Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

Flow field at collapse of a cavity

Employing Rayleigh’s method, the collapse of a vaporous bubble in an incompressible liquid with surface tension is analysed. The expressions of time versus radius, bubble-wall velocity and pressure developed at collapse are thus introduced.Finally, the numerical solution of velocity and pressure field in the liquid surrounding the cavity is also given.     

Vortex ring modelling of toroidal bubbles

During the collapse of a bubble near a surface, a high-speed liquid jet often forms and subsequently impacts upon the opposite bubble surface. The jet impact transforms the originally singly-connected bubble to a toroidal bubble, and generates circulation in the flow around it. A toroidal bubble simulation is presented by introducing a vortex ring seeded inside the bubble torus to account for the circulation. The velocity potential is then decomposed into the potential of the vortex ring and a remnant potential. Because the remnant potential is continuous and satisfies the Laplace equation, it can be modelled by the boundary-integral method, and this circumvents an explicit domain cut and associated numerical treatment. The method is applied to study the collapse of gas bubbles in the vicinity of a rigid wall. Good agreement is found with the results of Best (J. Fluid Mech. 251 79–107, 1993), obtained by a domain cut method. Examination of the pressure impulse on the wall during jet impact indicates that the high-speed liquid jet has a significant potential for causing damage to a surface. There appears to be an optimal initial distance where the liquid jet is most damaging.     

Heat and mass transfer of an individual vapor bubble in translational flow of an unbounded liquid volume

The condensational collapse of a spherical vapor bubble moving translationally through an unbounded incompressible liquid is investigated. The bubble moves at a varying velocity. The problem is solved within the framework of the axisymmetric formulation. The numerical investigation shows that the variability of the bubble rise velocity significantly affects the condensational collapse process. This effect is particularly prominent in the final stage, i.e., in the stage of thermal collapse. The results obtained agree well with the experimental data.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 109–115, May–June, 1994.     

nfluence of exciting parameters on the vibration of single cavitation bubble driven by intensive sound

根据考虑了液体可压缩性的改进的微气泡动力学方程,采用改进的初始半径对单泡超声空化现象进行了数值计算研究. 结果表明,微气泡振动对一些参量很敏感：微气泡振动半径与初始半径的比值随振动频率的增大而减小;提高声场声压会加剧气泡崩塌程度,但过高的声压又不能使微气泡崩塌;微气泡崩塌速率随气泡初始半径的增加而增大,在一定范围内能保证空化泡稳定振动,在初始半径为1.6\,$\mu$m 处空化程度最强,如果继续增大初始半径则空化程度减弱、甚至消失;微气泡崩塌程度随黏滞系数和表面张力的增大而减弱,过大的黏滞系数和表面张力会使微气泡崩塌难以发生. 计算结果与他人的实验数据相比,发现液体的可压缩性使单泡空化强度增强, 对最佳空化区域范围的确定有较大的影响.     

A numerical strategy for the direct 3D simulation of the expansion of bubbles into a molten polymer during a foaming process

In the framework of the foam process modelling, this paper presents a numerical strategy for the direct 3D simulation of the expansion of gas bubbles into a molten polymer. This expansion is due to a gas overpressure. The polymer is assumed to be incompressible and to behave as a pseudo‐plastic fluid. Each bubble is governed by a simple ideal gas law. The velocity and the pressure fields, defined in the liquid by a Stokes system, are subsequently extended to each bubble in a way of not perturbing the interface velocity. Hence, a global velocity–pressure‐mixed system is solved over the whole computational domain, thanks to a discretization based on an unstructured first‐order finite element. Since dealing with an Eulerian approach, an interface capturing method is used to follow the bubble evolution. For each bubble, a pure advection equation is solved by using a space–time discontinuous‐Galerkin method, coupled with an r‐adaptation technique. Finally, the numerical strategy is achieved by considering a global mesh expansion motion, which conserves the amount of liquid into the computational domain during the expansion. The expansion of one bubble is firstly considered, and the simulations are compared with an analytical model. The formation of a cellular structure is then investigated by considering the expansion of 64 bubbles in 2D and the expansion of 400 bubbles in 3D. Copyright © 2007 John Wiley & Sons, Ltd.     

