model and a hybrid incompressible–compressible method [23,24]. The agreements as well as differences between these two models are discussed. 2. Experiment This section will give a brief description of the experiment. A schematic of the experimental setup used for studying the motion of a spark-generated bubble near a rigid wall is shown in Fig. 1. Experiments are conducted in a water tank with dimension 500 mm × 500 mm × 500 mm, in which the water is filled up to 400 mm in depth. A steel plate, 300 mm × 300 mm × 8 mm, is placed at the bottom of the tank. The low-voltage spark bubble generation method can be found in Turangan’s work [18]. The circuit employed in current bubble generation is based on Zhang [25], including a 6600 ␮F capacitor and a 220 V DC power supply. A bubble is generated by burning the copper wire with its diameter about 0.25 mm, and captured by the Phantom V12.1 high-speed camera. The camera works at 30,010 frames per second with exposure time 10 ␮s. The whole experiment section is illuminated from the back with a 2 kW light. More detailed information about the experiment can be found in Zhang [25].

comparison to viscous diffusion times, the liquid surrounding the bubble is assumed inviscid and the motion irrotational [14]. Besides, the jet Mach number is larger than 0.1 for about 0.1% of the bubble lifetime [26], so we ignore the compressibility of the flow in the present model. Thus, the velocity potential ˚ satisfies the Laplace equation in the flow domain as follows

∇2˚ = 0

(1)

According to the Green function [28], the velocity potential at any point in the domain could be expressed as an integral equation

  S

1 ∂ ∂˚(q, t) − ˚(q, t) · |q − r| ∂n ∂n



× dS(q) = ε(r, t) · ˚(r, t)

1 |q − r|

 (2)

where ε(r, t) is the solid angle of a fixed point r with the integration variable q also situated on boundaries, ∂/∂n is the normal outward derivative from the boundary S. G(r, q) = (1/|r − q|) + (1/|r − q |) is the half-space Green function, with q being the reflected image of q across the wall. This Green function satisfies the zero flux condition through the wall. In order to solve Eq. (2) numerically, we take an axisymmetric model and discretize the bubble surface into M nodes and N elements. Then, Eq. (2) transforms into M equations M  

Wij

∂˚j ∂n

 =

j=1

M 

(Mij ˚j ) − ε(i)˚i

(3)

j=1

where Wij and Mij are influence coefficients. The calculation process can be found in Blake [27] and Wang [29]. The dynamic boundary condition on the bubble surface can be written as:



2

∇ ˚ D˚ P∞ P = + − − gz Dt 2  

(5)

where  is the density of the liquid, P∞ is the ambient pressure of the liquid at the inception point of the bubble, P is the pressure on the bubble surface, g is the gravity acceleration. The kinematic boundary condition on bubble surface is: dr = ∇˚ dt

(6)

The pressure inside the bubble is assumed to be uniform and consists of a constant vapor pressure and a volume-dependent noncondensable gas pressure [19]. Here, the surface tension is negligible compared to the high pressure caused by the bubble. Besides, the heat and mass transfers are also ignored [30,31]. Hence, the pressure inside the bubble Pg as a function of the volume can be described as: Pg = Pc + Pini

 V ϑ ini

V

(7)

where the subscript ini denotes initial quantities, ϑ is the ratio of the specific heats for the gas, Pc is the vapor pressure. 3.2. Toroidal bubble

3. Theory 3.1. Boundary-integral method Because of the large Reynolds number (Re ∼ 104 ) [26,27] associated to the bubble motion and the short bubble lifetime in

A toroidal bubble is formed after the jet impact upon the opposite bubble surface, i.e. the bubble is transformed from a singly-connected into a double-connected form. In order to handle this problem, some topology changes are made and a vortex ring is introduced inside the toroidal bubble [20]. The strength of

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229

Fig. 1. Experimental setup.

the vortex ring is the circulation of the flow along a closed path that threads through the torus [21].

