Abstract. Solution algorithms are presented for tracking interfaces with piecewise linear (PUC) volume-of-fluid (VOF) methods on fixed (Eulerian) two-dimensional (2-D) structured and three-dimensional (3-D) structured and unstructured grids. We review the theory of volume tracking methods, derive appropriate volume evolution equations, identify and present solutions to the basic geometric functions needed for interface reconstruction and volume fluxing, and provide detailed algorithm templates for modem 2-D and 3-D PUC VOF interface tracking methods. We discuss somekey outstanding issues for PUC VOF methods. namely the method used for time integration of fluid volumes (operator splitting, unsplit, Runge-Kutta, etc.) and the estimation of interface normals. We also present our latest developments in the continuum surface force (CSF) model for surface tension. namely extension to 3-D and variable surface tension effects. We identify and focus on key outstanding CSF model issues that become especially critical on fine meshes with high density ratio interfacial flows, namely the surface delta function approximation, the estimation of interfacial curvature, and the continuum surface force scaling and/or smoothing model. Numerical results in two and three dimensions are used to illustrate the properties of these methods.

A number of advances in modeling multiphase incompressible flow are described. These advances include high-order Godunov projection methods, unsplit piecewise linear interface reconstruction and tracking and the continuum surface force model. Various aspects of projection methods and high-order Godunov methods are expanded upon where they are important to modeling multiphase flow phenomena. Alternative approaches to interface tracking are presented including an innovative particle based approach. Examples are given that show the strengths and weaknesses of the methodology.

A finite-volume based computational model is developed to predict Marangoni con vection in a cavity with a curved and deforming free surface. The two-dimensional incompressible continuity, momentum, and energy equations are solved on a staggered Cartesian grid. The free surface location is computed using the volume-offluid transport equation. Normal and tangential boundary conditions at the free surface are modeled using respectively a surface pressure and a continuum surface force technique. Computational predictions of thermocapiUary flow in a shallow cavity are shown to be in good agreement with previously published asymptotic results. The new transient model is then used to study the influence of Marangoni number and Capillary number on thermocapiUary flows in a cavity for different static contact angles. The flows are characterized by streamline and isotherm patterns. The influence of the dimensionless parameters on heat transfer rate at the cavity walls is exposed by examination of local Nusselt number profiles.

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