A hybrid finite-element–volume-of-fluid method for simulating free surface flows and interfaces

A numerical technique is developed for the simulation of free surface flows and interfaces. This technique combines the strength on the finite element method (FEM) in calculating the field variables for a deforming boundary and the versatility of the volume-of-fluid (VOF) technique in advection of the fluid interfaces. The advantage of the VOF technique is that it allows the simulation of interfaces with large deformations, including surface merging and breaking. However, its disadantage is that is solving the flow equations, it cannot resolve interfaces smaller than the cell size, since information on the subgrid scale is lost. Therefore the accuracy of the interface reconstruction and the treatment of the boundary conditions (i.e. viscous stresses and surface tension forces) become grid-size-dependent. On the other hand, the FEM with deforming interface mesh allows accurate implementation of the boundary conditions, but it cannot handle large surface deformations occurring in breaking and merging of liquid regions. Combining the two methods into a hybrid FEM-VOF method eliminates the major shortcomings of both. The outcome is a technique which can handle large surface deformations with accurate treatment of the boundary conditions. For illustration, two computational examples are presented, namely the instability and break-up of a capillary jet and the coalescence collision of two liquid drops.     

Advection of axisymmetric interfaces by the volume-of-fluid method

A criterion is proposed for the advection of axisymmetric interfaces. The location of an interface is followed by a volume-tracking technique wherein a volume fraction parameter is assigned to each of the cells in a Eulerian grid system. The interface is discretized into a set of line segments fitted at the boundary of every pair of neighbouring computational cells. The orientation of a line segment is obtained by inspecting the volume fractions of two neighbouring cells. The volume fractions are then advected using the velocity components at the boundary of the two cells. The following advection criterion is proposed: for advection in the axial direction the axial velocity u is assumed constant in the vicinity of each cell face; for advection in the radial direction the radial velocity v times the radial distance r is assumed constant in the vicinity of each cell face, i.e. r^βv = const., where β = 0 for Cartesian and β = 1 for axisymmetric systems. The above criterion is used to develop an algorithm for the advection of axisymmetric interfaces which is referred to as the ‘axisymmetric flux line segment model for advection and interface reconstruction’ or A-FLAIR.     

Numerical study of bubble rise and interaction in a viscous liquid

In this article, the flow instabilities during the rise of a single bubble in a narrow vertical tube are studied using a transient two-dimensional/axisymmetric model. To predict the shape of the bubble deformation, the Navier-Stokes equations in addition to an advection equation for liquid volume fraction are solved. A modified volume-of-fluid technique based on Youngs' algorithm is used to track the bubble deformation. To validate the model, the results of simulations for terminal rise velocity and bubble shape are compared with those of the experiments. The effect of different parameters such as initial bubble radius, channel height, liquid viscosity and surface tension on the shape and rise velocity of the bubble is investigated.     

使用界面追踪对膜态沸腾的数值研究

提出一种计算带相变的自由界面的数值算法.基于分段线性界面重构(PLIC)的VOF(volume-of-fluid)方法用于追踪自由界面,并对汽液交界面上的相变导致的不连续速度场给出处理方法.此方法容易实施且被证明是有效的.流场的求解使用SIMPLE方法,表面张力使用连续表面力模型(CSF)进行计算.在三维直角坐标系下,模拟了水平壁面上的膜态沸腾,在二维适体坐标系下,模拟了竖直圆头柱体表面的自然对流膜态沸腾.计算结果与理论关系式符合较好.     

Second-order model for free surface convection and interface reconstruction

To improve the numerical analysis of free surface convection and interface reconstruction, both first- and second-order algorithms are developed based on the volume-of-fluid method. The methodology applied to the second-order model is to define the second-order linear curve having both face slopes as near horizontal as possible while satisfying the cell's defined volume fraction. The second-order method is compared with the FLAIR method and the first-order method through simulation of the convection for various sizes of circular liquid shapes and solitary waves. For small curvature of the free surface, e.g. circles with large diameter, linear methods such as the FLAIR method and the first-order method show relatively good predictions. However, for large-curvature configurations, e.g. circles with relatively small diameter or solitary waves, the linear approach shows large distortion of the free surface. In contrast, the second-order model always shows powerful prediction capabilities of free surface convection. © 1998 John Wiley & Sons, Ltd.     

