On study of non-spherical bubble collapse near a rigid boundary ＊

Xiao-jian Ma, Xin Zhao, Biao Huang, Xiao-ying Fu, Guo-yu Wang

1. School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China

2.Department of Mechanics, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081,China

3.State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065,China

Abstract: This paper applies numerical methods to investigate the non-spherical bubble collapse near a rigid boundary.A three-dimensional model, with the mass conservation equation reformulated for considering the compressibility effect, is built to deal with the coupling between the pressure and the flow velocity in the momentum and energy equations and to simulate the temporal evolution of the single bubble oscillation and its surrounding flow structure.The investigations focus on the global bubble patterns and its schlieren contours, as well as the high-speed jet accompanied when the bubble collapses and the counter jet is generated in the rebound stage.The results show that the robust pressure waves emitted due to the bubble collapse lead to substantial changes of the flow structures around the bubble, especially the formation of the counter jet generated in the rebound stage.Furthermore, compared with the high-speed jet when the bubble collapses, the counter jet in the rebound stage emits the momentum several times greater in the magnitude and in diametrically opposite direction at the monitoring point.

Key words: Bubble, jet, rigid boundary, numerical simulation, volume of fluid (VOF), pressure wave

Introduction

The bubble dynamics involves significant energetic actions, which can be destructive to the hydraulic machines and the propellers due to the pressure pulsations[1], the sudden changes of the loads[2], the vibration[3],the noise and the erosion[4](see the review paper[5]).They can also be beneficial in the industrial cleaning[6], the ice breaking[7], the medical diagnostics[8-10]and the microfluidics[11], due to the high fluctuating pressure.The energetic behaviors are closely related to the generation of the pressure waves and the high-speed liquid jet during the non-spherical bubble collapse.However, the generation of the bubbles near a rigid boundary remains a challenging task, since its mechanism has not yet been fully elucidated.

With the significant progress in experimental techniques of the bubble generation, such as with the laser pulse and the electric spark, a large number of observations demonstrate that the bubbles have usually a remarkable behavior of non-spherical collapse, especially, when affected by the rigid boundary.The related experimental results were reported by Hardy et al.[12], Li et al.[13].With the high-speed photographs and the particle image velocimetry (PIV), they find that the radius of the bubble grows quickly to the maximum size once it is generated, and then it begins to shrink, with radial flow toward the bubble center.During the collapse process, the distal surface of the bubble shrinks faster than the opposite surface closer to the wall.The asymmetric motion induces a high-speed jet towards the wall.As is known, the asymmetrical collapse and the formation of a jet is the result of the secondary Bjerknes force caused by a pressure gradient around the bubble near the rigid boundary.Ohl et al.[14]employed the high-speed photography with grating to investigate the detailed flow structure after the jet impacts the distal side of the bubble surface.A jet-impact-induced shock wave is clearly captured before the bubble reaches a minimum volume.As discussed above, the high-speed jet and the shock wave are generally regarded as a potential mechanism for the energetic actions of the bubble collapse.However, the bubble patterns and its flow structure in the rebound stage were not well studied, as compared with the high-speed jet in the stage of the bubble collapse.What is more, due to the small spatial and temporal scales of the bubble dynamic behaviors, it is difficult to visualize the transient bubble shapes and to make the measurements of its surrounding flow structures without any interference[15].Considering the difficulties in the experimental observations, numerical simulations become an obvious choice to get a further insight into the evolution of the bubble shape deformation and the development of the flow structure including the velocity and pressure fields.

