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Hirt and Niçois VOF (SLIC)

Youngs VOF (PLIC) FIGURE 7.9 VOF interface reconstruction methods.

techniques such as SLIC (simple line interface calculation) and PLIC (piecewise linear interface calculation) have been developed to minimize computational smearing of the interface (see Rider and Kothe, 1995, for a review).

One of the most popular VOF methods is that due to Hirt and Nichols (1981). This method uses an approximate interface reconstruction that forces the interface to align with one of the co-ordinate axis, depending on the prevailing direction of the interface normal. A schematic diagram of reconstruction of a two-dimensional interface is shown in Fig. 7.9. To compute fluxes in a direction parallel to the reconstructed interface, upwind fluxes are used. Fluxes in a direction perpendicular to the reconstructed interface are estimated using a donor-acceptor method. In a donor-acceptor method, a computational cell is identified as a donor of some amount of fluid from one phase and another neighbor cell is identified as the acceptor of that donated amount of fluid. The amount of fluid from one phase that can be convected (donated) across a cell boundary is limited by the minimum of the filled volume in the donor cell or the free volume in the acceptor cell. This minimizes numerical diffusion at the interface.

The PLIC method is much more accurate than the SLIC method discussed above (Rider and Kothe, 1995; Rudman, 1997). In PLIC, an interface within a computational cell is approximated by a straight-line segment with a slope determined from the interface normal (Fig. 7.9). This interface normal is calculated from the gradient of volume fraction (using a nine-point computational molecule in two dimensions). The line segment cuts the computational cell in such a way that the fractional fluid volume is equal to the value of marker function at that cell. The resulting fluid polygon is then used to determine fluxes through any cell face. Implementation of the PLIC method (calculation of face fluxes) proposed by Youngs (1982) is shown graphically in Fig. 7.10 for one specific interface configuration. Sixteen possible different configurations, depending on the orientation of the interface with respect to the co-ordinate axis, have been identified. For any specific configuration, formulae similar to those shown in Fig. 7.10 can be derived to calculate cell fluxes.

Apart from the identification of interface and calculation of cell fluxes, an appropriate computer implementation of interfacial tension is also one of the key elements in carrying out realistic VOF simulations. Most studies have implemented the continuum surface force (CSF) model of Brackbill et al. (1992) to describe interfacial tension. As discussed in Chapter 4, the CSF model replaces the interfacial force by a smoothly varying volumetric force acting on all fluid elements in the interface transition region (see Eq. (4.6)). The local surface curvature appearing in this equation can be calculated from the unit normal at the interface between phases (see Eq. (4.8); Kothe et al., 1991; Delnoij, 1999). For additional details of numerical implementation, the original references should be consulted. It is sufficient to emphasize here that the accuracy of the CSF model depends on sharpness of the interface maintained by the interface-tracking model. Preferably, the sharpness of the interface should be maintained at around the order of the cell width [O(5x)j.

Apart from the interface tracking and forces due to interfacial tension, conventional numerical techniques developed for single-phase flow simulations are used to solve the relevant transport equations. As mentioned earlier, controlling numerical diffusion and smearing of the interface are key issues in realistic VOF-based simulations. The size of computational cells and discretization schemes, therefore, significantly affect the predicted results. To illustrate VOF simulations, the formation of gas bubbles through an orifice sparger was simulated. The considered geometry, computational grid and other necessary data are shown in Fig. 7.11. Simulations were carried out in an unsteady mode. To simulate transients accurately, it is necessary to ensure adequate convergence at each time step. Typical predicted results 0.12 s after the introduction of gas from the nozzle are shown in Fig. 7.12. It can be seen that the discretization scheme has a substantial influence on the predicted shape and rise velocity of bubbles. Similar observations were made by Delnoij (1999). The computational requirements of VOF-based simulations are much higher than those of Eulerian-Eulerian or Eulerian-Lagrangian approaches. In most cases, accurate simulations of interfacial phenomena require simulations in three-dimensional domains, which further increases computational demands. For example, detailed two-dimensional VOF simulations carried out by Krishna and van Baten (1999) showed significant discrepancies in the observed and predicted bubble rise velocities. Most VOF-based simulations are, therefore, restricted to a few large fluid particles. Despite this, VOF-based simulations can be used as useful learning simulations to gain insight into the interaction of continuous flow field and deformation and interfacial processes of large fluid particles. Some examples are discussed in Part IV.

Side fractions:

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Fluxes through the four cell faces:

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Ftop = 2 [«top • St - (1 - bright ) • Sy]2 cot 3

0 0

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