Experimental and numerical investigation a bubble-driven laminar flow in a axisymmetric vessel

Measurements have been obtained, by laser-Doppler anemometry (LDA), of the axisymetric, recirculating liquid flow caused by a column of air bubbles ($\text{5–6}\text{1}\text{2}\text{mm dia.}$) rising through caster oil in a cylindrical enclosure (100 mm dia.). The liquid velocities correspond to creeping flow. Axial and radial liquid velocity profiles are reported at eight axial stations and, close to within the bubble column, as a function of time. The maximum liquid velocity found outside the bubble column is about 0.5 of that of the bubbles and a very rapid radical decay from this value is noted. The temporal variation of the velocity field, due to the passage of the air bubbles, is undetectable at radial locations greater than about $\text{1}\text{1}\text{2}$ bubble radii from the centreline.The variation of bubble velocity with axial distance was aise measured by LDA for liquid height to enclosure diámeter ratios of 0.98 and 2.78. The maximum bubble velocities were about 0.1–0.2 higher than the Strokes law terminal velocity. The increase is due to the convection of the bubble column by the liquid flow. The maximum bubble velocity is established within approximately three bubble diameters of the air inlet.The motion of the liquid has been calculated by the numerical solution of the steady form of the equations of motion, with the inner boundary of the area of integration lying 1.3 bubble radii from the centerline. The boundary conditions at this surface are assumed to be steady and are taken from measurements of the time-averaged velocity components. The assumption of steady flow at this boundary is supported by experimental observation and results in calculations which are generally in close agreement with the measurements. Discrepancies are confined to the immediate vicinity of the bubble column near to the top and bottom of the enclosure. These are ascribed to a combination of small asymmetries in the experiment and inadequate numerical resolution in these regions.     

On the modeling and simulation of a laser‐induced cavitation bubble

Single cavitation bubbles exhibit severe modeling and simulation difficulties. This is due to the small scales of time and space as well as due to the involvement of different phenomena in the dynamics of the bubble. For example, the compressibility, phase transition, and the existence of a noncondensable gas inside the bubble have strong effects on the dynamics of the bubble. Moreover, the collapse of the bubble involves the occurrence of critical conditions for the pressure and temperature. This adds extra difficulties to the choice of equations of state. Even though several models and simulations have been used to study the dynamics of the cavitation bubbles, many details are still not clearly accounted for. Here, we present a numerical investigation for the collapse and rebound of a laser‐induced cavitation bubble in liquid water. The compressibility of the liquid and vapor are involved. In addition, great focus is devoted to study the effects of phase transition and the existence of a noncondensable gas on the dynamics of the collapsing bubble. If the bubble contains vapor only, we use the six‐equation model for two‐phase flows that was modified in our previous work [A. Zein, M. Hantke, and G. Warnecke, J. Comput. Phys., 229(8):2964‐2998, 2010]. This model is an extension to the six‐equation model with a single velocity of Kapila et al. (Phys. Fluid, 13:3002‐3024, 2001) taking into account the heat and mass transfer. To study the effect of a noncondensable gas inside the bubble, we add a third phase to the original model. In this case, the phase transition is considered only at interfaces that separate the liquid and its vapor. The stiffened gas equations of state are used as closure relations. We use our own method to determine the parameters to obtain reasonable equations of state for a wide range of temperatures and make them suitable for the phase transition effects. We compare our results with experimental ones. Also our results confirm some expected physical phenomena. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical study of weak shock wave propagation in a reactive bubbly liquid

A numerical study is presented on the response of a weakly shock compressed liquid column that contains reactive gas bubbles. Both the liquid and gas are considered compressible. Compressibility of the liquid allows calculation of shock and rarefaction waves in the pure liquid as well as in the gas/liquid mixture. A microscopic model for local bubble collapse is coupled with a macroscopic model of wave propagation through the gas/liquid mixture. In the particular cases presented here, the characteristic times of propagation of the shock wave and bubble collapse are of the same order of magnitude. Consequently, the coupling between various phenomena is very strong. The present model based on fundamental principles of two-phase fluid mechanics takes into account the coupling of localized bubble oscillations. This model is composed of a microscopic one in the scale of a bubble size, and a macroscopic one which is based on the mixture theory. The liquid under study is water, and the gas is a reactive mixture of argon, hydrogen and oxygen. Received 18 December 1995 / Accepted 2 June 1996     

Bubble collapse in compressible fluids using a spectral element marker particle method. Part 1. Newtonian fluids

This paper is concerned with the development of a high‐order numerical scheme for the modelling of two‐phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two‐phase viscoelastic flows. The particular problem of the collapse of a two‐dimensional bubble in the vicinity of a rigid boundary is considered. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. The marker particle scheme is validated using the Zalesak slotted disk rotation test problem. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscous effects are shown to inhibit bubble collapse and prevent jet formation and are therefore likely to have a mitigating effect on cavitation damage.Copyright © 2011 John Wiley & Sons, Ltd.     