  =

(8)

u dl C

3.3. The auxiliary function method for calculating pressure The dynamic pressure Pd at any point in the flow domain can be evaluated using the Bernoulli equation Pd = P − P∞ = −

This circulation equals the difference between ˚ at the north and south poles of the bubble just before impact. Then, the velocity potential ˚ is decomposed into two parts: a single-valued remnant potential ˚r and the potential of the vortex ring ˚v . ˚ = ˚v + ˚r

(9)

The vortex ring (rv , zv ) is located at the farthest point from the toroidal bubble surface. The potential of the vortex ring ˚v can be evaluated by integrating the velocity field



nodei

˚v (nodei ) =

uv dl + ˚v (node1 )

(10)

node1

node1

˚v (node1 ) =

uv dr

(11)

∞

=

∂˚ ∂t

(15)

The auxiliary function  has the similar mathematical properties with velocity potential ˚, thus it satisfies the Laplace equation in the flow domain [22] (16)

The kinematic boundary condition for  on the wall can be written as follows ∂ =0 ∂n

(17)

The dynamic boundary condition for  on bubble surface is

After the decomposition of velocity potential, the velocity is also decomposed into two parts. The velocity induced by a circular vortex ring uv can be calculated from Bio-Savart law:  uv (r, z) = 4

(14)

Generally, the finite difference approximation [32] was adopted to calculate ∂˚/∂t in Eq. (14). However, this method needs several velocity potentials at different times, which imports some errors into result. In this study, the term ∂˚/∂t can be obtained by solving a boundary value problem for an auxiliary function  defined by Wu [22].

∇2 = 0

where ˚v (node1 ) is the potential of the vortex ring at node 1 of the bubble, which is obtained by



∂˚ |∇ ˚|2 − 2 ∂t



dl × r − r3 C

 C

dl × r 



r3

(12)

where C is the image of the vortex ring C across the wall. The single-valued remnant potential ˚r satisfies the Laplace equation on the flow domain, and therefore satisfies (2). The dynamic boundary condition for ˚r on bubble surface yields d˚r |ut |2 P∞ − P =− + ut · ∇ ˚r − − gz 2  dt

(13)

The total velocity ut = uv + ur is used to update the location of the bubble surface.

2

=−

(∇ ˚) P∞ P + − − gz 2  

(18)

The right hand side of Eq. (18) can be evaluated easily after solving Eq. (3). Then the value of  on the wall or in the flow domain can be calculated by Eq. (2). The pressure in Eq. (14) can be obtained afterward. Therefore, the dynamic pressure on the wall is directly calculated without making finite difference of the velocity potential. For the toroidal bubble, the term ∂˚/∂t is also decomposed into two parts ∂˚v ∂˚ ∂˚r + = ∂t ∂t ∂t

(19)

The term ∂˚r /∂t can be obtained from the method introduced above. As the position of the vortex ring is fixed in the current time step, the term ∂˚v /∂t equals zero.

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Fig. 2. Comparison between high-speed photographic record of bubble motion and numerical simulation results (cross-section of the bubble, the pressure contours and velocity fields in the domain). The letters beneath each frame represents the sequence of events. The dimensional times for experiment are: 199, 633, 2832, 4199, 4331, 4498, 4631, 4998 ␮s. The nondimensional times for simulation are: 1e−5, 0.1028, 1.106, 2.1228, 2.1833, 2.1963, 2.2203, 2.3052.

