Simulation of bubble motion in a compressible liquid based on the three dimensional wave equation

A simulation model for bubble motion in a compressible liquid is developed based on the linear wave equation. At the initial stage, the bubble is assumed to be spherical and the wave equation is simplified as a one dimensional ordinary equation in the radial direction through the prophase approximation. When time becomes much larger than that required for a disturbance to travel across the bubble at the speed of the sound, the obtained integral equation is approximated by keeping the term of the first order in terms of the characteristic Mach number, through the anaphase approximation. An equation is introduced to unify the approximations in these two phases, which is then used over the entire simulation period. The problem at each time step is solved by a three dimensional boundary element method. The convergence study has been first taken with meshes and time steps. Comparison is made with the analytical solution for spherical bubble in compressible liquid and good agreement is found. Further comparison is made for a bubble in an incoming acoustic wave. Extensive simulations are then made for a bubble in various conditions, including the cases with solid boundary effect, free surface effect, buoyancy effect, as well as for interactions between bubbles.     

Boundary integral equations as applied to an oscillating bubble near a fluid-fluid interface

A new method is presented to describe the behaviour of an oscillating bubble near a fluid-fluid interface. Such a situation can be found for example in underwater explosions (near muddy bottoms) or in bubbles generated near two (biological) fluids separated by a membrane. The Laplace equation is assumed to be valid in both fluids. The fluids can have different density ratios. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be used to update the co-ordinates of the new interface at the next time step. The boundary integral method is then used for both fluids. With the resulting equations the normal velocities on the interface and the bubble are obtained. Depending on initial distances of the bubble from the fluid-fluid interface and density ratios, the bubbles can develop jets towards or away from this interface. Gravity can be important for bubbles with larger dimensions.     

Application of the boundary integral method to the interaction of rising two-dimensional deformable gas bubbles

The boundary integral method is applied to model the initial motion of two-dimensional or cylindrical deformable gas bubbles in an inviscid, incompressible fluid. Following the success of recent boundary integral studies to predict the qualitative behaviour of a single gas bubble, this numerical study is extended to consider the interaction of several bubbles. Surface tension, relative initial position and volume are all found to be important factors affecting the bubble interaction, jet formation, trapping of fluid between bubbles and bubble shedding. As well as computing the evolution of the bubble surfaces, consideration of the pressure fields and resulting instantaneous streamlines is given.     

A modified Rayleigh–Plesset model for a non-spherically symmetric oscillating bubble with applications to boundary integral methods

In this article the analytical solution to the Rayleigh–Plesset equation for a spherically symmetric oscillating bubble is extended to apply to the much more general (non-spherically symmetric) bubble configuration. An equivalent bubble radius and an equivalent bubble wall velocity are introduced in order to do so. The influence of gravity, surface tension, nearby solid walls, vapor bubbles, bubbles filled with adiabatic or isothermal gas have been considered in the model. An interesting outcome is that the equivalent bubble wall velocity is no longer the time derivative of the equivalent bubble radius. This observation can possibly explain why in various numerical and experimental observations the oscillation time of a bubble changes when compared to that of a standalone bubble; near a solid surface it increases while it decreases when the bubble is placed near a free surface. The current developed theory can be further employed to ascertain the accuracy of a numerical scheme simulating bubble dynamics in an incompressible surrounding flow approximation. An often used numerical technique to simulate such bubble dynamics is the boundary integral method (BIM).     

Numerical solution of the Dirichlet initial boundary value problem for the heat equation in exterior 3-dimensional domains using integral equations

A numerical method for the Dirichlet initial boundary value problem for the heat equation in the exterior and unbounded region of a smooth closed simply connected 3-dimensional domain is proposed and investigated. This method is based on a combination of a Laguerre transformation with respect to the time variable and an integral equation approach in the spatial variables. Using the Laguerre transformation in time reduces the parabolic problem to a sequence of stationary elliptic problems which are solved by a boundary layer approach giving a sequence of boundary integral equations of the first kind to solve. Under the assumption that the boundary surface of the solution domain has a one-to-one mapping onto the unit sphere, these integral equations are transformed and rewritten over this sphere. The numerical discretisation and solution are obtained by a discrete projection method involving spherical harmonic functions. Numerical results are included.     

