Model flames in the Boussinesq limit: Rising bubbles

Using the Boussinesq buoyancy approximation, we study a bubble of reaction products rising in the reactant fluid under the influence of gravity. Reaction on the surface of the bubble (the flame) results in an increase of the volume of the bubble. We consider fluids with low Prandtl and high Froude numbers (heat diffusion dominates over viscous dissipation, and burning dominates over gravitational effects). We show that, under these conditions, all initially small bubbles follow the same growth pattern, regardless of the flame speed, the reaction type, the gravity, the viscosity, the initial size, and, to some extent, the initial shape of the bubble. In the initial stage of this similarity solution a bubble grows radially in an essentially motionless fluid until it reaches some critical size, which is determined by the laminar flame speed, the gravitational acceleration, and the Atwood number. Once the bubble reaches the critical size, convection becomes significant and the bubble evolves into a more complicated, mushroom-like shape. The similarity solution is expressed using the critical bubble size for the unit length and the critical size divided by the laminar flame speed as the unit time.     

Arrested Bubble Rise in a Narrow Tube

If a long air bubble is placed inside a vertical tube closed at the top it can rise by displacing the fluid above it. However, Bretherton found that if the tube radius, R, is smaller than a critical value \(R_{c}=0.918 \; \ell _c\), where \(\ell _c=\sqrt{\gamma /\rho g}\) is the capillary length, there is no solution corresponding to steady rise. Experimentally, the bubble rise appears to have stopped altogether. Here we explain this observation by studying the unsteady bubble motion for \(R<R_{c}\). We find that the minimum spacing between the bubble and the tube goes to zero in limit of large t like \(t^{-4/5}\), leading to a rapid slow-down of the bubble’s mean speed \(U \propto t^{-2}\). As a result, the total bubble rise in infinite time remains very small, giving the appearance of arrested motion.     

Numerical investigation of the deformation mechanism of a bubble or a drop rising or falling in another fluid

A numerical method for simulating the motion and deformation of an axisymmetric bubble or drop rising or falling in another infinite and initially stationary fluid is developed based on the volume of fluid （VOF） method in the frame of two incompressible and immiscible viscous fluids under the action of gravity, taking into consideration of surface tension effects. A comparison of the numerical results by this method with those by other works indicates the validity of the method. In the frame of inviseid and incompressible fluids without taking into consideration of surface tension effects, the mechanisms of the generation of the liquid jet and the transition from spherical shape to toroidal shape during the bubble or drop deformation, the increase of the ring diameter of the toroidal bubble or drop and the decrease of its cross-section area during its motion, and the effects of the density ratio of the two fluids on the deformation of the bubble or drop are analysed both theoretically and numerically.     

Numerical simulation of bubble dynamics in a Phan-Thien–Tanner liquid: Non-linear shape and size oscillatory response under periodic pressure

We study non-linear bubble oscillations driven by an acoustic pressure with the bubble being immersed in a viscoelastic, Phan-Thien–Tanner liquid. Solution is provided numerically through a method which is based on a finite element discretization of the Navier–Stokes flow equations. The proposed computational approach does not rely on the solution of the simplified Rayleigh–Plesset equation, is not limited in studying only spherically symmetric bubbles and provides coupled solutions for the velocity, stress fields and bubble interface. We present solutions for non-spherical bubbles, with asphericity being addressed by means of Legendre polynomials or associated Legendre functions. A parametric investigation of the bubble dynamical oscillatory response as a function of the fluid rheological properties shows that the amplitude of bubble oscillations drastically increases as liquid elasticity (quantified by the Deborah number) increases or as liquid viscosity decreases (quantified by the Reynolds number). Extensive numerical calculations demonstrate that increasing elasticity and/or viscosity of the surrounding liquid tend to stabilize the shape anisotropy of an initially non-spherical bubble. Results are shown for pressure amplitudes 0.2–2 MPa and Deborah, Reynolds numbers in the intervals of 1–8 and 0.094–1.256, respectively.     