4. Results and discussion Here we have scaled length with respect to the equivalent maximum radius of the bubble Rmax ; and pressure with respect to P = P∞ − Pc ; and time by Rmax / P. This scaling leads to the following dimensionless parameters

the bubble motion at jet impact in the dimensionless form, which is confirmed by Blake [12]. Besides, the bubble surface is discretized into 100 nodes and 99 elements. The forward time integration of Eqs. (5) and (6) is carried out using the fourth-order Runge–Kutta method. In this study, the time-step size is chosen as t =

Strength parameter: ˛ = Pini / P Stand-off parameter: ˇ = d/Rm Buoyancy parameter: = (gRm / P)0.5 In our calculations, the parameters for water are chosen as  = 1000 kg/m3 , Pc = 2338 Pa, and the ratio of the specific heats for the gas is ϑ = 1.25, and the gravity acceleration g = 9.8 m/s2 . 4.1. Bubble dynamic behavior and flow field for ˇ = 1.51 In this case, the initial distance between the bubble and the rigid wall is 14.88 mm, and the equivalent maximum radius of the bubble is 9.88 mm. According to the experiment, the stand-off parameter and buoyancy parameter are chosen as ˇ = 1.51 and = 0.03, respectively. The strength parameter is chosen as ˛ = 50 in this paper for the initial internal pressure condition does not significantly affect

˚ max |1 + 0.5|∇ ˚|2

ϑ

− 2 z − Pini (Vini /V ) |

(20)

˚ is chosen as 0.01 before the jet impacts upon the opposite side of the bubble. After that, ˚ is chosen as 0.005 because the toroidal bubble is more instable and complicate. It should be stated that the spark-generated bubble is different from an UNDEX bubble [3]. Firstly, the contents of a sparkgenerated bubble are derived from water. During the collapse phase, all the reaction products may dissolve in the surrounding water under the high pressure inside the bubble [33], while most of the contents of an UNDEX bubble will not dissolve in the water at the collapse phase [34]. Secondly, the copper wire continues to burn for a relatively long time after the bubble is initiated. On the contrary, nearly all the explosive reacts soon after the detonation, forming an UNDEX bubble. Generally, the evolution of the UNDEX bubble radius has excellent agreement with analytical solution (R–P equation) [34]. But the period of a spark-generated

S. Li et al. / Applied Ocean Research 50 (2015) 227–236

231

6

1 jet impact

5

0.8

4 E

Radius

E

k

E

3

p

E

total

2

Numerical Result R-P Equation

0.4

1 0 0

0.6

0.2 0.5

1

1.5

2

0 0

t

0.5

1

1.5

2

t

Fig. 3. Time history of variations of kinetic, potential and total energy.

Fig. 4. Time history of equivalent bubble radius.

during collapse phase cannot reach its initial value, so is the pressure inside the bubble. Fig. 4 shows the evolution of the equivalent bubble radius versus time, and the solution of R–P equation with the same initial condition (˛ = 50) is also plotted. The trends of these two curves are similar, but the period and minimum radius have much difference. Owing to the retardation of the rigid wall, the period of the bubble will be longer. The minimum radius during the collapse phase will be larger because of the circulation of the flow occupies some energy, which have been discussed above. Variation of dynamic pressures at several moments on the wall versus r coordinate are shown in Fig. 5. The corresponding pressure contours in the domain can refer to Fig. 2. Because of the high pressure inside the bubble at t = 1e − 5, the pressure induced on the wall is very high. The pressure in the middle of the wall reaches 10.88, and decreases as the r coordinate is gained. During the expansion (e.g. t = 0.1028), the pressure inside the bubble decreases, resulting in the decrease of pressure on the wall. When the bubble is over-expanded, the pressure inside the bubble drops below the ambient pressure (e.g. t = 1.1067). The dynamic pressure on the wall becomes negative, and tends to zero at infinity. During the collapse phase (e.g. t = 2.1228, 2.1833), the bubble inside pressure rises again, and the dynamic pressure on the wall increases. After the formation of the toroidal bubble, the pressure increases rapidly due to the jet impact. The maximum pressure on the wall reaches about 22.77 at t = 2.1963, which is about 2.1 times the pressure at t = 0. After the bubble reaches its minimum volume at t = 2.209,