Multiphase flow modeling in centrifugal partition chromatography

The separation efficiency in Centrifugal Partition Chromatography (CPC) depends on selection of a suitable biphasic solvent system (distribution ratio, selectivity factor, sample solubility) and is influenced by hydrodynamics in the chambers. Especially the stationary phase retention, the interfacial area for mass transfer and the flow pattern (backmixing) are important parameters. Their relationship with physical properties, operating parameters and chamber geometry is not completely understood and predictions are hardly possible. Experimental flow visualization is expensive and two-dimensional only. Therefore we simulated the flow pattern using a volume-of-fluid (VOF) method, which was implemented in OpenFOAM^®. For the three-dimensional simulation of a rotating FCPC^®-chamber, gravitational centrifugal and Coriolis forces were added to the conservation equation. For experimental validation the flow pattern of different solvent systems was visualized with an optical measurement system. The amount of mobile phase in a chamber was calculated from gray scale values of videos recorded by an image processing routine in ImageJ^®. To visualize the flow of the stationary phase polyethylene particles were used to perform a qualitative particle image velocimetry (PIV) analysis. We found a good agreement between flow patterns and velocity profiles of experiments and simulations. By using the model we found that increasing the chamber depth leads to higher specific interfacial area. Additionally a circular flow in the stationary phase was identified that lowers the interfacial area because it pushes the jet of mobile phase to the chamber wall. The Coriolis force alone gives the impulse for this behavior. As a result the model is easier to handle than experiments and allows 3D prediction of hydrodynamics in the chamber. Additionally it can be used for optimizing geometry and operating parameters for given physical properties of solvent systems.     

A volume-of-fluid method for incompressible free surface flows

This paper proposes a hybrid volume-of-fluid (VOF) level-set method for simulating incompressible two-phase flows. Motion of the free surface is represented by a VOF algorithm that uses high resolution differencing schemes to algebraically preserve both the sharpness of interface and the boundedness of volume fraction. The VOF method is specifically based on a simple order high resolution scheme lower than that of a comparable method, but still leading to a nearly equivalent order of accuracy. Retaining the mass conservation property, the hybrid algorithm couples the proposed VOF method with a level-set distancing algorithm in an implicit manner when the normal and the curvature of the interface need to be accurate for consideration of surface tension. For practical purposes, it is developed to be efficiently and easily extensible to three-dimensional applications with a minor implementation complexity. The accuracy and convergence properties of the method are verified through a wide range of tests: advection of rigid interfaces of different shapes, a three-dimensional air bubble's rising in viscous liquids, a two-dimensional dam-break, and a three-dimensional dam-break over an obstacle mounted on the bottom of a tank. The standard advection tests show that the volume advection algorithm is comparable in accuracy with geometric interface reconstruction algorithms of higher accuracy than other interface capturing-based methods found in the literature. The numerical results for the remainder of tests show a good agreement with other numerical solutions or available experimental data. Copyright © 2009 John Wiley & Sons, Ltd.     

A study on the extension of a VOF/PLIC based method to a curvilinear co-ordinate system

The conventional volume-of-fluid method has the potential to deal with large free surface deformation on a fixed Cartesian grid. However, when free-surface flows are within or over complex geometries of industrial relevance, such as free-surface flows over offshore oil platforms, it is advantageous to extend the method originally written in Cartesian forms into non-Cartesian forms. In the present study, an algorithm similar to the algorithm described by Rudman in 1997 is proposed for use with curvilinear co-ordinates. This extension results in the ability to model complex geometries which could not be modelled using the original algorithm. Excellent agreement between the solutions obtained on both orthogonal and non-orthogonal meshes is achieved, although in general the L ₁ error, based on the exact solution, on the non-orthogonal mesh is slightly higher than that on the orthogonal mesh. The extended fluid flow solving capacity of the present method is demonstrated through its application to a non-orthogonal Rayleigh–Taylor instability problem.     

Finite element analysis of transient fluid flow with free surface using VOF (volume-of-fluid) method and adaptive grid

The VOF method is adopted for the finite element analysis of transient fluid flow with a free surface. In particular, an adaptation technique for generating an adaptive grid is incorporated to capture a higher resolution of the free surface configuration. An adaptive grid is created through the refinement and mergence of elements. In this domain the elements in the surface region are made finer than those in the remaining regions for more efficient computation. Also, three techniques based on the VOF method are newly developed to increase the accuracy of the analysis, namely the filling pattern, advection treatment and free surface smoothing techniques. Using the proposed numerical techniques, radial flow with a point source and the collapse of a dam are analysed. The numerical results agree well with the theoretical solutions as well as with the experimental results. Through comparisons with the numerical results of several cases using different grids, the efficiency of the proposed technique is verified. © 1998 by John Wiley & Sons, Ltd.     

Extension of the volume-of-fluid method for analysis of free surface viscous flow in an ideal gas

The volume-of-fluid (VOF) method is a simple and robust technique for simulating free surface flows with large deformations and intersecting free surfaces. Earlier implementations used Laplace's formula for the normal stress boundary condition at the interface between the liquid and vapour phases. We have expanded the interfacial boundary conditions to include the viscous component of the normal stress in the liquid phase and, in a limited manner, to allow the pressure in the vapour phase to vary. Included are sample computations that show the accuracy of added third-order-accurate differencing schemes for the convective terms in the Navier-Stokes equation (NSE), the viscous terms in the normal stress at the interface and the solution of potential flow in the vapour phase coupled with the solution of the NSE in the liquid phase. With these modifications we show that the VOF method can accurately predict the instability of a thin viscous sheet flowing through a stagnant vapour phase.