Direct numerical simulation of single gas bubbles in pure and contaminated liquids

Disperse gas bubbles play an important role in many industrial applications. Knowing the rising velocity, the interfacial area, or the critical size for break–up or coalescence in different systems can be crucial for the process design. Hence, knowing the fundamental behaviour of a single bubble appears mandatory for the examination of bubble swarms and for the Euler–Lagrange or Euler–Euler modelling of disperse systems. In the present work a level–set–based volume–tracking method is implemented into the CFD–code OpenFOAM to follow the free interface of a single bubble. The volume–tracking method is coupled with a transport model for surfactants on the interface, including adsorption and desorption processes. The dependency of surface tension on the local surfactant concentration on the interface is modelled by a non–linear (Langmuir) equation of state. Marangoni forces, resulting from surface tension gradients, are included. The rise of a single air bubble (i) in pure water and (ii) in the presence of surfactants of different strengths is simulated. The results show good agreement with available correlations from literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bubble dynamics: Long-time behavior and interaction in viscous liquids

A mathematical two-phase model is used to numerically investigate physical and rheological effects on small, individual bubbles in high-viscosity liquids under pressure impact. It is found out that bubbles remain stable over time at high viscosity and surface tension. The steady case is considered and connected to the stability behavior of the bubble. An upper bound for the bubble radius is derived and the new equilibrium state of the bubble can be predicted by means of stability theorems of differential equations. Finally, the interaction of a limited number of well separated bubbles in an Hele-Shaw flow is mathematically analyzed to visualize and physically interpret their trajectories. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.     

Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods.

A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [ J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another.

The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution.

The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

     

Numerical modelling of bubbly flows with and without heat and mass transfer

In this paper, modelling gas–liquid bubbly flows is achieved by the introduction of a population balance equation combined with the three-dimensional two-fluid model. For gas–liquid bubbly flows without heat and mass transfer, an average bubble number density transport equation has been incorporated in the commercial code CFX5.7 to better describe the temporal and spatial evolution of the geometrical structure of the gas bubbles. The coalescence and breakage effects of the gas bubbles are modelled according to the coalescence by the random collisions driven by turbulence and wake entrainment while for bubble breakage by the impact of turbulent eddies. Local radial distributions of the void fraction, interfacial area concentration, bubble Sauter mean diameter, and gas and liquid velocities, are compared against experimental data in a vertical pipe flow. Satisfactory agreements for the local distributions are achieved between the predictions and measurements. For gas–liquid bubbly flows with heat and mass transfer, boiling flows at subcooled conditions are considered. Based on the formulation of the MUSIG (multiple-size-group) boiling model and a model considering the forces acting on departing bubbles at the heated surface implemented in the computer code CFX4.4, comparison of model predictions against local measurements is made for the void fraction, bubble Sauter mean diameter, interfacial area concentration, and gas and liquid velocities covering a range of different mass and heat fluxes and inlet subcooling temperatures. Good agreement is achieved with the local radial void fraction, bubble Sauter mean diameter, interfacial area concentration and liquid velocity profiles against measurements. However, significant weakness of the model is evidenced in the prediction of the vapour velocity. Work is in progress through the consideration of additional momentum equations or developing an algebraic slip model to account for the effects of bubble separation.     

Stokes flow response to a localised change in interfacial tension

The interfacial tension contribution to the normal stress balance leads to a rather complicated low-Reynolds number hydrodynamic response to the imposed surface tension variation. It is shown that the assumptions dictated by the model prevent the steady state of zero response from ever being attained and thereby demand that the forcing be switched off, after which the interface displacement decays algebraically in time with a small amplitude.     

Equilibria for anisotropic surface energies with wetting and line tension

We study the stability of surfaces trapped between two parallel planes with free boundary on these planes. The energy functional consists of anisotropic surface energy, wetting energy, and line tension. Equilibrium surfaces are surfaces with constant anisotropic mean curvature. We study the case where the Wulff shape is of “product form”, that is, its horizontal sections are all homothetic and have a certain symmetry. Such an anisotropic surface energy is a natural generalization of the area of the surface. In particular, we study the stability of parts of anisotropic Delaunay surfaces which arise as equilibrium surfaces. They are surfaces of the same product form of the Wulff shape. We show that, for these surfaces, the stability analysis can be reduced to the case where the surface is axially symmetric and the functional is replaced by an appropriate axially symmetric one. Moreover, we obtain necessary and sufficient conditions for the stability of anisotropic sessile drops.     