25 Pd

2 20

0 -2 0

15 Pd

bubble is apparently lager than Rayleigh oscillation time (Tosc = 1.83Rmax / P) [6]. However, for the bubble near a boundary, the UNDEX and spark-generated bubble display similar behaviors, and the jet tip velocities of both are in the same order [34]. More discussion about the differences and similarities between the UNDEX and spark-generated bubbles can be found in Hung [34]. Due to the simplicity and low cost, spark-generated bubbles are often used to validate numerical model though some difference in time scaling [18]. Some typical phenomena of the experiment are shown in Fig. 2, while the numerical simulation results (cross-section of the bubble, the pressure contours and velocity fields) are also shown below the experimental results. Each sequence shows the bubble just after it is created (frame a), during expansion (frame b), at maximum expansion (frame c), during collapse (frame d) and jet impact upon the opposite side of the bubble (frame e). In Fig. 2(d), a high pressure region is located above the bubble, which drives a high speed liquid jet into the bubble [27]. These phenomena before the formation of a toroidal bubble have been discussed in many literatures, which are not the main purpose in this study. Because of the highspeed liquid jet, a protrusion appears at the jet tip, and a ‘sideways’ jet [12] moves upward on the bubble surface (frame f–h). Besides, a high pressure region appears around the jet tip, which covers only a small area. Similar phenomenon can be found in Brujan’s [14] work. The variation of the pressure on the wall will be discussed further below. It is observed that the bubble is drawn toward the rigid wall during its rebounding process, and a good portion of the bubble is ‘stretched out’ in the vertical direction (frame g). It is clear that the toroidal bubble splits into two parts in experiment (frame h). In our numerical simulation, smoothing and redistribution of nodes are applied on bubble surface, avoiding the numerical instabilities. Although these treatments may smooth away some information in the flow, the main physical features are preserved [13]. In the calculation process, the smoothing technique is applied every 4 time steps and redistribution technique every time step. Overall, our numerical results correlate well with the experimental results. In the current incompressible calculation, the total energy should always be a constant. The variations of kinetic, potential and total energy versus time are plotted in Fig. 3. As can be seen from the figure, the total energy is almost a constant except for a little fluctuation after the jet impact, and the maximum energy loss is 2.67%. The reasons for total energy change are topology changes and strong instabilities of the toroidal bubble [35]. As a whole, the total energy keeps a constant number, indicating the high precision of the present simulation. The initial kinetic energy of the liquid is zero due to the zero value of the bubble velocity. Yet, when bubble reaches its minimum volume, the dimensionless kinetic energy is about 0.82 (13.9% of the total energy) due to the circulation of the flow around the toroidal bubble. Thus, the potential energy

0.05 r

0.1

10

t=1e-5 t=0.1028 t=1.1067 t=2.1228 t=2.1833 t=2.1963 t=2.2203 t=2.3052

5 0 0

0.5

1

1.5 r

2

2.5

Fig. 5. Variation of dynamic pressure on the wall versus r coordinate.

3

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S. Li et al. / Applied Ocean Research 50 (2015) 227–236

Fig. 6. Variations of dynamic pressures on the wall and the inner pressure of the bubble versus time.

the second cycle of oscillation begins and the bubble rebounds immediately. Then, the pressure on the wall drops once again during the rebounding process (e.g. t = 2.2203, 2.3052). Fig. 6 shows the variations of dynamic pressures at different positions on the wall versus time for ˇ = 1.51, in which the instants