The bubble behavior near the rigid boundary is a classic problem of the multiphase flow.Therefore, the accurate simulation of the bubble behaviors and its surrounding flow structures depends on a correct mathematical formulation of the physical characteristics of each fluid phase, as well as the interface between two phases.In the recent years, significant developments have been made in understanding and modeling complex multiphase flows by advanced flow simulation methods.In the term of solving the governing equations, the Naiver-Stokes equations, the main methods include the Eulerian method[16], the lattice Boltzmann method[17], the boundary element method[18-19], the smoothed particle hydrodynamics[20],and the moving particle semi-implicit method[21].Johnsen and Colonius[22]employed the Eulerian method to compare the 2-D bubble dynamics of the shock-induced and Rayleigh collapses.They found that the maximum average bubble pressure in the shock-induced collapse is much smaller than that in the Rayleigh collapse.Zhang et al.[23]used the BEM coupling vortex ring model to study the splitting of a toroidal bubble near a rigid boundary.They found that the sideway jet continues propagating on a sub-bubble surface, to cause a partial breakup of the splash film into droplets.Zhang et al.[24]applied the three dimensional multiphase SPH model to study the bubble rising and coalescing.The background pressure is shown to play an important role in keeping the multiphase interface stable.The advantages and the disadvantages of these methods were reviewed by Deen et al.[25].In principle, the multi-fluid systems could also be numerically modeled using one set of governing equations for a single-phase flow.However,two problems will be cropped up in using this numerical scheme to simulate the bubble behaviors near the rigid boundary: (1) the time-consuming solution of the Poisson equation, (2) the weak compressibility in the bubble collapse.When solving the Naiver-Strokes equation, the solution of a partial differential equation of elliptic type, namely, the Poisson equation, takes much time and computing resource.In addition, the robust high-speed jet will have a weak compressibility, which significantly affects the bubble dynamics.In the present work, a general model is built to simulate the interaction between the gas and liquid phases within the framework of the non-spherical bubble collapse.The motion equations are for a deterministic description of the compressible two-phase flow, as done by Caltagirone et al.[26].In the model, the mass conservation equation is reformulated by developing the partial derivative of the pressure, to avoid the solution of the Poisson equation.The reformulated momentum and energy equations are solved in the whole computational domain, being discretized using the finite-difference scheme.A stationary and fixed grid is used for the fluid flow, and the VOF method is used to identify the interface.The fluid properties such as the density and the viscosity are calculated according to the position of the interface.As a result,this robust method could simulate the compressible effect caused by the bubble collapses near the rigid boundary[27].

Although the bubbles were quite widely studied in the past years, the dynamic behavior of a single bubble near a rigid boundary is still not well understood, especially in the rebound stage, and hence a further study is still needed.The present paper will(1) introduce and validate the compressible model to capture the transient behaviors of the single bubble shape; and (2) compare the difference between the high-speed jet when the bubble collapses and the counterjet in the rebound stage.

1.Numerical methodology

The motion equations are for a deterministic description of the compressible two-phase flow, as done by Caltagirone et al.[26].The governing equations are as follows:

wherepis the pressure,tis the time,uis the velocity,gis the gravitational acceleration,χris the adiabatic compressibility,Tis the temperature,Cvis the heat capacity at the constant volume, andFTSis the surface tension force.The density, the viscosity, and the conductivity of the fluid are typically defined as=Cρl+(1 -C)ρg,=Cμl+(1 -C)μgand=Cλl+(1 -C)λg, where the subscriptslandgdenote the liquid phase and the gas phase, respectively.Cis the local volume fraction, taking the value of 1 for the liquid phase and 0 for the gas phase.The interface ∑ between the liquid phase and the gas phase is defined asC=0.5.The volume of fluid (VOF) method is adopted to track the gas-liquid interface, since it is of interest to maintain a sharp interface between the two involved phases along with the topological changes of the interface.

Compared to the classical governing equations in the cases of incompressibility, the mass, momentum and energy equations are reformulated to consider the compressible effect of the fluid in terms of dilatation effects.For the mass conservation, the adiabatic compressibility coefficientχTis small and the ratio 1/χTis large enough to neglect the effect of ?p/?tin the liquid phase.On the other hand,χTin the gas phase is larger than that in the liquid phase and the mass conservation equation turns to the standard compressible pressure equation where the isobaric dilatation term can be discarded due to its negligible magnitude.With the dilatation term 1/χT?·uone can take into account the dilatation effects directly in the momentum equations whereas these terms ask for the incompressibility when 1/χTis large in the liquid.The derivation of the numerical model is briefly presented in the following section.

