Page 307 - Process Modelling and Simulation With Finite Element Methods

294 Process Modelling and Simulation with Finite Element Methods

conservation law for volume fraction and the Navier-Stokes equations
simultaneously. Since the interface is represented in terms of volume fraction,
mass is always conserved, while maintaining a sharp representation of the
interface. The VOF method needs to have accurate reconstruction algorithms to
solve for the advection of volume fraction. A disadvantage of the VOF method is
that it is difficult to compute accurate local curvature from volume fraction. This
is due to the sharp transition in volume fraction near the interface.
The Lattice Boltzman method (LBM) is a mesoscopic approach to the
numerical simulation of fluid motions based on the assumption that a fluid
consists of many particles whose repeated collision, translation, and distribution
converge to a state of local equilibrium, yet always remaining in flux. LBM has
advantages such as implementation on a complex geometry, very efficient
parallel processing, and ease of reproduction of the interface between the phases.
However, LBM is not yet a widely used computational method to track the fluid
motion in multiphase systems, due to its computational intensity.
The level set approach is another potential numerical method to solve
incompressible two-phase flow incorporating surface tension term. In the level
set method, the interface is represented as the zero level set of a smooth function.
This has the effect of replacing the advection of physical properties with steep
gradients at the interface with advection of level set function that is smooth in
nature. Although level set method does not have the same conservation
properties as of VOF method or front tracking method, the major strength of
level set method lies in its ability to compute curvature of the interface easily.
Furthermore, level set method does not require complicated front tracking
regridding algorithms or VOF reconstruction algorithms. Level set method is
based on continuum approach in order to represent surface tension and local
curvature at the interface as a body force. This facilitates the computations in
capturing any topological change due to change in surface tension.
The diffuse interface method is a kindred notion to the level set method and
VOF in that it computes the transport of another function that varies between the
phases - the chemical potential. As is well known (see [7]), the surface tension
between two fluids is also the excess partial molar Gibbs free energy per unit
surface area, so that the change of chemical potential across an interface between
immiscible fluids is treated by the notion of surface tension as infinitely steep.
The diffuse interface method permits this condition to be merely relaxed to be
steep, and then a field equation for chemical potential is tracked, rather than the
imposition of topology and stress balance equations implied by the notion of
surface tension. The latter method still requires grid adaption, which in state of
the art computational models (see [8] and references therein) employ auxiliary
equations for elliptic mesh diffusion, but are fragile in the face of topological
change, e.g. coalescence or breakage phenomena [9]. Whether greater accuracy
at the same computational intensity is available by the topological method of