Transient bubble domain configuration in garnet materials observed using high speed photography

The dynamic domain configurations of bubbles in garnet materials have been studied using a sampling optical microscope capable of single exposure photographs with a 10 nsec exposure time. The microscope is an integral part of a sampling system so that the transient shape of the bubble is recorded at various times after a field pulse or, for bubbles in field access devices, during a clock cycle. A triggerable flowing nitrogen gas laser pumping a low Q Rodamine 6G Dye laser is used as an illumination source giving light pulses of ∼1.5 KW for 10 nsec. This light is sufficient to expose Kodak 4 × 16 mm movie film. Standard pulse generators (HP 214A) are used to make free bubble radial velocity measurements. A modified generator allows free bubble collapse measurement to be made. For bubbles propagating at operating frequency within field access devices, a standard bubble exerciser is used, synchronized to the sampling system. In this case, special samples with an internal mirror and epi-mode illumination are used. Illustrative results of bubble domain shapes in a chevron propagating structure and a 90° chevron expander detector are included.     

A numerical study of periodic slug flow at zero gravity and normal gravity

This paper reports numerical simulations of slug flow at zero and normal gravity. The particular experimental results chosen for validation were obtained at microgravity under conditions which resulted in evenly-spaced and evenly-sized Taylor bubbles facilitating a simulation with periodic boundary conditions. The numerical technique was a free-surface method which explicitly tracked the motion of the gas-liquid interface using a volume-of-fluid specification and a finite volume discretisation of the solution domain. The large scale features of the bubble such as the classic bullet-shaped nose were well predicted by the model. Unsteady features of the bubble shape such as waves in the film and fluctuations of the bottom surface were also predicted but are harder to compare quantitatively to the experiments. The velocity field predictions reveal several interesting features of the flow. When viewed by an observer moving with the bubbles, the liquid slug is dominated by a large recirculating region with the flow travelling from the leading to the trailing bubble along the tube centreline. In this frame of reference, the near-wall region features a jet of fluid issuing from the film of the leading bubble which entrains fluid in the slug. As the film of the trailing bubble begins to form, the entrained fluid must be ejected since the flowrate in the film of each bubble must be the same. It appears to be this process that drives the main recirculation.     

Disappearing bodies and ghost vortices

In many dispersed multiphase flows bubbles, droplets, and particles move and disappear due to a phase change. Practical examples include vapour bubbles condensing in subcooled liquids, fuel droplets evaporating in a hot gas and ice crystals melting in water. After these 'bodies' have disappeared, they leave behind a remnant 'ghost' vortex as an expression of momentum conservation. A general framework is developed to analyse why and how a ghost vortex is generated. A study of these processes is incomplete without a detailed discussion of the concept of momentum for unbounded flows. We show how momentum can be defined unambiguously for unbounded flows and show its connection with other expressions, particularly that of Lighthill. We apply our analysis to interpret new observations of condensing vapour bubbles and discuss droplet evaporation. We show that the use of integral invariants, widely applied in turbulence, introduces a new perspective to dispersed multiphase flows.     

High‐order artificial boundary conditions for ideal axisymmetric irrotational flow around 3D obstacles

In this paper the numerical solution for ideal irrotational incompressible flow around axisymmetric 3D obstacles is discussed with the artificial boundary method. By introducing an artificial boundary, we divide the exterior unbounded domain into a bounded part and an unbounded part. After setting up a proper artificial boundary condition, we get an approximate reduced problem defined on the bounded part. Both non‐local and local artificial boundary conditions are designed. Numerical experiment is also presented, and its result demonstrates the effectiveness of these artificial boundary conditions. Copyright © 2002 John Wiley & Sons, Ltd.     