1.1 Mass conservation equation

The single bubble consists of the ideal gas and the liquid that obeys an appropriate equation of state,which can be expressed as

whereγgis a constant similar to the exponent of the density in the equation of state,Bis the stiffness of the liquid, Therefore, the state of equation can be re written as one in terms of the pressure corresponding to the time in the function of temperature and density

whereβis the coefficient of the thermal expansion.Substituting the equation of the compressible mass conservation dp/dt+ρ? ·u=0 into Eq.(3), we have

So Eq.(3) can be rewritten as

To express in a concise way, the characteristic variables in the important terms of Eq.(6) are defined as the following dimensionless variables:

whereV0is the reference velocity,Rthe current radius of the bubble, andTRa reference temperature.And the non-dimensional equation for the pressure can be expressed as

whereMais the Mach number,Peis the Peclet number, andGais the Gay-Lussac number.To further simplify the equation for the pressure, the three terms in the right hand of Eq.(8) are considered in details to get the expression suitable for the case of the bubble dynamics near the rigid boundary.Figure 1 shows the relatedR,Ma, 1/MaandGa/(Ma·Pe) in the case of a single bubble near the rigid boundary.As observed,Rrepresents the top margin of the bubble obtained from the experimental observation in our previous work[28], and the reference velocityV0can be calculated asV0=dR/dt.Furthermore, compared with the absolute value of 1/Ma～O(103), the values ofMa～O(10-2) andGa/(Ma·Pe) ～O(100) are so small in magnitude and could be neglected in Eq.(8).So the final equation of the mass conservation can be written as

And it can be written in the original variables again as

1.2 Momentum equation

The momentum equation for the incompressible fluid model is

With Eqs.(10), (11) can be rewritten as

Equation (12) provides a mathematical framework for multiphase compressible flows.In the model, the mass conservation equation is reformulated by developing the partial derivative of the pressure according to the time in the function of density and by assuming that the gas phase behaves like an ideal gas.

1.3 Energy equation

The energy equation can be expressed as

where the second term on the right hand of this equation can be expressed in the formation of the dilatation based on Eq.(3)

where the first term on the right hand of this equation can be expressed as

Fig.1 (Color online) The related in the case of bubble dynamics near the rigid boundary

With dp/dt=-ρ?·uand Eqs.(15), (14) can be expressed as

As a result, the energy equation with the new formulation can be expressed as

In the present work, the time step is selected according to a stability criterion, called the Courant-Friedrich-Levy (CFL) criterion, such that the inertial terms do not propagate a distance more than one grid cell in one calculation step.In this paper, a uniform and staggered mesh is used to discretize the governing equations.The velocity vector is evaluated at the cell edge, while the pressure and the density are evaluated at the cell center.This grid configuration is suitable for the “marker-and-cell (MAC)” method used to solve the temporal evolution of the pressure and velocity fields.The convective terms are discretized by an explicit second-order upwind scheme, and the viscous terms are discretized by a central difference scheme[27].We employ our in-house code in Fortran to simulate the bubble dynamics near the rigid boundary.

2.Results and discussions

2.1 Numerical setup and validation

To investigate the temporal evolution of the single bubble near a rigid boundary, the calculated domain is set as a 3-D cuboid.Figure 2 shows the schematic description of the solution domain, the initial condition and the boundary condition for the numerical simulation.As observed, the bottom line represents the rigid boundary, which is set as the reflecting boundary.The boundary is constructed into three layers, namely,N-1,NandN+1.If the no-through-flow condition is satisfied, the scalar quantities, such as the density and the pressure, must be mirrored across the wall, while the normal velocity component is reflected withuN+1=-uN.Other five boundaries except for the bottom one are all set as the non-reflecting boundaries.In addition, the computational domain is assumed to be full with a bubble and the liquid, which are initially quiescent.So the initial velocities of the gas and liquid phases,ugandulare both zero.

