Numerical study of the behavior of a near-wall bubble under the influence of an acoustic wave

Acoustic cavitation has been applied to many ultrasonic fields, but its characteristics are not fully understood, especially near the wall. To analyze the dynamics of a near-wall acoustic bubble, an acoustic pressure wave boundary condition is developed in the OpenFOAM platform. We first conducted the model’s validation with an experiment to prove its accuracy. The bubble’s radius increases when the bubble is excited by the lower-pressure wave, and it decreases under the higher-pressure wave. When the bubble collapses, its pressure and temperature will reach the maximum value. In accordance with the non-dimensional distance, the dynamics of the acoustic-driven bubble are classified into three types, i.e., the liquid jet not touching the nearby wall, the liquid jet touching the wall, and no obvious liquid jet. The maximum value of pressure and temperature increases with the dimensionless distance decreasing if the dimensionless distance is small. However, there is no variation of these peak values with dimensionless distance if the dimensionless distance is large.


Introduction
Acoustic cavitation, a process in which a bubble driven by an acoustic wave undergoes expansion and rapid collapse, leading to extremely high pressure and temperature, has been extensively investigated in several engineering fields, i.e., medical treatments, surface cleaning, and water treatment.Specifically, the potential energy is stored inside the bubble as the bubble expands, and the energy is subsequently released during the last collapse phase, which is also associated with sonochemistry and sonoluminescence [1].
The bubble is hypothesized to be spherical when it oscillates in a free field, and its internal pressure is a uniform field.The Rayleigh-Plesset (R-P) equation, which considers the viscosity of the surrounding liquid and becomes a second-order nonlinear ordinary differential equation (ODE), is usually adopted to investigate bubble dynamics in its surrounding liquid under the excitation of an ultrasonic pressure wave.The simple form of the R-P equation makes it widely applicable in the field of ultrasonic cavitation.Due to the incompressible liquid assumption, the R-P equation will overpredict the rebound radius.Since the velocity of the bubble wall is high, the compressibility of the liquid cannot be ignored.The Keller-Miksis (K-M) equation takes into account the liquid compressibility to overcome the R-P equation's shortage.Another equation for acoustic-driven bubble dynamics is the Gilmore equation, which also considers the liquid compressibility by incorporating the sound emission into the surrounding liquid from the pulsating cavitation bubble.These two equations are widely used in many theoretical investigations.Zhang et al. [2] proposed a new equation to investigate bubble dynamics by considering bubble migration and the interaction of multiple bubbles.These methods have been widely used to investigate the dynamics of acoustic-driven bubbles since they can be easily and efficiently implemented.
Since the bubble is treated as the sphere model, the R-P family equations, as an ordinary equation, cannot address the complex change of bubble shape, i.e., liquid jet and toroidal bubble.Thus, to deal with nonlinear aspherical bubble dynamics, i.e., a near-wall collapsing bubble, the finite volume method (FVM) has been developed for the computation.It includes the liquid compressibility, viscosity, surface tension, and sharp interface between the internal gas and surrounding liquid.With the improvement of computer performance, the FVM can be adopted in the numerical simulation at a minor cost.The Euler equations is used to study the shock-induced bubble dynamics in shockwave lithotripsy and concluded that the wall pressure caused by the final collapse reaches on the order of 1 GPa.It has been proven that the asymmetry bubble dynamics can be obtained using the Euler equations, and the shock wave prorogation is also observed in the liquid field.The near-wall asymmetric bubble dynamics is also modeled using the Euler equations.The topology formations/evolutions of asymmetry bubbles near a rigid wall, i.e., collapse characteristics, liquid jets, rebound, and pressure acting on the wall, are intensely related to the dimensionless distance.The more common N-S equations in the OpenFOAM platform are used to investigate the evolutions and shock waves emitted by the spherical and near-wall bubble.It is pointed out that the collapse pressure attenuates with the reciprocal of distance from the bubble center when it propagates in liquid.Besides, the high-speed jet formation toward the wall and asymmetric shape during the first cycle is validated by the experimental data.Besides, a shear flow along with its shear stress deduced by a near-wall bubble is discussed using a similar method, and the variation of the wall shear stress with the dimensionless distance and viscosity is further investigated.The findings can be adopted to find a suitable fluid for surface cleaning.Yin et al. [3] pointed out that the near-wall bubble dynamics, the maximal temperature, and the maximal pressure are heavily influenced by the dimensionless distance.Three different types of bubbles are classified according to the dimensionless distance.Nguyen et al. [4] analyzed the characteristics of a cavitation bubble near a static wall and an oscillating wall and found the moving wall leads to the movement of the surrounding fluid.The mutual interaction between the bubble and oscillating wall will provide a new insight into complex bubble evolution.Tian et al. [5] proposed a modified phase-change model by considering the non-condensable gas effects.They systematically studied the wall deterioration induced by a collapsing bubble using the pressure impulse.There are two peaks of the pressure impulse, which is different from the pressure and means the time of acting pressure is important to acoustic erosion.The bubble's evolution is arranged in the position near a flat wall with a gas-containing opening (or hole), and it is pointed out that the complex wall boundary highly affects the bubble evolution, movement, and oscillation period.Due to a hole, the velocity of a high-speed jet induced by an oscillating bubble was lower than that under a flat, rigid wall.The findings may be unitized in the design of hydro-machines.Yin et al. [6] analyzed the influence of the non-dimensional distance, the relative size between hole diameter size, and bubble radius on collapse time and maximum temperature.Hu et al. [7] investigated the high-speed jet induced by a collapsing bubble influenced by a spherical particle.Ren et al. [8] studied the bubble's evolution in a cone-shaped tube.They pointed out that the jet velocity is related to the position of the bubble's inception, the internal tube's diameter, and the maximal distance of the bubble surface.Thus, many investigations of bubble dynamics near the wall or complex boundaries are conducted using the FVM since it can deal with the complex evolution of bubble topology.These references provide inspiration for the investigation of the acoustic cavitation near the wall.
In many applications of the acoustic-driven bubble, the acoustic pressure wave is not only one factor that influences bubble dynamics, but the wall also highly affects the bubble characteristics, including bubble-induced pressure, temperature, and liquid jet.Thus, in the present work, an acoustic pressure wave boundary condition is developed to investigate the acoustic-driven near-wall bubble.