A boundary condition in Padé series for frequency‐domain solution of wave propagation in unbounded domains

A boundary condition satisfying the radiation condition at infinity is frequently required in the numerical simulation of wave propagation in an unbounded domain. In a frequency domain analysis using finite elements, this boundary condition can be represented by the dynamic stiffness matrix of the unbounded domain defined on its boundary. A method for determining a Padé series of the dynamic stiffness matrix is proposed in this paper. This method starts from the scaled boundary finite‐element equation, which is a system of ordinary differential equations obtained by discretizing the boundary only. The coefficients of the Padé series are obtained directly from the ordinary differential equations, which are not actually solved for the dynamic stiffness matrix. The high rate of convergence of the Padé series with increasing order is demonstrated numerically. This technique is applicable to scalar waves and elastic vector waves propagating in anisotropic unbounded domains of irregular geometry. It can be combined seamlessly with standard finite elements. Copyright © 2006 John Wiley & Sons, Ltd.     

Gas bubbles generated by thermal aging in thin film bicrystals of silver I: Observations and quantitative analysis

Gas bubble generation in thin film bicrystals of silver under thermal and room temperature aging was studied by transmission electron microscopy. Bicrystals were prepared by superposing two epitaxial films obtained by evaporation in vacuum. After aging they were separated by a (001) twist grain boundary with an angle ranging from 0° to 45°. Gas bubbles only appeared in bicrystals in which welding had produced the boundary and they were absent from bicrystals that were badly welded and from single films which had undergone the same aging. A quantitative analysis of the gas bubbles showed that their density decreases if the aging temperature or the sample thickness are increased. A correlation with the mean size of the gas bubbles was also found, and a relation between the gas bubble density and the growth rate of the films was established.     

A boundary element study of the effect of surface dissolution on the evolution of immiscible viscous fingering within a Hele-Shaw cell

Fingering instabilities at the interface between two immiscible fluids can appear during the displacement of a fluid of higher viscosity by another one of lower viscosity. The evolution of the finger structures is determined by the interface kinematic and dynamic matching conditions, which describe mass and momentum conservation across the interface. In the case when the injected fluid is a gas and the resident one is a liquid, dissolution of the injected gas into the displaced liquid can occur at the interface between the two phases. In this case, the transfer velocity of the dissolved gas reduces the interface displacement velocity as described by the kinematic matching condition, delaying the evolution of the fingering. In addition, the momentum flux across the interface, due to the dissolution, modifies the dynamic matching condition with possible changes in the patterns of the fingers structures.This work studies the effects of gas dissolution on the evolution of fingering instabilities during the displacement of a viscous liquid by an immiscible injected gas in a Hele-Shaw cell. A boundary element numerical simulation of the growth of the injected gas bubble is developed and implemented. This numerical model takes into account the dissolution across a sharp interface between the two phases. Our numerical simulations suggest that the inclusion of gas dissolution can lead to the eventual breaking of the fingers. These “shed fingers” become individual bubbles, which move away from the injection source with the velocity of the surrounding fluid and eventually will dissolve into the ambient fluid. New fingers evolve, with their concurrent breaking, resulting in the possibility of a cascade of travelling and dissolving bubbles, instead of a continuous fingering structure.     

Condensation and Collapse of Vapor Bubbles Injected in Subcooled Pool

We focus on condensation and collapse processes of vapor bubble(s) in a subcooled pool. We generate the vapor in the vapor generator and inject it/them to form vapor bubble(s) at a designated temperature into the liquid at a designated degree of subcooling. In order to evaluate the effect of induced flow around the condensing/collapsing vapor bubble, two different boundary conditions are employed; that is, the vapor is injected through the orifice and the tube. We also focus on interaction between/among the condensing/collapsing vapor bubbles laterally injected to the pool. Through this system we try to simulate an interaction between the vapor bubble and the subcooled bulk in a complex boiling phenomenon, especially that known as MEB (microbubble emission boiling) in which a higher heat flux than critical heat flux (CHF) accompanying with emission of micrometer-scale bubbles from the heated surface against the gravity is realized under a rather high subcooled condition.     

Exact solutions for two steady inviscid bubbles in the slow viscous flow generated by a four-roller mill

The nonlinear free-boundary problem of finding the equilibrium shapes of two equal-sized two-dimensional inviscid bubbles with surface tension situated in a polynomially-singular slow viscous flow is solved in terms of closed-form formulae. The singular flow is taken to be within the class of those realizable at the centre of a four-roller mill apparatus. The associated flow field is also found explicitly. These solutions allow investigation of the bubble shapes and associated streamline patterns as functions of the far-field asymptotic conditions. In certain regimes, the bubbles are found to exhibit both near-cusps and a characteristic dimpling as they draw closer together. The results provide the first instances of exact solutions involving two interacting bubbles in an unbounded Stokes flow.     