上浮气泡在壁面处的弹跳特性研究

本文针对毫米量级的上浮气泡在壁面处的弹跳现象进行数值研究.基于势流方法求解气泡的运动,同时考虑气泡的表面张力作用.在伯努利方程中,对气泡与壁面之间水膜中因黏性引起的压力梯度进行修正,开发相应的计算程序,计算值与实验值符合良好.从气泡弹跳的基本现象入手,研究了特征参数对气泡弹跳过程的动态特性以及最终平衡形态的影响.发现随着泡在撞击壁面之前上浮距离增大,气泡回弹距离和弹跳周期增加,但是当上浮距离增加到一定程度后将不会影响气泡的弹跳特性;表面张力是影响气泡弹跳特性的重要因素,气泡的弹跳周期随其增大逐渐减小,但回弹距离却呈现先增后减的规律;最后,影响气泡最终平衡形态的主要因素是气泡的浮力参数与韦伯数.     

黏性液体中单个气泡上升的形状特性

采用基于Level Set方法的直接数值模拟技术对黏性液体中单个气泡的上升运动进行三维模拟.数值模拟采用拟单相流模型处理气泡内外的气液两相流动,应用Level Set方法捕捉运动气泡的变形.针对Eo数从O(0)～O(2),Mo数从O(-11)～O(2)的流动范围,重点研究了上升气泡的形状特性,并与经典的气泡形状图谱进行了比较.模拟结果表明,上升气泡的形状与无量纲参数(Eo、Mo和Re)密切相关.在高Re的扁椭球区域,数值发现了气泡形状的周期性振荡行为.     

The impact of methanol mass transport on its conversion for the production of hydrogen and oxygenated reactive species in sono-irradiated aqueous solution

This study aims principally to assess numerically the impact of methanol mass transport (i.e., evaporation/condensation across the acoustic bubble wall) on the thermodynamics and chemical effects (methanol conversion, hydrogen and oxygenated reactive species production) of acoustic cavitation in sono-irradiated aqueous solution. This effect was revealed at various ultrasound frequencies (from 213 to 1000 kHz) and acoustic intensities (1 and 2 W/cm²) over a range of methanol concentrations (from 0 to 100%, v/v). It was found that the impact of methanol concentration on the expansion and compression ratios, bubble temperature, CH₃OH conversion and the molar productions inside the bubble is frequency dependent (either with or without consideration of methanol mass transport), where this effect is more pronounced when the ultrasound frequency is decreased. Alternatively, the decrease in acoustic intensity decreases clearly the effect of methanol mass transport on the bubble sono-activity. When methanol mass transfer is eliminated, the decrease of the bubble temperature, CH₃OH conversion and the molar yield of the bubble with the rise of methanol concentration was found to be more amortized as the wave frequency is reduced from 1 MHz to 213 kHz, compared to the case when the mass transport of methanol is taken into account. Our findings indicate clearly the importance of incorporating the evaporation and condensation mechanisms of methanol throughout the numerical simulations of a single bubble dynamics and chemical activity.     

I. Formation and rise of a bubble stream in a viscous liquid

The continuous emission of gas bubbles from a single ejection orifice immersed in a viscous fluid is considered. We first present a semi empirical model of spherical bubble growth under constant flow conditions to predict the bubble volume at the detachment stage. In a second part, we propose a physical model to describe the rise velocity of in-line interacting bubbles and we derive an expression for the net viscous force acting on the surrounding fluid. Experimental results for air/water-glycerol systems are presented for a wide range of fluid viscosity and compared with theoretical predictions. An imagery technique was used to determine the bubble size and rise velocity. The effects of fluid viscosity, gas flow rate, orifice diameter and liquid depth on the bubble stream dynamic were analyzed. We have further studied the effect of large scale recirculation flow and the influence of a neighbouring bubble stream on the bubble growth and rising velocity. Received: 23 July 1997 / Revised: 16 December 1997 / Accepted: 11 May 1998     