corresponding to (d)–(h) in Fig. 2 are also marked. Besides, the inner pressure of the bubble is also plotted. At the instant of (d), the pressure at the wall center is relatively small. The pressure increases to 6.45 when the jet is impacting on the opposite side of the bubble (instant (e)). After the formation of the toroidal bubble, the pressure on the wall increases rapidly and reaches its maximum value at the instant (f). The pressure decreases afterwards, indicating the “jet power” is decreasing. Wang [36] demonstrated that the bubble is migrating toward the wall rapidly when it is near its minimum volume. So the distance between the bubble and the wall is reduced, resulting in a second peak on the pressure curves (instant (g)). At the instant of (h), the toroidal bubble becomes much closer to the wall and the high pressure region is also near the wall. However, the maximum pressure has decreased a lot at this instant. Therefore, the pressure on the wall decreases gradually. As shown in Fig. 6, the inner pressure of the bubble reaches its maximum between the instants of (f) and (g), indicating that the inner pressure of the bubble is not the key factor to the two peaks on the pressure profile. From the dicussion above, we can conclude that the first peak is mainly caused by the jet impact, and the occurrence of the second peak is induced by the rapid migration of the toroidal bubble toward the wall.

Fig. 7. Comparison between high-speed photographic record of bubble motion and numerical simulation results (cross-section of the bubble). The letters beneath each frame represents the sequence of events. The dimensional times for experiment are: 0, 600, 2898, 4198, 4465, 4532, 4598, 4764 ␮s. The nondimensional times for simulation are: 1e−5, 0.1312, 0.9538, 2.1152, 2.2308, 2.274, 2.306, 2.3254.

S. Li et al. / Applied Ocean Research 50 (2015) 227–236

120

233

3

100 Pd

2

80

1 0

Pd

60

-1 0

40

0.05 r

0.1

1.5 r

2

t=1e-5 t=0.1312 t=0.9538 t=2.1152 t=2.2308 t=2.2740 t=2.3060 t=2.3254

20 0 -20 0

0.5

1

2.5

3

Fig. 8. Variation of dynamic pressure on the wall versus r coordinate.

4.2. Bubble dynamic behavior and flow field for ˇ = 1.05 In this case, the initial distance between the bubble and the rigid wall is 10 mm, and the equivalent maximum radius of the bubble is 9.52 mm. In numerical calculation, the dimensionless parameters are chosen as ˛ = 50, ˇ = 1.05, = 0.03, and the other parameters are the same as those in Section 4.1. Some typical phenomena of the experiment and numerical simulation results are shown in Fig. 7. The interaction between pre-toroidal bubble and rigid wall won’t be discussed here. As shown in Fig. 7(e), when the jet is about to impact upon the lower side of the bubble, the bubble surface has not been in contact with the wall. A protrusion is drawn toward the rigid wall soon after the impact and a high pressure region caused by the jet impact is located at the wall center, shown in Fig. 7(f). In Fig. 7(g), the jet impacts on the rigid wall afterward and the bubble assumes a mushroom shape. From the numerical results, the width of the jet increases as the bubble continues to shrink. The maximum pressure is decreasing while the area of the high pressure region is increasing. As the ‘sideways’ jet propagates upward along the bubble surface and meets the incoming flow due to the shrinking toroidal bubble, a ring-shape high pressure region emerges around the bubble, as shown in Fig. 7(h). This is the so-called splashing effect in Tong’s [13] work. Due to the copper wire burnt in experiment, the gas inside the bubble is dark, so the jet cannot be seen in the photographic records. As a whole, the numerical results have excellent agreement with the experimental results in shapes, and the characteristic of the pressure on the wall will be discussed as follows. Fig. 8 shows the variation of dynamic pressures at several moments on the wall versus r coordinate. Before the formation of the toroidal bubble, the pressure on the wall is closely associated with the inner pressure of the gas bubble. When the jet is on the point of impacting upon the opposite side of the bubble at t = 2.2308, the pressure at the center of the wall is only 2.04. After the liquid jet threading the bubble, the bubble is contact with the wall directly and the pressure reaches as much as 107.7 at t = 2.2740. This is because the high velocity jet decelerates around the jet tip, leading to the formation of a stagnation point and a high pressure region along the z-axis on the wall. However, the high pressure region has a small area. It can be seen from Fig. 8 that the pressure drops rapidly along the r-direction at t = 2.2740. Brujan [14] also pointed out that the high pressure region on the rigid boundary covers only a small area localized around the jet tip. The rigid wall will redirect jet to form a radial flow outward along the boundary from the jet axis, which results in a broader jet and a larger high pressure region. At