A novel numerical scheme for the investigation of surface tension effects on growth and collapse stages of cavitation bubbles

In the present study the effects of surface tension on the growth and collapse stages of cavitation bubbles are studied individually for both spherical and nonspherical bubbles. The Gilmore equation is used to simulate the spherical bubble dynamics by considering mass diffusion and heat transfer. For the collapse stage near a rigid boundary, the Navier–Stokes and energy equations are used to simulate the flow domain, and the VOF method is adopted to track the interface between the gas and the liquid phases. Simulations are divided into two cases. In the first case, the collapse stage alone is considered in both spherical and nonspherical situations with different conditions of bubble radius and surface tension. According to the results, surface tension has no significant effects on the flow pattern and collapse rate. In the second case, both the growth and collapse stages of bubbles with different initial radii and surface tensions are considered. In this case surface tension affects the growth stage considerably and, as a result, the jet velocity and collapse time decrease with increasing surface tension coefficient. This effect is more significant for bubbles with smaller radii.     

INVESTIGATION OF BUBBLE-BUBBLE INTERACTION EFFECT DURING THE COLLAPSE OF MULTI-BUBBLE SYSTEM

通过直接数值模拟方法对多泡在压力驱动下的溃灭过程进行了研究。气相满足理想气体正压模型,液相为不可压 流体,采用基于压力的方法求解多泡的两相流场。数值研究表明,在多泡流场中,中心气泡的溃灭过程明显不同于单泡,存在总体溃灭延迟现象和后期加速现象。随着周围气泡数的增多或气泡间距的减小,中心气泡的溃灭时间增长,溃灭时的压力峰值增大。结合理论定性分析发现,气泡运动不仅受远场压力的驱动,还受周围气泡诱导压力场的影响。周围气泡的诱导压力经历先减小后增大的过程,从而使受其影响的中心气泡产生先延迟后加速的特征。     

Numerical simulation of a single bubble by compressible two‐phase fluids

The present work deals with the numerical investigation of a collapsing bubble in a liquid–gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non‐conservative transport equations of the pressure law coefficients. The original first‐order discretization is extended to higher order applying second‐order ENO reconstruction to the primitive variables. The derivation of the non‐conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale‐based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2‐D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser‐induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd.     

A calculation of the parameters of the high-speed jet formed in the collapse of a bubble

As is known, the collapse of vapor bubbles in a liquid can cause the intensive destruction of solid boundary surfaces. Experimental and theoretical investigations of bubble collapse have led to the conclusion that the surface of a bubble can deform and a liquid jet directed toward the solid surface can form in the process [1, 2]. In the theoretical reports [3, 4] too low jet velocities were obtained, inadequate to explain the destruction of the surface in a single impact. In [5] it was found as a result of numerical calculations that the formation of jets possessing enormous velocities is possible. It was also found that two fundamentally different schemes of jet formation are possible in the collapse of a bubble near a wall. The transition from one scheme to the other occurs upon a relatively small change in the initial shape of the bubble. In the present report we investigate the case of sufficiently small initial deformations of a bubble when the region occupied by the bubble remains simply connected during the formation of the jet; i.e., the separation of a small bubble from the bubble does not occur. In the case of the second scheme of bubble collapse near a wall the connectedness of the free boundary is disrupted and a small bubble separates off during the formation of the jet.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 94–99, May–June, 1979.     