Numerical simulation of single gas bubbles under shear flow conditions

Disperse gas bubbles play an important role in many industrial applications. Knowing the rising velocity, the interfacial area, or the critical size for break-up or coalescence in different systems can be crucial for the process design. Usually the flow experienced by bubbles is not uniform but sheared. Under shear-flow conditions bubbles develop a lift force perpendicular to the flow direction. In the present work direct numerical simulations are applied to examine the dependency of the lift force on the shear rate for bubbles in pure liquids. A level-set based volume-tracking method is implemented into the CFD-code OpenFOAM, to follow the free interface of the gas bubble. Results show good agreement with available experimental results from single bubbles in a rotating chamber. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Prediction of initial motion of a gas bubble in liquids

Numerical predictions are presented for the motion and distortion of a single gas bubble rising through the liquid. The computations were made with an implicit finite-difference procedure which solves the transient equations of motion throughout the bubble and the liquid, such that the free surface between the gas bubble and the liquid is not a boundary of the computational domain.The predictions compare well with the experimental results of others. Computations are presented for bubble sizes from 0.02 to 0.05 m radius and for bubbles of different gas densities rising in liquids of different densities. Surface tension effects are neglected.     

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

The influence of electric fields and surface tension on Kelvin–Helmholtz instability in two-dimensional jets

We consider nonlinear aspects of the flow of an inviscid two-dimensional jet into a second immiscible fluid of different density and unbounded extent. Velocity jumps are supported at the interface, and the flow is susceptible to the Kelvin–Helmholtz instability. We investigate theoretically the effects of horizontal electric fields and surface tension on the nonlinear evolution of the jet. This is accomplished by developing a long-wave matched asymptotic analysis that incorporates the influence of the outer regions on the dynamics of the jet. The result is a coupled system of long-wave nonlinear, nonlocal evolution equations governing the interfacial amplitude and corresponding horizontal velocity, for symmetric interfacial deformations. The theory allows for amplitudes that scale with the undisturbed jet thickness and is therefore capable of predicting singular events such as jet pinching. In the absence of surface tension, a sufficiently strong electric field completely stabilizes (linearly) the Kelvin–Helmholtz instability at all wavelengths by the introduction of a dispersive regularization of a nonlocal origin. The dispersion relation has the same functional form as the destabilizing Kelvin–Helmholtz terms, but is of a different sign. If the electric field is weak or absent, then surface tension is included to regularize Kelvin–Helmholtz instability and to provide a well-posed nonlinear problem. We address the nonlinear problems numerically using spectral methods and establish two distinct dynamical behaviors. In cases where the linear theory predicts dispersive regularization (whether surface tension is present or not), then relatively large initial conditions induce a nonlinear flow that is oscillatory in time (in fact quasi-periodic) with a basic oscillation predicted well by linear theory and a second nonlinearly induced lower frequency that is responsible for quasi-periodic modulations of the spatio-temporal dynamics. If the parameters are chosen so that the linear theory predicts a band of long unstable waves (surface tension now ensures that short waves are dispersively regularized), then the flow generically evolves to a finite-time rupture singularity. This has been established numerically for rather general initial conditions.     

Steady Gravity-Capillary Waves on Deep Water—II. Numerical Results for Finite Amplitude

Finite amplitude capillary-gravity waves of permanent form on deep water are studied numerically. Bifurcation and limit lines are calculated. Pure and combination waves are continued to maximum amplitude. It is found that the height is limited in all cases by the surface enclosing one or more bubbles.     

Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension

We consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments.     

The effect of surfactant on bubble and thread dynamics

Two examples from the dynamics of surfactant-laden interfacial flow are considered. In the first example, a bubble is rapidly stretched by an imposed flow to form a dumb-bell shape, then the imposed flow is relaxed, so that the bubble evolves solely under the action of surfactant-modified surface tension. In the second example, a surfactant-coated bubble is continually stretched by a steady extensional flow. At sufficiently small strain rates, steady bubble shapes are found, whereas at larger strain rates a long-wave model of the dynamics predicts behavior that is reminiscent of experimentally observed tip-streaming. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