Governing equations
In this work, two phases (i.e., gas and liquid) are immiscible, with an obvious interface between them.Thus, a compressible two-phase solver implemented in the Open-source OpenFOAM platform is employed to deal with acoustic-driven bubble dynamics.The conservation equations of continuity and momentum for the flow are given by: (2) where t, ρ, U, p, and g are the time, the mixture density, the mixture velocity, the pressure, and the gravitational constant, respectively.The viscous stress tensor τ is given by: ( ) ( ) where μ is the mixture dynamics viscosity, and I is the unit tensor.The continuum surface force (CSF) approach [3,5] is employed to compute the surface tension, and the expression Fs is written as: (4) where σ is the surface tension coefficient.To distinguish the interface between two phases, the equation for the liquid volume fraction can be given by:

Dp t Dt
(5) where ψl and ψg are the compressibility of two phases.To keep a distinguishable interface and decrease numerical diffusion, Equation ( 5) is solved using the volume of fluid (VOF) method, and it is supplied an artificial term, namely, the left-hand side's second term.Ur is an artificial compression velocity given by [3,5]: where nf, , Sf, Cγ are the normal vector of the cell surface, mass flux across the cell surface, the cell surface area, and modifiable compression parameter, respectively.
The energy conservation equation is given by: where T, K = |U| 2 /2, Cv, l, Cv,g, λl, λg are the temperature, the kinematic energy, the heat capacity, and the thermal conductivity of two phases, respectively.
Finally, the equations of state are used to close the model.The interior gas is regarded as an ideal gas, and its equation of state is given by:  =  g g p T (8) where ℛg is the gas constant.The exterior liquid is compressible, and its equation of state is given by: ( )

Acoustic pressure wave boundary condition
In the present study, to investigate the acoustic-driven bubble evolution, a boundary condition of an oscillatory pressure wave is developed and implemented in the OpenFOAM software.Here, an acoustic pressure wave used by Barber and Putterman [9] is employed: (10) where p∞ is far-field pressure, pa is the acoustic pressure amplitude, and f is the driving frequency.Other boundary conditions are documented in our earlier studies [5].

Computational zone
In the present study, a 3D rectangular zone is adopted to be the computational zone, as shown in Figure 1.The upper surface is set as the wall boundary or far boundary, and the far boundary is an acoustic pressure wave.If the bubble is located at the infinite liquid, the height of the computational zone will be set as 50R0.This setup will eliminate the influence of the size of the computational zone on the bubble evolution.Otherwise, the height is chosen as 25R0, when the lower boundary is configured as the wall boundary condition.To clearly observe the bubble wall, the iso-surface is adopted in the following.The angle (θ) at the right corner is the sight angle in the following 3D maps.

Dimensionless form
In the present work, to obtain results in dimensionless form, the characteristic time (t * ) is defined as 1/f, the characteristic length chosen as R0, the characteristic pressure regarded as p∞, and the characteristic temperature given by T∞.Other physical quantities, i.e., velocity, are nondimensionalized with a combination of these characteristic value.As shown in Figure 1, the dimensionless distance is given by D = d/R0, where d is the distance from the center of the bubble to the wall.

Acoustic-driven bubble in an infinite field
An acoustic-driven bubble dynamic in an infinite field is used to validate the numerical method in the present work, and it is excited by a pressure amplitude (pa) of 1.075 atm at a frequency (f) of 26.5 kHz.