两空泡运动特性研究

本文应用边界方法研究了两个相邻空泡的运动特性,得到了空泡的演化规律,以及空泡溃灭时的射流速度与溃灭时间的变化趋势,对于两个空泡之间的距离和半径比的影响进行了讨论。计算结果表明：不同大小的空泡在一起时则小泡会先溃灭,且人泡的存在时间与两泡的半径比成正比;大泡对小泡来说其作用相当于－固壁面,小泡会形成－指向大泡的溃灭射流。相同大小的空泡在一起溃灭时,会同时形成指向中间的射 流,与单空泡在固壁面附近的溃     

Single cavitation bubble generation and observation of the bubble collapse flow induced by a pressure wave

This study utilizes a U-shape platform device to generate a single cavitation bubble for a detailed analysis of the flow field characteristics and the cause of the counter jet during the process of bubble collapse caused by sending a pressure wave. A high speed camera is used to record the flow field of the bubble collapse at different distances from a solid boundary. It is found that a Kelvin–Helmholtz vortex is formed when a liquid jet penetrates the bubble surface after the bubble is compressed and deformed. If the bubble center to the solid boundary is within one to three times the bubble’s radius, a stagnation ring will form on the boundary when impinged by the liquid jet. The fluid inside the stagnation ring will be squeezed toward the center of the ring to form a counter jet after the bubble collapses. At the critical position, where the bubble center from the solid boundary is about three times the bubble’s radius, the bubble collapse flow will vary. Depending on the strengths of the pressure waves applied, the collapse can produce a Kelvin–Helmholtz vortex, the Richtmyer–Meshkov instability, or the generation of a counter jet flow. If the bubble surface is in contact with the solid boundary, the liquid jet can only move inside-out without producing the stagnation ring and the counter jet; thus, the bubble collapses along the radial direction. The complex phenomenon of cavitation bubble collapse flows is clearly manifested in this study.     

Investigation on the collapse behavior of a cavitation bubble near a conical rigid boundary

The collapse stage of cavitation bubble development near a conical rigid boundary is investigated in detail by a finite-volume method and the volume of fluid method. The obtained results reveal the effect of the angle of the conical boundary on the bubble shape and the collapse time, as well as liquid jet formation. The degree of departure of the bubble shape from spherical one and the collapse time are found to increase with the increase of cone angle. The relationship between the prolongation factor of the collapse time near a conical boundary and the cone angle is proposed, and theoretical values of the collapse time are calculated. Good agreement is found between the theoretical values and the values obtained from simulations using a finite-volume method.     

Nonconforming local projection stabilization for generalized Oseen equations

A new method of nonconforming local projection stabilization for the gen- eralized Oseen equations is proposed by a nonconforming inf-sup stable element pair for approximating the velocity and the pressure. The method has several attractive features. It adds a local projection term only on the sub-scale （H ≥ h）. The stabilized term is simple compared with the residual-free bubble element method. The method can handle the influence of strong convection. The numerical results agree with the theoretical expectations very well.     

Noise at inception and collapse of a cavity

The paper analyzes the noise at inception and collapse of an isolated bubble cavity filled with gas and vapour. The expressions and their numerical solutions of the sound pressure and the vibration velocity are presented. The results indicate that the noise occurs at every stage of a cavity. The noise has comparatively big value only at the late period of collapse. The sound pressure is of magnitude 100db.     

Numerical analysis of gas bubbles in close proximity to a movable or deformable body

The growth and collapse of gaseous bubbles near a movable or deformable body are investigated numerically using the boundary element method and fluid–solid coupling technique. The fluid is treated as inviscid, incompressible and the flow irrotational. The unsteady Bernoulli equation is applied on the bubble surface as one of the boundary conditions of the Laplace’s equation for the potential. Good agreements between the numerical and experimental results demonstrate the robustness and accuracy of the present method. The translation and rotation of the rigid body due to the bubble evolution are captured by solving the six-degrees-of-freedom equations of motion for the rigid body. The fluid–solid coupling is achieved by matching the normal component of the velocity and the pressure at the fluid–solid interface. Compared to a fixed rigid body, the expansion of the bubble is not affected too much but much faster collapsing velocities during the collapsing phase of bubble can be observed when considering the motion of the rigid body. The rigid body is pushed away as the bubble grows and moved toward the bubble as the bubble collapses. The motion of two bubbles near a movable cylinder is also simulated. The large rotation of the cylinder and obvious deformation and distortion for the bubble in close proximity to a curved wall are observed in our codes. Finally, the growth and collapse of bubble near a deformable ellipsoid shell are also simulated using the combination of boundary element method (BEM) and finite element method (FEM) techniques. The oscillations of the ellipsoid shell can be observed during the growth and collapse of bubble, which much differs from the results obtained by only considering effects of a rigidly movable body on the bubble evolution.