Influence of orifice spacing on twin bubbles formation in shear-thinning fluids: Laser image measurement

A laser image system for investigating the influence of orifice spacing on twin bubbles formation in shear-thinning fluid was established. The bubbles formation process at two orifices could be directly visualized and real-time recorded through computer by means of He–Ne laser as light source using the beam expanding and light amplification technology. The shape and size of bubbles generating in carboxymethylcellulose (CMC) aqueous solutions were studied experimentally at the orifices spacing 1D_o, 2D_o and 3D_o (D_o orifice diameter). The results reveal that the minute bubbles can be imaged clearly and amplified without distortion in the higher concentration solutions, and therefore the shape and size of bubbles are obtained accurately. With the increase of orifice spacing, bubble instantaneous volume in shear-thinning fluid decreases at the radial expansion stage but then increases at vertical elongation period, and the deformation and deviation of bubbles forming goes down due to the reduced interaction of adjacent bubble.     

A combined resonance-Doppler technique for studying bubble evolution in liquids

Gas bubbles in liquids have been studied for decades with a variety of optical and acoustic techniques. The evolution of a bubble consists of several stages, including formation and growth at a nozzle, detachment and resonance, and rise towards terminal velocity. Most existing techniques can monitor only a single aspect of the bubble behaviour. This work describes an acoustic technique to monitor all stages of an air bubble's evolution. The technique uses a combination of passive acoustic listening and active ultrasonic Doppler observation to study millimetre-sized air bubbles in liquid. A hollow cylindrical piezoelectric transducer, located around the nozzle used to produce the bubbles, detects the resonance of the bubble following its detachment. An ultrasonic Doppler system, positioned several centimetres above the nozzle, monitors both the growth and the rise of the bubble, including shape oscillations and the terminal velocity through the use of joint time-frequency analysis. Because all aspects of the bubble evolution are affected by the properties of the liquid, by monitoring the bubble evolution with this technique the rising bubble can potentially be used as a tool to characterize the liquid.     

How do dissolved gases affect the sonochemical process of hydrogen production? An overview of thermodynamic and mechanistic effects – On the “hot spot theory”

Although most of researchers agree on the elementary reactions behind the sonolytic formation of molecular hydrogen (H₂) from water, namely the radical attack of H₂O and H₂O₂ and the free radicals recombination, several recent papers ignore the intervention of the dissolved gas molecules in the kinetic pathways of free radicals, and hence may wrongly assess the effect of dissolved gases on the sonochemical production of hydrogen. One may fairly ask to which extent is it acceptable to ignore the role of the dissolved gas and its eventual decomposition inside the acoustic cavitation bubble? The present opinion paper discusses numerically the ways in which the nature of dissolved gas, i.e., N₂, O₂, Ar and air, may influence the kinetics of sonochemical hydrogen formation. The model evaluates the extent of direct physical effects, i.e., dynamics of bubble oscillation and collapse events if any, against indirect chemical effects, i.e., the chemical reactions of free radicals formation and consequently hydrogen emergence, it demonstrates the improvement in the sonochemical hydrogen production under argon and sheds light on several misinterpretations reported in earlier works, due to wrong assumptions mainly related to initial conditions. The paper also highlights the role of dissolved gases in the nature of created cavitation and hence the eventual bubble population phenomena that may prevent the achievement of the sonochemical activity. This is particularly demonstrated experimentally using a 20 kHz Sinaptec transducer and a Photron SA 5 high speed camera, in the case of CO₂-saturated water where degassing bubbles are formed instead of transient cavitation.     