Fig. 9. Variations of dynamic pressures on the wall and the inner pressure of the bubble versus time.

t = 2.3254, a little fluctuation of the pressure occurs around r = 0.5, which is induced by the splashing effect. Fig. 9 shows the variation of dynamic pressure on the wall at different positions versus time, in which the instants corresponding to (d)–(h) in Fig. 7 are also marked. Besides, the inner pressure of the bubble is also plotted. At the instant of (e), the pressure at the wall center is only 2.1. At the instant of (f), the pressure at the wall center reaches its maximum value of 107.7 soon after the jet touchdown, which is about 6.9 times the pressure at t = 0. The dimensionless inner pressure of the gas bubble is only 6.4 at this moment, indicting the inner pressure has relatively little effect on the highly-pressure at the wall center. In this sense, the liquid jet can cause serious damage to the close wall. However, the high pressure region is restricted around the jet tip, so the pressure curves for r = 0.5, 1, 2 have no evident peaks. The velocity of the liquid jet cannot maintain at a high level, so the pressure at r = 0 decreases rapidly afterward. At the instant of (g), the pressure at the wall center rises again due to the migration of the toroidal bubble centroid. The splashing effect occurs at the instant of (h), and the ‘sideways’ jet continues propagating into the liquid jet after t = 2.3254. The ring-shape high pressure region is also moving along with the ‘sideways’ jet. At the instant of (i), the high pressure region has moved to the wall center, resulting in the emergence of another peak. Besides, the liquid inside the jet is gathered by the rebounding toroidal bubble at this instant, which is also contributing to the high pressure region. The inner pressure of the toroidal bubble reaches its maximum at t = 2.331, indicating the gas pressure is not the key factor to the several peaks on the pressure profile. 4.3. Discussion In the works by Chahine [23] and Hsiao [24], a hybrid incompressible–compressible method is adopted to simulate the dynamic behaviors of a cavitation bubble. Their incompressible model and the compressible Euler equation solver GEMINI are based on the potential flow theory and a finite difference scheme, respectively. The incompressible solver is used during most of the bubble dynamics where the liquid velocities are very small compared to the sound speed. The compressible Euler solver is used during shock formation and the jet impact stage. In this section, a comparison will be made between the present model and the hybrid incompressible–compressible method. The calculation parameters are set the same with a case in Hsiao [24], which will be briefly described as follows. A bubble of initial radius Rini = 50 m is at equilibrium in the liquid at 1 atm (105 Pa), located at a distance of 1.5 mm from a rigid wall. The pressure then suddenly decreases to 103 Pa, stays there for t = 2.415 ms, and then rises sharply to 10 MPa. The bubble expands when the ambient pressure drops to the low 103 Pa. The maximum radius of the

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Fig. 10. Pressure contours and velocity fields at different instants during the the bubble collapse near the wall. (a) t − tjet = 0; (b) t − tjet = 0.06 s; (c) t − tjet = 0.12 s; (d) t − tjet = 0.17 s.

bubble is around 2 mm. After the sudden pressure rise to 10 MPa, the bubble will collapse and jet toward the wall. Fig. 10 shows the pressure contours and velocity fields of some instants during the collapse phase of the bubble. Fig. 10(a) shows the moment just before the jet touches the opposite side of the bubble, which is denoted by tjet . The maximum pressure in the flow field is about 230 MPa, which is close to the result of 250 MPa in Hsiao [24]. After the jet threading the bubble, a high pressure region is located around the jet tip, shown in Fig. 10(b). A protrusion is stretched downward in the vertical direction (Fig. 10(c)), and the toroidal bubble splits into two parts afterwards (Fig. 10(d)), which could also be found in Hsiao [24]. However, the afterward process is not simulated in our model because the multi-vortexring model is still a difficult problem. The results of the hybrid incompressible–compressible could be found in Hsiao [24] (Fig. 8, p.154). Overall, the pressure contour levels and bubble dynamics are similar with those in Hsiao [24]. To display a better comparison between these two models, the pressure variations at the wall center are plotted in Fig. 11 simultaneously, in which the instants corresponding to (a)–(d) in Fig. 10 are also marked. All pressure histories versus time have been shifted to the same starting point. It is noted that the pressure of the