Wavelet BEM for large‐scale Stokes flows based on the direct integral formulation

This paper describes a new wavelet boundary element method (WBEM) for large‐scale simulations of three‐dimensional Stokes problems. It is based on a Galerkin formulation and uses only one set of wavelet basis. A method for the efficient discretization and compression of the double‐layer integral operator of Stokes equation is proposed. In addition, a compression strategy for further reducing the setting‐up time for the sparse system matrix is also developed. With these new developments, the method has demonstrated a high matrix compression rate for problems with complicated geometries. Applications of the method are illustrated through several examples concerning the modeling of damping forces acting on MEMS resonators including a cantilever resonator oscillating in an unbounded air and a perforated plate resonator oscillating next to a fixed substrate. The numerical results clearly illustrate the efficiency and accuracy of the developed WBEM in these large‐scale Stokes flow simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

The Effect of Gas Properties on Bubble Formation,Growth, and Detachment

Bubble formation and growth play an important role in various processes and industries, where the dispersion of gas bubbles in a liquid medium occurs frequently. In this paper, the formation, growth, and detachment of gas bubbles produced from a submerged needle in water are numerically and experimentally investigated. The effect of injected gas properties on bubble characteristics, including bubble diameter, contact angle, and the frequency of bubble formation, is evaluated. In particular, the changes in bubble characteristics during the injection process are investigated for three different gases to evaluate the effect of density and surface tension on the bubble detachment criteria. The present numerical results show an acceptable agreement with experiments under different operating conditions. The results show that the increase in surface tension, and the decrease in gas density result in larger bubble sizes before detachment occurs. Moreover, the bubble generation frequency is found to strongly depend on the contact angle and the surface tension.     

Finite element analysis of unbounded and incompressible fluid domains

A finite element technique of analysing the hydrodynamic pressure, resulting from the vibration of a structure submerged in an unbounded fluid domain, is presented. A suitable boundary condition is proposed for the surface where the unbounded fluid domain is truncated. Pressure is assumed to be the nodal unknown and the fluid is treated as being incompressible and inviscid. The proposed method is computationally very efficient and its implementation in a finite element program is quite straightforward. The efficiency of the method is demonstrated by analysing some two-dimensional problems.     

超重力场下球形炸药水下爆炸实验及数值模拟

常重力条件下,Mach数和Froude数无法同时满足相似。采用离心模型实验,研究水下爆炸冲击波特性及气泡脉动规律,并采用数值模拟方法重现了超重力场下的水下爆炸过程。实验及数值模拟的结果表明,重力的改变基本不影响冲击波的峰值,COLE理论在超重力环境下仍然适用,而超重力场下,小当量炸药可以模拟气泡的脉动迁移过程,其结果可以用于预测大当量炸药深水爆炸的特性。此外,为减少计算时间并提高计算精度,采用将二维冲击波作为初始条件映射到三维模型的建模方法。对于冲击波计算,网格尺寸宜取药包半径的1/20¹/10;对于气泡脉动,宜取药包半径的1/21/10;对于气泡脉动,宜取药包半径的1/2¹。     

Potential of earthing grid in heterogeneous soil

Calculation of the scalar electric potential of an earthing grid in heterogeneous soil is obtained by the finite element technique combining differential and integral formulations of the problem. Such a combined formulation made it possible to generate easily the mesh of finite elements and to divide the problem domain into a sufficiently small number of large finite elements. The unbounded domain of calculation is solved by using mapped infinite elements. The contribution of singular sources is represented by the analytical expression, and included in the approximation function of the potential. A sufficiently satisfactory approximation is realized with a small number of singular source images, obtained by local imaging of sources within a finite element on its boundary surfaces. The calculation accuracy is tested on examples solvable by an average potential method which is based on an integral formulation of the problem.