Volume of fluid methods for immiscible-fluid and free-surface flows

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Abstract

This article reviews and analyzes a number of numerical methods to track interfaces in multiphase flows. Several interface tracking methods can be found in literature: the level-set method, the marker particle method, the front tracking method and the volume of fluid method (VOF) to name a few. The volume of fluid method has an advantage of being conceptually simple, reasonably accurate and phenomena such as interface breakup and coalescence are inherently included. Over the years a number of different techniques to implement the VOF method have been devised.

This article gives a basic introduction to the VOF method and focuses on four VOF methods: flux-corrected transport (FCT) by Boris et al. [J.P. Boris, D.L. Book, Flux-corrected transport. I: SHASTA, a fluid transport algorithm that works, J. Comput. Phys. 11 (1973) 38–69], Lagrangian piecewise linear interface construction (L-PLIC) by van Wachem and Schouten [B.G.M. van Wachem, J.C. Schouten, Experimental validation of 3-d Lagrangian VOF model: bubble shape and rise velocity, AIChE 48 (12) (2002) 2744–2753], Compressive interface capturing scheme for arbitrary meshes (CICSAM) by Ubbink [O. Ubbink, Numerical prediction of two fluid systems with sharp interfaces, Ph.D. Thesis, Imperial College of Science, Technology and Medicine, 1997] and inter-gamma scheme by Jasak and Weller [H. Jasak, H.G. Weller, Interface-tracking capabilities of the InterGamma differencing scheme, Technical Report, Imperial College, University of London, 1995]. A detailed description of these schemes is given and implemented into an in-house fully coupled solver. Further, the performance of these schemes is examined employing a number of tests to analyze their strengths and weaknesses. Their advantages and limitations are discussed.

Introduction

Interfacial multiphase flows are frequently encountered both in nature and industries. Processes such as extraction, chemical reaction, mass-transfer, separation, etc., involve interfacial flows. To understand the basic hydrodynamic phenomena associated with such processes requires a proper and sharp definition of the interface between two phases. These phenomena include bubble transport, formation, breakup, coalescence, etc.

A detailed computation of immiscible-fluid and free-surface flows requires an accurate representation of the interface separating the two fluids. Immiscible-fluid flows are commonly encountered in nature as well as in industries. The applications include processes involving separation, extraction, mixing and chemical reactions. Free-surface flows such as water waves and splashing droplets are encountered in nature and industrial processes. These flow problems include phenomena like fluid coalescence and breakup which further increases the need for an accurate and sharp interface definition. A number of techniques to track the interface have been developed in the last few decades. The most important techniques are shown in Table 1. The existing methods for the computation of free surfaces and fluid interfaces can be classified into two groups (Fig. 1): (a) surface methods (surface fitting) and (b) volume methods (surface capturing).

With surface methods, the interface is represented by special marker points. Interpolation is used to approximate the points between these points, usually using a piecewise polynomial. The advantage of this approach is that the interface position is known throughout the flow field and remains sharp as it is advected across the domain. This enables the accurate calculation of the interface curvature which is needed for the inclusion of the surface tension force. Limitations arise while simulating coalescence and breakup of the interface surface, as the particles might tend to either move apart or very close to each other leading to lower resolution of the interface. Several surface methods exist, two such methods are explained below:

Front tracking method: In front tracking methods (Unverdi and Tryggvason [15]), the interface is tracked explicitly on a fixed Eulerian mesh by marking the interface with a set of connected massless marker particles. The local velocities are used to advect these massless particles in a Lagrangian manner. The method is sensitive to the spacing between the marker particles, i.e. when the particles are far apart, the interface is not well resolved and when they are too close, the curvature is over-estimated. Therefore it is necessary to add or delete marker particles dynamically. Also, difficulties arise when multiple interfaces interact with each other as in coalescence and breakup requiring a proper sub-grid model.

Level-set method: In level-set methods (Osher and Sethian [10]), the interface is defined as a zero level set of a distance function from the interface. To distinguish between the two fluids on either side of the interface a negative sign is attached to the distance function for one of the fluids. The distance function γ is a scalar property and is advected with the local fluid velocity by solving the scalar advection equation:γt+Uiγxi=0The level-set methods are conceptually simple and relatively easy to implement yielding accurate results when the interface is advected parallel to one of the co-ordinate axis. However, in flow fields with appreciable vorticity or in cases where the interface is significantly deformed, level-set methods suffer from loss of mass.

In volume methods, the fluids on either side of the interface are marked by either massless particles or an indicator function. Thus the exact position of the interface is not known explicitly and special techniques are needed to reconstruct the well-defined interface, which is one of the main drawbacks of this technique. A number of volume methods exist, two such volume methods are explained here.

Marker particle methods: In the marker and cell (MAC) method of Harlow and Welch [3] marker particles are scattered initially to identify each material region in the calculation. These particles are transported in a Lagrangian manner along with the materials. Their presence in a computational cell indicates the presence of the marked material. The material boundary is reconstructed using the marker particle densities in the mixed cells with marker particles of two or more materials. Marker particle methods are extremely accurate and robust and can be used successfully to predict the topology of an interface subjected to considerable shear and vorticity in the fluids sharing the interface. However, this method is computationally expensive due to the requirement of many particles, especially in three dimensions. Moreover, difficulties arise when the interface stretches considerably which requires the addition of fresh marker particles during the flow simulation.

Volume of fluid method (VOF): A scalar indicator function between zero and one, known as volume fraction is used to distinguish between two different fluids. In this study four different volume of fluid schemes are implemented into a fully coupled in-house finite volume, boundary fitted code. The next section gives an introduction to the volume of fluid method and the schemes implemented are explained in detail. Further, the performance of these schemes is reviewed with the help of various test cases.

Section snippets

Volume of fluid method

The volume of fluid method was first proposed by Hirt and Nichols [4]. In the volume of fluid method, the flow equations are volume averaged directly to obtain single set of equations and the interface is tracked using a phase indicator function γ (also known as color function or volume fraction) which is defined as:

  • γ=1 control volume is filled only with phase 1

  • γ=0 control volume is filled only with phase 2

  • 0<γ<1 interface present

The flow equations are volume averaged using an averaging

Test cases

To evaluate the performance of the implemented VOF schemes, a number of standard test cases were setup. These test cases can be generally classified into theoretical ones, where comparisons are made against analytical solutions and real cases, where comparisons are made against theoretical/experimental data.

Error estimation: The fractional error resulting from the simulations is calculated using:E=i,jγi,jnγi,jei,jγi,jo,where γn is the calculated solution after n time steps, γe is the exact

Summary of methods

The FCT method is a simple scheme to implement. It uses a combination of a lower order stable scheme and a higher order monotonic scheme to calculate anti-diffusive fluxes. The fluxes are limited using limiters calculated from the possible theoretical extrema in the solution. Direction splitting is used to extend the scheme into multi-dimensions. The results of the simple advection tests show that the performance of FCT is reasonably good, but, the results of more practical tests, i.e. sloshing

Conclusions

This paper discusses four volume of fluid (VOF) schemes in detail, showing the derivation, implementation, advantages and limitations. The schemes presented are: flux-corrected transport (Boris et al. [1]), Lagrangian piecewise linear interface construction (van Wachem and Schouten [16]), Compressive interface capturing scheme for arbitrary meshes (Ubbink [14]) and inter-gamma scheme (Jasak and Weller [5]). The performance of the four schemes is tested initially using simple analytical velocity

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