界面滑移流体动压膜承载能力的形成

运用界面滑移可在两平行平板表面间形成具有承载能力的流体动压膜．在流体入口区,静止平板表面上流体-接触表面的界面剪切强度具有较低值,以在该界面处产生界面滑移,而在流体出口区,静止平板表面上流体-接触表面的界面剪切强度具有足够高的值,以避免在该界面处出现界面滑移．整个运动平板表面上流体-接触表面的界面剪切强度具有足够高的值,以避免在运动平板表面上出现界面滑移．分析表明,这种流体动压接触区具有显著承载能力．使整个接触区具有最大承载能力的流体出口区宽度与入口区宽度的比值为0.5．     

Numerical study of dynamics of single bubbles and bubble swarms

A systematic computational study of the dynamics of gas bubbles rising in a viscous liquid is presented. Two-dimensional simulations are carried out. Both the dynamics of single bubbles and small groups of bubbles (bubble swarms) are considered. This is a continuation of our previous studies on the two-bubble coalescence and vortex shedding [A. Smolianski, H. Haario, P. Luukka, Vortex shedding behind a rising bubble and two-bubble coalescence: a numerical approach, Appl. Math. Model. 29 (2005) 615–632]. The proposed numerical method allows us to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining the mass conservation. The computed time-evolution of bubble’s position and rise velocity shows a good agreement with the available experimental data. At the same time, the results on the dynamics of bubble interface area, which are, up to our knowledge, presented for the first time, show how much the overall mass transfer would be affected by the interface deformation in the case of the bubble dissolution. Another set of experiments that are of interest for chemical engineers modelling bubbly flows concerns the bubble swarms and their behavior in different bubble-shape regimes. The ellipsoidal and spherical shape regimes are considered to represent, respectively, the coalescing and non-coalescing bubble swarms. The average rise velocities of the bubble swarms are computed and analyzed for both regimes.     

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.     

Electromagnetic force on the wall of cylindrical waveguide

When transverse electric (TE) wave or transverse magnetic (TM) wave propagates inside a cylindrical waveguide, the electromagnetic force on the wall is investigated. The characteristics of surface charge, current, electric force, magnetic force and electromagnetic force are studied. The results show that the electric force is tension and magnetic force is press. The surface density of electromagnetic force on the wall can be calculated by the difference between magnetic and electric energy density there. For TE wave, the electromagnetic force distribution on the walls may be either tension or pressure in general. However, the electromagnetic force is always pressure for TM wave.     

Vortex shedding behind a rising bubble and two-bubble coalescence: A numerical approach

In this work, we present the computational results on the wake instability in wobbling bubble regime as well as on the coalescence of two bubbles in different shape regimes. This is a continuation of our previous studies on the dynamics of a single gas bubble rising in a viscous liquid (see [A. Smolianski, H. Haario, P. Luukka, Computational Study of Bubble Dynamics, Research Report 86, Lappeenranta University of Technology, Finland]), and we use the same, finite-element/level-set/operator-splitting method that was proposed in [A. Smolianski, Numerical Modeling of Two-Fluid Interfacial Flows, Ph.D. Thesis, University of Jyväskylä, 2001]. The numerical method allows to simulate a wide range of flow regimes, accurately capturing the shape of the deforming interface of the bubble and the surface tension effect, while maintaining a good mass conservation. Due to the highly unstable and small-scale nature of the considered problems there are very few experimental investigations, but the comparison with available experimental data confirms a good accuracy of our numerical predictions. Our studies show that plausible results can be obtained with two-dimensional numerical simulations, when a single buoyant bubble or a coalescence of two bubbles is considered.     

有流存在下两层密度分层流体毛细重力波的三阶Stokes波解

以小振幅波理论为基础,利用奇异摄动方法研究了有背景流存在下两层密度成层状态下的毛细重力波,求得了两层密度成层状态下各层流体速度势的三阶解及毛细重力波波面位移的三阶Stokes波解,并讨论了毛细重力波的kelvin-Helmholtz的不稳定性.结果表明在有流存在的情况下,两层密度成层流体毛细重力波的一阶渐近解、频散关系,二阶渐近解及三阶渐近解不仅依赖于各层流体的厚度和密度,也依赖于表面张力和各层流体的背景流流场;毛细重力波的三阶解描述了背景流场与毛细重力波之问的三阶非线性相互作用.对于给定的波数k(实数)毛细重力波可能出现kelvin-Helmholtz不稳定性.