AFME-2023 Journal of Physics: Conference Series 2756 (2024) 012015
The initial bubble radius is 10.5 µm, and its radius variation is shown in Figure 2. The bubble initially enlarges until it reaches its maximum radius prior to collapsing and then oscillates many times.
Compared with the experimental data [9], the bubble oscillation can be captured by the model in the present work.As shown in Figure 3, the bubble keeps its spherical shape, and its maximum velocity is located near the bubble wall.Overall, the characteristics of the bubble can be captured using the method.
After the quantitative analysis of the bubble's radius, the numerical method in the present study is appropriate to deal with the acoustic-driven bubble.

Variation of radius, pressure, and temperature inside the bubble over time
Figure 4 shows that the radius, interior pressure, and interior temperature (D = 3.0) vary with time.The bubble's radius increases when the bubble is excited by a lower-pressure wave, and it decreases under the higher-pressure wave.When the bubble collapses, its pressure and temperature will reach the maximum value.The peak pressure reaches about 38 atm, and the peak temperature is 1.65 times the far-field temperature.The maximal value of pressure and temperature decreases when the bubble collapses more times.The high temperature and pressure field caused by the collapse is utilized in sonochemistry fields.

Numerical results
Experimental data [9]   t

Collapse of acoustic-driven near-wall bubble under the stand-off distances
When a near-wall bubble is subjected to an acoustic pressure wave, its shape evolution may be quietly different under the infinite liquid.Firstly, the bubble will expand under the low liquid pressure, collapse under the high liquid pressure, and continue to oscillate.Due to the wall, there is a non-spherical bubble shape.However, this evolution is quite different from the bubble near the wall.
Figure 5 shows acoustic-driven bubble evolution during the last collapse and rebound phase and pressure distribution on the wall.Under D = 1.2, the acoustic bubble shrinks uniformly in the circumferential direction, and when it collapses for the last time, no visible liquid jet is presented.The bubble touches the nearby wall at the expansion stage and oscillates nonviolently.After the oscillation, the bubble becomes a hemisphere shape, which is the reason that there is no obvious liquid jet.Besides, as shown in Figure 6, a shock wave formed during the final collapse is observed.When the bubble undergoes the rebound stage, it also grows uniformly in the circumferential direction.Under D = 3.0, the upper surface will sink in the bubble since it moves more quickly than the lower one, forming a liquid jet and causing a high-pressure region on the wall.Under D = 4.6, the bubble quickly contracts during the final collapse stage, and its upper surface also pierces the interior, which means the highspeed jet forms.Subsequently, the jet finally penetrates the bubble.However, the jet did not touch the wall.According to the dimensionless distance, we classify the evolution of the bubble into three types, i.e., the liquid jet not touching the wall (D > 4.4), the liquid jet touching the wall (1.4 < D < 4.4), and no obvious liquid jet (D < 1.4).The two former phenomena are similar to the bubble inertial collapse [12], but the stand-off threshold is different.

Variation of interior pressure and temperature over the stand-off distances
Figure 7 shows that the pressure and temperature peak value changes with the stand-off distance.If D < 2.0, the maximal value of pressure and temperature increases with the stand-off distance decreasing.However, if D > 2.0, there is no variation of these peak values with stand-off distance.

Discussions
The acoustic bubble dynamics are significantly affected by the wall, and the temperature and pressure induced by its collapse also vary with the stand-off distance.In practical engineering fields, i.e., medical treatments, surface-cleaning, and water treatment, the mechanism of the acoustic bubble is important.

Conclusions
A numerical model is developed to investigate the dynamics of an acoustic-driven near-wall bubble.After fulfillment of validation of the numerical model, the complex bubble dynamics under the different dimensionless distances are analyzed, and some conclusions are given: (1) The bubble's radius increases when the bubble is excited by a lower-pressure wave, and it decreases under the higher-pressure wave.When the bubble collapses, its pressure and temperature will reach the maximum value.The peak value of pressure and temperature decreases when the bubble collapses more times.
(2) According to the dimensionless distance, the bubble dynamics are classified into three types, i.e., the liquid jet not touching the wall (D > 4.4), the liquid jet touching the wall (1.4 < D < 4.4), and no obvious liquid jet (D < 1.4).This phenomenon is different from a near-wall bubble.
(3) If D < 2.0, the maximal value of pressure and temperature increases with the stand-off distance decreasing.However, if D > 2.0, there is no variation of these peak values with stand-off distance.

Figure 1 .
Figure 1.Computational zone and boundary conditions.

Figure 2 .
Figure 2. Comparison of the experimental data [9] and simulations.

Figure 3 .
Figure 3. Bubble shape covered by the velocity vector at different times (Frame size: 19R0 × 19R0).

Figure 7 .
Figure 7. Variation of pressure and temperature peak value with the stand-off distance.