Fig.2 (Color online) Schematic description of the solution domain, the initial condition and the boundary condition for the numerical simulation

To demonstrate the effect of the side wall on the bubble shape patterns, Fig.3(a) shows the relationship between the bubble patterns and the different domain sizes, namely,R/Rm=2, 3, 4 and 5, whereRis the distance between the bubble center and the side wall.It is clearly shown that the side wall effect can be ignored when the radial radius of the solution domain is about 4 times of the bubble radius.In addition, Fig.3(b) shows the effect of the number of the calculated gridsn3on the bubble patterns, wherenis the number of grids in each dimension.Four different grid resolutionsn=80, 130, 160 and 180 are applied for the simulations.With the coarse grid resolutionsn=80, 130, the patterns of the single bubble near the rigid boundary are significantly different, especially, the top margin of the bubble cannot be captured well.When the grid resolution is finer (n=160, 180), the differences of the bubble patterns can be very small.Hence, the simulation results reported in this work are for the grid resolutionn=160 to save the calculation time.

Fig.3(a) (Color online) The effect of the different solution domain size on the bubble shapes for R/Rm=2, 3, 4 and 5 under the simulation condition of γ=1.2

Fig.3(b) (Color online) The effect of the different grid numbers on the bubble shapes for n=80, 130, 160 and 180

Fig.4 (Color online) Comparisons of the numerical (solid line)and experimental (point) observations of bubble collapse close to a rigid boundary for the model

In order to validate the numerical method discussed in the last section, Fig.4 shows the comparisons of the numerical results (solid line) and the experimental observations (point) of the bubble shapes close to a rigid boundary in a quiescent liquid.The detailed information about the experiment setup can be found in our previous paper[29].To quantitatively describe the initial position of the bubble in the infinite fluid, the dimensionless standoff distance between a bubble and the boundary is defined as

whereLis the distance from the bubble center to the boundary, andRmis the maximum radius of a bubble.And the initial parameters in the numerical simulation are set as: the radius of bubbleRm=17.80mm , the initial standoffL=21.36 mm ,the ambient pressureP0=100atm, the inner pressurePv=1atm, the liquid densityρl=980 kg/m3, and the sound velocityc=1480 m/s.As observed,compared to the spark-generated bubble experiment, it is shown that there is a general agreement between the numerical and experimental temporal bubble shapes.

2.2 Temporal evolution of bubble patterns and flow structures

The evolution of the bubble dynamics near the rigid boundary involves two scales, namely the temporal and spatial scales.In this section, the temporal scale, namely, the time evolution of the bubble shape and the flow structure, will be discussed in detail.Figure 5 shows the comparisons of the experimentally observed bubble pattern and the numerically predicted interface against the normalized time.The dimensionless collapse timeτis defined as

wheretBis the time from the bubble generation to the first collapse,tOSCis defined as the Rayleigh oscillation time.In Fig.5, the numerically predicted bubble interfaces near the rigid boundary withγ=1.2 are compared with the experimentally observed bubble pattern, in three typical stages,namely, the shrink stage (frames 1, 2), the collapse stage (frames 3-6) and the rebound stage (frames 7-8).As observed in frame 1, the bubble pattern near the rigid boundary at the lowest limit of the frame is initially in a spherical shape.In frame 2, it then shrinks with the lower bubble margin almost attached to the rigid boundary and the rest of the bubble surface shrinks rapidly.Especially the left and right bubble margins contract faster than the top part of the bubble, which makes the bubble assume an ellipsoid shape.In the frame 3, the lower part of the bubble surface is flattened, while the upper part of the bubble moves inwards.Compared with the lower part, the top surface of the bubble has a greater collapse speed,making the top surface continue to move downward(the frame 4).The top surface of the bubble impacts the opposite bubble surface, to form an entrance inside the bubble, in a toroidal shape (the frame 5).In the final stage of the collapse, the tip of the entrance penetrates through the bubble to form a relatively sharp protrusion at the bottom part of the bubble, and eventually the bubble becomes a stagnation ring (the frame 6).In the rebound stage of the bubble, the bubble reaches the rigid boundary, the volume of the bubble expands again, and the entrance width of the stagnation ring decreases significantly (the frame 7).Eventually, the bubble begins to shrink and collapse for the second time (the frame 8).The numerical predictions of the bubble shapes compare well with the experimental measurements with only small differences as shown in the frame 7 due to the rebound of the bubble.