present model begins to increase soon after the jet touchdown, resulting from the infinite sound speed in the present model. However, it takes some time for the shock wave traveling to the wall in the hybrid model. In the present model, the pressure reaches its first peak 1068 MPa at t − tjet = 0.06 s. In the hybrid model,

Fig. 11. Variation of dynamic pressure on the wall versus time. (The triangle blocks denote the hybrid model result from Hsiao [23], and the circles denote the result of the present model.)

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the pressure reaches a higher peak 1359 MPa by using a shorter time t − tjet = 0.036 s. Chahine [23] pointed out the pressures due to bubble collapse are scaled by the water hammer impact pressure [37], in which the compressibility plays an important role. The results of the present model are just hydrodynamic pressures, which is based on the incompressible assumption. Therefore, the differences between these two models are mainly derived from the compressibility of the flow. Due to the rapid migration of the bubble, the pressure reaches its second peak at the instant of (c), shown in Fig. 11. Similar phenomenon could be found in the result of the hybrid method. Besides, pressure or shock waves bounce back and forth between the target wall, the bubble surface, and any other daughter bubbles in the near-wall flow field, causing more peaks in the hybrid method. In addition, Hsiao [24] demonstrated that another peak emerges when the bubble reaches its minimum volume. This peak is not observed in the present result because the process after the split of the toroidal bubble is not simulated. In a conclusion, the differences between these two models show that the compressibility is another influence factor of the jet impact. However, the agreements show that the main feathers of the pressures and the toroidal bubble dynamics could be well simulated using the present model.

5. Conclusions The process and mechanism of bubble jet impact on a rigid wall have been studied both experimentally and numerically. The dynamic behavior of spark-generated bubbles is captured by a high-speed photography. The numerical study is performed based on BIM, and a vortex ring model is introduced to handle the toroidal bubble. The numerical results compare favorably with the experimental results in bubble shapes. Besides, auxiliary function method is adopted to calculate the pressure for the improvement of accuracy. At last, a comparison is made between the present model and a hybrid incompressible–compressible model [24]. From the foregoing work, some useful conclusions can be made. As to the case ˇ = 1.51, the jet is non-contact with the wall during the collapse phase. A double-peaked structure occurs in the pressure profile (r = 0, 0.5). The pressure on the wall reaches its first peak soon after the jet impact, and the occurrence of the second peak is caused by the rapid migration of the toroidal bubble toward the wall. In addition, the total energy is almost a constant except for a little fluctuation after the jet impact, indicating a high precision of the simulation. As to the case ˇ = 1.05, the bubble jet is in direct contact with the wall. The maximum pressure on the wall caused by the jet impact is approximately 6.9 times the pressure at t = 0 (caused by high inner pressure of the bubble). This indicates that the liquid jet in direct contact with the wall can also cause much serious damage. Multiple-peaked structure occurs in the pressure profile during the collapse and rebounding phase. The mechanisms of the first two peaks are the same with the case ˇ = 1.51, and the subsequent peaks are associated with the splashing effect and rebounding of the toroidal bubble. Both agreements and differences are found in the comparison between the present model and a hybrid incompressible–compressible model [24]. The differences between these two models show that the compressibility is another influence factor of the jet impact. However, the pressure contour levels, toroidal bubble dynamics and variations of the pressures on the wall are similar with each other, indicating the main feathers of the jet impact pressure and the toroidal bubble dynamics could be simulated by BIM.

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