Study on the spatial distribution of the liquid temperature near a cavitation bubble wall

A simple new model of the spatial distribution of the liquid temperature near a cavitation bubble wall (T_li) is employed to numerically calculate T_li. The result shows that T_li is almost same with the ambient liquid temperature (T₀) during the bubble oscillations except at strong collapse. At strong collapse, T_li can increase to about 1510 K, the same order of magnitude with that of the maximum temperature inside the bubble, which means that the chemical reactions occur not only in gas-phase inside the collapsing bubble but also in liquid-phase just outside the collapsing bubble. Four factors (ultrasonic vibration amplitude, ultrasonic frequency, the surface tension and the viscosity) are considered to study their effects for the thin liquid layer. The results show that for the thin layer, the thickness and the temperature increase as the ultrasonic vibration amplitude rise; conversely, the thickness and the temperature decrease with the increase of the ultrasonic frequency, the surface tension or the viscosity.     

测量单泡声致发光中气泡R(t)曲线的前向Mie散射技术

光干涉原理和Mie理论计算结果表明,单泡声致发光中气泡前向Mie散射的振荡信号,主要是由于气泡的透射光束和表面反射光束之间光干涉产生的. 这些干涉波峰形成了测量气泡半径的空间标尺,标尺的单位长度δR由散射角θ,检测光波波长和流体光折射率确定,而每个波峰就是标尺的刻线,它们与该时刻的气泡半径大小一一对应. 在30°—50°散射角范围内,利用前向Mie散射实验测定了单泡声致发光中气泡的最大半径,R(t)曲线及平衡半径,表明前向Mie散射是一种便捷的测定气泡运动特性的有效方法. 关键词： 单泡声致发光 前向Mie散射 光干涉     

Evolution of the shape of a bubble deformed at the zero time in a viscous fluid

The boundary-value problem of the shape of a charged bubble in a viscous incompressible insulating fluid as a function of time is solved in an approximation linear in the initial deformation amplitude. Both centrosymmetric pulsations and vibrations at constant volume are considered. It is shown that the time evolution of the shape of the bubble, as well as of the velocity and pressure fields of the fluid in its neighborhood, can be represented by finite sums over the numbers of initially excited modes, which involve two terms. The first term is a sum over the roots of the dispersion relation; the second, an improper integral. In the low-and high-viscosity limits, the relevant analytical expressions simplify, i.e., become free of integrals. It turns out that the damping constant of bubble surface vibrations varies with viscosity nonmonotonically in the case of radial pulsations and this dependence is different in the different limits. The frequencies of radial pulsations and surface vibrations vary with viscosity monotonically.     

声波在含气泡液体中的线性传播

为了探讨含气泡液体对声波传播的影响, 研究了声波在含气泡液体中的线性传播. 在建立含气泡液体的声学模型时引入气泡含量的影响,建立气泡模型时引用 Keller的气泡振动模型并同时考虑气泡间的声相互作用,得到了经过修正的气泡振动方程. 通过对含气泡液体的声传播方程和气泡振动方程联立并线性化求解,在满足 (ω R₀)/c << 1 的前提下,得到了描述含气泡液体对声波传播的衰减系数和传播速度. 通过数值分析发现,在驱动声场频率一定的情况下,气泡含量的增加及气泡的变小均会导致衰减系数增加和声速减小;气泡的体积分数和大小一定时, 驱动声场频率在远小于气泡谐振频率的情况下,声速会随驱动频率的增加而减小; 气泡间的声相互作用对声波传播速度及含气泡液体衰减系数的影响不明显.最终认为气泡的大小、 数量和驱动声场频率是影响声波在含气泡液体中线性传播的主要因素. 关键词： 含气泡液体 线性声波 声衰减系数 声速     

多量子位Heisenberg模型中纠缠的时间演化特性

研究了多量子位Heisenberg模型中纠缠的时间演化特性, 并给出了平均纠缠度〈C〉和多体纠缠度Q的解析表达式. 结果发现无论是对〈C〉还是对Q随着时间t的不断增长, 它们均先线性的增大, 而后达到一近似稳定状态, 并绕一平衡值做无规则的上下震荡. 若进一步考察N〈C〉则还可以发现, 纠缠上下震荡的平衡值与Heisenberg链的长度几乎无关, 而仅由它们的次近邻耦合常数J决定.     