Fig.5 (Color online) The comparisons of temporal evolution between the liquid and gasphases when γ=1.2,Rm=21mm

Figure 6 shows the numerical schlieren contours of the pressure waves emitted from the bubble during the shrink, collapse and rebound stages.Plots of the density gradient are generated in order to compare with the experimental schlieren images.This allows the waves and the interfaces to be visualized on the same plots.The definition of the density gradient is as follows

Fig.6 Numerical schlieren contours of the pressure waves emitted from the non-spherical bubble collapse near the rigid boundary for standoff parameter γ=1.2.The first row indicates the shrink stage of the bubble behaviors, the second row represents the collapse stage,and the last row shows the rebound stage.The normalized times for numerical results are as follows: 0,0.55, 0.71, 1.02, 1.03, 1.04, 1.10, 1.31 and 2.52,respectively

whereθ=40 for the air andθ=400 for the water,in order to observe the pressure waves in both the air and the water simultaneously.As observed in the initial shrink phase of the bubble atτ=0, a progressive pressure wave is emitted from the bubble surface in all directions.Due to the rigid boundary, the progressive wave toward the wall is reflected atτ=0.55.Then the wave distribution in the superimposed region is chaotic and irregular because of the interaction between the reflected and progressive waves.During the collapse stage of the bubble dynamics atτ=1.02-1.04, the robust compression waves are generated and radiated from the bubble to the far field.It is observed that the formation of the compression waves is closely related with the presence of the high-speed jet.The jet-impact-induced pressure wave is clearly captured before the bubble reaches a minimum volume.When the high-speed jet and the compression wave push the liquid surrounding the bubble to impact the rigid boundary atτ=1.10, it is observed that the blast wave is emitted by the bubble non-spherical collapse due to the release of a high pressure.Meanwhile, a secondary reflected wave is formed again due to the rigid boundary.The water-hammer pressure associated with the high-speed jet impacting the boundary is regarded as a potential damage mechanism.The water-hammer pressure is generated by the high-speed jet and it is expressed as

whereρsis the solid density,csis the sound speed in the solid.In the problems of interest, the high-speed jet of water always first impacts the bottom part of the bubble (as shown in Fig.6(f)), and moves at a velocityvd.Thus, the right-hand side of the equation is reduced to, as proposed by Johnson and Colonius[22].So the computed water-hammer pressure reaches the value of1.8GPa.When the bubble stays in the rebound stage and with the presence of an entrance atτ=1.31, 2.52, a progressive wave is observed with a smaller curvature radius above the topmost part of the bubble margin, as compared with the larger one atτ=0.55.It is understood that the progressive wave with a smaller curvature radius is radiated, because the high pressure is released along the entrance inside the bubble (as discussed in the next section), which is closely related with the formation of the counter jet.

In order to further demonstrate the detailed information of the flow structure, Fig.7 shows the temporal profiles of the schlieren value, the normalized vertical component of the velocity and the pressure, as well as the maximum temperature for the standoff parameterγ=1.2.The normalized velocity and pressure are defined as:

