Douglas B. Kothe

A continuum method for modeling surface tension

Abstract A new method for modeling surface tension effects on fluid motion has been developed. Interfaces between fluids of different properties, or “colors,” are represented as transition regions of finite thickness, across which the color variable varies continuously. At each point in the transition region, a force density is defined which is proportional to the curvature of the surface of constant color at that point. It is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero. The continuum method eliminates the need for interface reconstruction, simplifies the calculation of surface tension, enables accurate modeling of two- and three-dimensional fluid flows driven by surface forces, and does not impose any modeling restrictions on the number, complexity, or dynamic evolution of fluid interfaces having surface tension. Computational results for two-dimensional flows are given to illustrate the properties of the method.

A balanced-force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework

A new balanced-force algorithm is presented for modeling interfacial flow with surface tension. The algorithm is characterized by a pressure-correction method with the interfaces represented by volume fractions. Within this flow algorithm, we devise a continuous (e.g., continuum surface tension model) and a sharp (e.g., a ghost fluid method) interface representation of the surface-tension-induced interfacial pressure jump condition. The sharp interface representation is achieved by temporarily reconstructing distance functions from volume fractions. We demonstrate that a flow algorithm designed to legislate force balance retains an exact balance between surface tension forces and the resulting pressure gradients. This balance holds for both continuous and sharp representations of interfacial surface tension. The algorithm design eliminates one of the elusive impediments to more accurate models of surface tension-driven flow, the remaining of which is accurate curvature estimation. To validate our formulation, we present results for an equilibrium (static) drop in two and three dimensions having an arbitrary density jump across the interface. We find that the sharp surface tension method yields an abrupt pressure jump across the interface, whereas the continuous surface tension method results in a smoother transition. Both methods, however, yield spurious velocities of the same order, the origin of which is due solely to errors in curvature. Dynamic results are also presented to illustrate the versatility of the method.

Volume tracking of interfaces having surface tension in two and three dimensions

Solution algorithms are presented for tracking interfaces with piecewise linear (PLIC) 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 modern 2-D and 3-D PLIC VOF interface tracking methods. We discuss some key outstanding issues for PLIC 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.

Accurate solution algorithms for incompressible multiphase flows

A number of advances in modeling multiphase incompressible flow are described. These advances include high-order Godunov projection methods, piecewise linear interface reconstruction and tracking and the continuum surface force model. Examples are given.

Perspective on Eulerian Finite Volume Methods for Incompressible Interfacial Flows

Incompressible interfacial flows here refer to those incompressible flows possessing multiple distinct, immiscible fluids separated by interfaces of arbitrarily complex topology. A prototypical example is free surface flows, where fluid properties across the interface vary by orders of magnitude. Interfaces present in these flows possess topologies that are not only irregular but also dynamic, undergoing gross changes such as merging, tearing, and filamenting as a result of the flow and interface physics such as surface tension and phase change. The interface topology requirements facing an algorithm tasked to model these flows inevitably leads to an underlying Eulerian methodology. The discussion herein is confined therefore to Eulerian schemes, with further emphasis on finite volume methods of discretization for the partial differential equations manifesting the physical model.

Modeling High Density Ratio Incompressible Interfacial Flows

We present an approach to modeling incompressible interfacial flows on fixed meshes that yields solutions at any density ratio. There are two aspects of the methodology that are crucial for obtaining accurate high density ratio solutions: a consistent approach to mass and momentum conservation, by using mass flux information from an interface advection algorithm as the basis for the momentum advection calculation, and a careful evaluation of pressure gradients near the interface. Our particular implementation couples a volume tracking algorithm with a predictor/projection solution of the flow equations on unstructured meshes. We present the methodology, and then the results of several calculations.Copyright

Accurate and Robust Methods for Variable Density Incompressible Flows with Discontinuities

We are interested in the solution of incompressible flows which are characterized by large density variations, interfacial physics, arbitrary material topologies and strong vortical content. The issues present in constant density incompressible flow are exacerbated by the presence of density discontinuities. A much greater premium requirement is placed the positivity of computed quantities The mechanism of baroclinc vorticity generation exists ({gradient}p x {gradient}p) to further complicate the physics.

Constrained minimization for monotonic reconstruction

The authors present several innovations in a method for monotonic reconstructions. It is based on the application of constrained minimization techniques for the imposition of monotonicity on a reconstruction. In addition, they present extensions of several classical TVD limiters to a genuinely multidimensional setting. In this case the linear least squares reconstruction method is expanded upon. They also clarify data dependent weighting techniques used with the minimization process.

BALANCED FORCE IMPLEMENTATION OF THE CONTINUUM SURFACE TENSION FORCE METHOD INTO A PRESSURE CORRECTION ALGORITHM

A consistent formulation is presented for modeling surface tension driven flow with the continuum surface tension force (CSF) model within a volume of fluid (VOF) method using a pressure-correction projection method. We show that a flow algorithm whose inherent design is motivated by legislating force balance gives an exact (to round off) balance between surface tension forces and pressure gradients that arise as a result. This design eliminates one of the elusive impediments to more accurate CSF-based surface tension models, the remaining of which is curvature estimation accuracy. To validate our formulation, we present results for an equilibrium (static) drop in two and three dimensions having an arbitrary density ratio and demonstrate in the process that scaling effects within the CSF framework are insignificant.Copyright

Robust Finite Volume Modeling of 3-D Free Surface Flows on Unstructured Meshes *

The numerical simulation of incompressible fluids possessing multiple distinct, immiscible fluids. are of great interest to the engineering community. These flows contain interfaces -which can merge and tear as a result of interface physics such as phase change and surface tension. The arbitrarily complex interface topologies inevitably requires an underlying Eulerian formulation. Volume tracking methods are in wide use today, having proven themselves to be topologically robust and relatively easy to implement. The basis in volume fractions also allows the straightforward incorporation of interfacial physics. The formulation of kernel-based continuum surface tension models for these methods has demonstrated acceptable results for orthogonal grids. Attaining accurate, high order results however, on unstructured meshes is necessary if surface tensiondriven flows are to. be reliably modeled within the confined; complex geometries of most indusl trial processes. In the following, we investigate the accuracy and convergence of our interface topology estimates and surface tension model on 3-D tetrahedral meshes. We use various ellipsoids to scrutinize our algorithm, which is based on a convolution (hybridj method to determine the interface unit normal and mean curvature.