波浪破碎气体的卷入过程及相关统计量的估计

基于实验观测,导出了波浪破碎能量耗散ε_ed、气泡云卷入深度z_b、气体卷入速率Q(z)和湍流动能耗散率ε_T(z)的表达式,在此基础上建立了一种简单、实用的气泡粒径谱参数化模式N(a,z),揭示了波浪破碎气泡云卷入过程能量耗散、气泡破碎临界Hinze特征尺度和气泡粒径谱在不同海况下的变化. 研究表明：气泡云卷入过程能量耗 关键词： 波浪破碎能量耗散 气泡云卷入深度 气泡粒径谱     

Vortex dynamics of a trapezoidal bluff body placed inside a circular pipe

The influence of Reynolds number and blockage ratio on the vortex dynamics of a trapezoidal bluff body placed inside a circular pipe is studied experimentally and numerically. Low aspect ratio, high blockage ratio, curved end conditions (junction of pipe and bluff body), axisymmetric upstream flow with shear and turbulence are some of the intrinsic features of this class of bluff body flows which have been scarcely addressed in the literature. A large range (200:200,000) of Reynolds number (Re_D) is covered in this study, encompassing all the three pipe flow regimes (laminar, transition and turbulent). Four different flow regimes are defined based on the distinct features of Strouhal number (St)–Re_D relation: steady, laminar irregular, transition and turbulent. The wake in the steady regime is stationary with no oscillations in the shear layer. The laminar regime is termed as irregular owing to irregular vortex shedding. The vortex shedding in this regime is observed to be symmetric. The emergence of separation bubble downstream of the bluff body on either side is another interesting feature of this regime, which is further observed to be symmetric. Two pairs of mean streamwise vortices are noticed in the near-wake regime, which are termed as reverse dipole-type wake topology. Beyond the irregular laminar regime, the Strouhal number falls gradually and vortex shedding becomes more periodic. This regime is named transition and occurs close to the Reynolds number at which transition to turbulence takes place in a fully developed pipe. The turbulent regime is characterised by a nearly constant Strouhal number. Typical Karman-type vortex shedding is noticed in this regime. The convection velocity, wake width formation length and irrecoverable pressure loss are quantified to highlight the influence of blockage ratio. These results will be useful to develop basic understanding of vortex dynamics of confined bluff body flow for several practical applications.     

Two-dimensional Euler flows in slowly deforming domains

We consider the evolution of an incompressible two-dimensional perfect fluid as the boundary of its domain is deformed in a prescribed fashion. The flow is taken to be initially steady, and the boundary deformation is assumed to be slow compared to the fluid motion. The Eulerian flow is found to remain approximately steady throughout the evolution. At leading order, the velocity field depends instantaneously on the shape of the domain boundary, and it is determined by the steadiness and vorticity-preservation conditions. This is made explicit by reformulating the problem in terms of an area-preserving diffeomorphism g_Λ which transports the vorticity. The first-order correction to the velocity field is linear in the boundary velocity, and we show how it can be computed from the time derivative of g_Λ.The evolution of the Lagrangian position of fluid particles is also examined. Thanks to vorticity conservation, this position can be specified by an angle-like coordinate along vorticity contours. An evolution equation for this angle is derived, and the net change in angle resulting from a cyclic deformation of the domain boundary is calculated. This includes a geometric contribution which can be expressed as the integral of a certain curvature over the interior of the circuit that is traced by the parameters defining the deforming boundary.A perturbation approach using Lie series is developed for the computation of both the Eulerian flow and geometric angle for small deformations of the boundary. Explicit results are presented for the evolution of nearly axisymmetric flows in slightly deformed discs.