wherevis the horizontal component of the velocity,cis the sound speed in the water.The monitoring point is on the top of the bubble, two times ofRmfrom the center of the bubble.The three gray regions marked in the figure correspond to the processes depicted in Fig.7, namely, the formation of the progressive waves, the high-speed jet and the counter jet.Black, red, blue and green solid lines indicate the temporal evolutions of the schlieren value, the normalized vertical component of the velocity, the pressure and the maximum temperature, respectively.In the shrink stage of the bubble, the vertical component of the velocity increases significantly and the pressure decreases sharply due to the interaction of the progressive waves and the temperature keeps stable asTm=300 K.In the collapse stage of the bubble, the vertical component of the velocity reaches the maximum value asv*=-0.014, the pressure peakP*=0.018 is formed at aroundτ=1.00, and the maximum temperature of the bubble also has a significant peak of aboutTm=2 517 K.During the rebound stage of the bubble, the fluctuation of the pressure and the vertical component of the velocity after τ=1.10 and the secondary temperature peakTm=750 K atτ=1.50 are observed due to the bouncing back and forth of the pressure caused by the combined effect of the daughter bubbles, the bubble surface and the boundary wall.It is interesting to see that the vertical component of the velocity is negative beforeτ=1.15 and positive after that time, which indicate that the motions of the high-speed jet and the counter jet are in opposite directions.In order to measure the relative relationship between them in a unified and brief way, a parameter similar to the momentum ratio is proposed as

Fig.7 Temporal profiles of the schlieren value, the normalized vertical component of the velocity, as well as the normalized pressure for standoff parameter γ=1.2.1,2 and 3 represent the shrink, collapse, and rebound stages of the bubble behaviors, respectively

whereεaxialandεjetare the momentums caused by the counter jet and the high-speed jet, respectively.Substituting the data in Fig.7 into Eq.(22), the momentum ratio between the counter jet and the high-speed jet isα=-3.9, where the negative sign means the opposite direction.Although the high-speed jet can release an enormous momentum in a very short period of time, the momentum released by the counter jet is larger than that in general, at the monitoring position.In the practical applications, such as the medical delivery and the underwater explosion, this condition should be avoided as much as possible,because the tremendous momentum caused by the counter jet is not released at the targeted location.To further investigate the difference of the high-speed jet in the collapse stage and the counter jet in the rebound stage, the pressure and velocity contours in both typical stage’s patterns would be analyzed in the next section.

2.3 Flow structures in typical stages

In order to gain a detailed understanding of the bubble shape variation corresponding to the flow structure characteristics, Figs.8, 9 show the numerical results for the high-speed jet in the collapse stage and the counter jet in the rebound stage, respectively.The contours in the right represent the normalized velocity,the contours in the left represent the normalized pressure, and the arrow represents the velocity vector.The normalized velocity is defined as

Fig.8 (Color online) Numerical results about temporal evolution of the bubble dynamic behavior and flow characteristics in the collapse stage (the contours in the right represent the velocity, the contours in the left represent the pressure, and the arrow represents the velocity vector) in the collapse stage near a solid wall with γ=1.2.The non-dimensional times for numerical results are: 1.00, 1.02, 1.04 and 1.08, respectively

Fig.9 (Color online) Numerical results about temporal evolution of the bubble dynamic behavior and flow characteristics in rebound stage (the contours in the right represent the velocity, the contours in the left represent the pressure, and the arrow represents the velocity vector) in the rebound stage near a solid wall with γ=1.2.The non-dimensional times for numerical results are 1.15, 1.25, 1.31 and 2.65, respectively

whereuis the vertical component of the velocity.Figure 8 shows the numerical results for the temporal evolution of the bubble dynamic behavior and the flow structure characteristics in the collapse stage near a rigid boundary.As shown in Fig.8(a), a highpressure region appears above the top margin of the bubble, with a high-speed area inside the bubble and with the topmost part of the bubble wall being pushed downwards.As the time progresses toτ=1.02 in Fig.8(b), the high-pressure region further enlarges,with a large deformation curvature of the top margin of the bubble surface, relative to the bottom margin in a flattened shape.As shown in Fig.8(c), the top margin of the bubble transfers a high-speed jet with a maximum velocity of 0.52, which moves rapidly towards the bottom margin of the bubble.The pressure inside the bubble is enhanced evidently due to the formation of the high-speed jet and the adiabatic compression of the bubble wall.It is notable that the pressures of the liquid surrounding the bubble and of the gas inside the bubble are increased to 0.75 and 0.36, respectively.As shown in Fig.8(d), the high-speed jet penetrates through the bottom margin of the bubble, making toroidal bubble.The velocity vectors clearly show that the sub-bubbles assume the shape of a vortex ring after the jet impact.The high-speed jet impacts the rigid wall, to generate a high-pressure region with a maximum pressure of 0.045 at the center of the rigid boundary.The velocity vectors turn from the vertical direction to the radial direction due to the restriction of the wall.It is notable that the pressure difference surrounding the bubble before the collapse is one of important factors for the non-spherical collapse and the migration towards the rigid wall.The bubble near the rigid boundary with the initial standoffγis treated as two similar bubbles separated with a distance of 2γ.Due to the in-phase oscillation motion of those bubbles, the two bubbles move to each other (or rather, move to the rigid boundary).

To demonstrate the formation of the counter jet caused by the rebound, Fig.9 gives the temporal evolution of the pressure and velocity contours in the rebound stage of the bubble.As shown in Fig.9(a) atτ=1.15, the protrusion of the high-speed jet impacts the rigid boundary and an entrance is generated inside the bubble.The velocity vectors inside the entrance still turn towards the rigid boundary due to the inertial force, to form a relatively high-pressure region near the surface of the rigid boundary, relative to the pressure of the liquid surrounding the bubble.As shown, when the time progresses toτ=1.25 in Fig.9(b), due to the reflection on the rigid boundary, the area of the high-pressure region enlarges along the entrance inside the bubble, with the velocity vectors turning away from the rigid boundary, forming a high-speed region at the topmost end of the entrance.Whenτ=1.31, as shown in Fig.9(c), this high-speed region and the velocity vectors are emitted from the topmost end of the entrance due to the large pressure difference between the topmost and nethermost parts of the entrance.At the timeτ=2.65 in Fig.9(d), the high-speed region and the velocity vectors, under action of the extrusion by the stagnation ring with deceasing the width of the entrance, continue to move away from the rigid boundary under the influence of the high-pressure region, to form the counter jet.When the value of the pressure inside the entrance decreases, the pressure difference between the inner bubble and the entrance decreases simultaneously and the entrance decreases its size and finally disappears.

2.4 The spatial effect of standoff on bubble dynamics

Section 2.3 is devoted to a detailed discussion of the temporal scales, namely, the time evolution of the bubble shapes and its surrounding flow structures.In this section, the effect of a spatial parameter on the bubble dynamics will be analyzed.It is well known that the initial standoff between the bubble and the rigid boundary has a profound effect on the bubble shape and the flow structure.Hence, the detailed numerical data are used to elucidate the physics.Comparisons of the experimentally observed bubble patterns and the numerically predicted interfaces for three different standoff values are shown in Fig.10.It is found that the distance between the bubble and the boundary has a significant effect on the form of the high-speed jet.With the increase of the standoff, the pattern and the size of the re-entrant jet diminish remarkably, until finally it disappears.The bubbles are almost unaffected by the boundaries and make Rayleigh oscillations whenγ=2.5.In the rebound stage shown in Fig.10(b), the counter jet is found under the conditions ofγ=1.2, 1.8, but not in the case ofγ=2.5, while the Rayleigh oscillation is always observed.

To further demonstrate the effect of the standoff distance on the high-speed jet and the counter jet, Fig.11 shows the comparisons of the maximum temperature, the vertical component of the velocity,and the pressure for different standoff values, namely,γ=1.2, 1.8 and 2.5.Figure 11(a) shows the temporal evolution of the maximum temperature caused by the bubble oscillation, and in all three cases, one sees the obvious peak in the collapse stage during the period of aboutτ=0.95-0.97and the fluctuation stays on in the rebound stage.The maximum temperature,respectively, is about 2 517K, 2 762 K and 2 795 K forγ=1.2, 1.8 and 2.5, which is about half the solar surface temperature.Significantly different trends are observed in the collapse and rebound stages, and with the increase of the standoffγ, the maximum temperatureTmkeeps in an increasing tendency in the collapse stage and in a decreasing tendency in the rebound stage.Furthermore, the normalized vertical component of the velocityv*has a same tendency as the temperature for these standoff values andv*increases in the collapse stage and decreases in the rebound one, as shown in Fig.11(b).As for the normalized pressure shown in Fig.11(c), the maximum normalized pressure drops from 0.018 to 0.015 with the increase of the distance in the collapse stage, as is different from the trends of the temperature and the velocity.And the different tendencies of the temperature, the velocity and the pressure are easy to explain.The pressure is inversely proportional to the speed and directly proportional to the temperature, according to the state of equation.To clearly compare the differences between the high-speed jet and the axial jet, the momentum ratio, as proposed in Eq.(24), is calculated with the data from Figs.11.The momentum ratiosα=-3.9, -2.4 and-1.8 forγ=1.2, 1.8 and 2.5, respectively, show that the momentum caused by the counter jets is larger than that generated by the high-speed jet in all three cases.

Fig.10 Temporal evolution of the bubble shapes from experimental and numerical results with different standoff parameters

Fig.11 (Color online) The comparisons of the flow field parameters for different standoff distances, namely, τ=1.2, 1.8 and 2.5, respectively To demonstrate the detailed information in the stage of the high-speed jet,each parameter is displayed from τ=0.94-1.00in insets

3.Conclusions

This paper studies the bubble dynamics near a rigid boundary for different standoff values, namely,γ=1.2, 1.8 and 2.5.A 3-D volume of fluid model with a reformulated mass conservation equation for considering the compressible effect, is used to deal with the coupling between the pressure and the flow velocity in the momentum and energy equations, as is critical to accurately capture the temporal evolution of the bubble shapes, and the corresponding fluid density and pressure dynamics.Statistics of the schlieren contours, the velocity distributions, as well as the pressure and temperature fluctuations are also presented to quantify the temporal evolution of the bubble dynamics.The primary findings are as follows:

(1) The bubble behaviors near the rigid boundary forγ=1.2 involve three typical stages, namely the shrink stage, the collapse stage and the rebound stage.A progressive pressure wave is emitted from the bubble surface towards all directions in the beginning.Due to the rigid boundary, the progressive wave toward the wall is reflected.Then the wave distribution in the superimposed region is chaotic and irregular because of the interaction between the reflected and progressive waves.In the collapse stage of the bubble dynamics, the robust compression waves are generated and radiated from the bubble to the far field.When the high-speed jet and the compression wave push the liquid surrounding the bubble to impact the rigid boundary, a blast wave is emitted by the bubble’s non-spherical collapse due to the release of a high pressure.

(2) In the rebound stage, the counter jet is found to be emitted from the entrance inside the bubble.Due to the reflection on the rigid boundary, the high-pressure region is enlarged along the entrance inside the bubble, making the velocity vectors turn away from the rigid boundary, to form the counter jet,which is much larger in the momentum and is opposite in direction, as compared with the high-speed jet.

(3) The evolution of the bubble dynamics near the rigid boundary is closely related with the spatial scale.With the increase of the standoff, the patterns of the high-speed jet and the counter jet change remarkably and their sizes are significantly reduced,until they finally disappear.The bubbles are almost unaffected by the boundaries and are in the Rayleigh oscillations whenγ=2.5.The values of the temperature, pressure and velocity peaks are all affected greatly with the increase of the standoff distance.

Furthermore, in the initial bubble position within the range ofγ=0-3.0, it is found that the bubble is accompanied with no counter jet whenγ＜1 and a significant counter jet appears whenγ＞1.Additional experimental research is needed for the condition of the counter jet occurrence to further advance the understanding of the bubble dynamics,especially in the rebound stage.Additional numerical studies for the simulation method proposed in this work are also needed to improve the understanding of the different mechanism between the counter jet and the case without the counter jet.Such research is important because an accurate prediction of the rebound stage is critical in the applications of the drug delivery, the underwater explosion, and the ultrasonic cleaning.

Acknowledgements

This work was supported by the Open Foundation of State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, the Chinese Advanced Research of Equipment Fund(Grant Nos.61402070401, 61402070501).