. We describe a three dimensional front tracking algorithm, discuss its numerical implementation, and present studies to validate the correctness of this approach. Based on the results of the two dimensional computations, we expect three dimensional front tracking to improve significantly computational efficiencies for problems dominated by discontinuities. In some cases, for which the interface computations display considerable numerical sensitivity, we expect a greatly enhanced capability. 1. Introduction Front tracking is a numerical method in which surfaces of discontinuity are given explicit computational degrees of freedom; these degrees of freedom are supplemented by degrees of freedom representing continuous solution values at regular grid points. This method is ideal for solutions in which discontinuities are an important feature, and especially where their accurate computation is difficult by other methods. Computational continuum mechanics abounds in such problems, which in...

We present a front tracking algorithm for the solution of the 2D incompressible Navier-Stokes equations with interfaces and surface forces. More particularly, we focus our attention on the accurate description of the surface tension terms and the associated pressure jump. We consider the stationary Laplace solution for a bubble with surface tension. A careful treatment of the pressure gradient terms at the interface allows us to reduce the spurious currents to the machine precision. Good results are obtained for the oscillation of a capillary wave compared with the linear viscous theory. A classical test of Rayleigh-Taylor instability is presented.

This article is devoted to the description and assessment of a numerical procedure for the simulation of flows with interfaces between viscous Newtonian fluids. The interfaces are modeled as discontinuities with constant surface tension. This physical model is relevant for many applications. Of particular interest to us are phenomena such as droplet formation and breakup where interface topology may change through the reconnection of the interface. The method may also be useful to study complex multiphase flows, when for instance the fluid particles undergo three-dimensional perturbations.

Numerical simulations describing plunging breakers including the splash-up phenomenon are presented. The motion is governed by the classical, incompressible, two-dimensional Navier-Stokes equation. The numerical modelling of this two-phase flow is based on a piecewise linear version of the volume of fluid method. Capillary effects are taken into account as a stress tensor computed from gradients of the volume fraction function. Preliminary results concerning the time evolution of liquid--gas interface and vorticity field are given for short waves, showing how an initial steep wave undergoes breaking and successive splash-up cycles. Different evolutions of the wave energy are observed during the breaking stage. The energy dissipation due to viscosity becomes significantly important at each time of the impact of jet and the formation of splash-up. It is found that nearly 70% of the wave energy is lost after about three periods. 1 Plunging breakers are due to the formation o...

. In this paper, we establish a Front Tracking method to solve Rayleigh-Taylor (RT) and RichtmyerMeshkov (RM) Instabilities in axi-symmetric cylindrical and spherical geometries. Validation is carried out by comparing the computed single mode bubble velocity with various theoretical models and experimental results. We also validate our results by comparing three different front propagation algorithms, mesh refinement and the comparison of the asymptotic limit of the minimum radius r min ! 1 to a pure planar computation in two dimensions. In the cylindrical RT simulations, we study the influence of the geometry on the bubble velocities. We achieve convergence of bubble velocities as the minimum radius r min ! 0. We observe an interesting monotonic dependence of the bubble velocity on r min . For the RM simulations, we perform a detailed study of the growth rate of fingers at an unstable shell driven by an imploding spherical shock. A qualitative understanding of this system has...

This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [69]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper (Part II [66]), we analyze the behavior of the models in the fully nonline...

A space-time discontinuous Galerkin finite element method for two fluid flow problems is presented. By using a combination of level set and cut-cell methods the interface between two fluids is tracked in space-time. The movement of the interface in space-time is calculated by solving the level set equation, where the interface geometry is identified with the 0-level set. To enhance the accuracy of the interface approximation the level set function is advected with the interface velocity, which for this purpose is extended into the domain. Close to the interface the mesh is locally refined in such a way that the 0-level set coincides with a set of faces in the mesh. The two fluid flow equations are solved on this refined mesh. The procedure is repeated until both the mesh and the flow solution have converged to a reasonable accuracy. The method is tested on linear advection and Euler shock tube problems involving ideal gas and compressible bubbly magma. Oscillations around the interface are eliminated by choosing a suitable interface flux. Key words: cut-cell method, discontinuous Galerkin finite element method, interface tracking, level set method, space-time, two fluid flows. 1

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INTRODUCTION In numerical approximations of physical systems with discontinuous coefficients, often the largest numerical errors are introduced in a neighborhood of the discontinuities. These errors are often greatly reduced if the grid is aligned with the discontinuities and special formulas are used to incorporate the jump conditions directly into the numerical model, for example, in solving the equations governing the conservation of mass, momentum, and energy in multimaterial or multiphase flows, such as a liquid--gas or liquid--solid interface where the normal and tangential stresses must be matched at the interface or the equations of state are drastically different at an interface. When the strain and stress of materials are being modeled and the coefficients are discontinuous, the numerical solution is often extremely sensitive to the proper alignment of the control volume with the boundary. In numerical approximations of wave equations, discontinuities (e.g., the tensori

. An attractive approach for simulation of large deformation solid dynamics is to combine Eulerian finite differencing with material interface tracking. The Eulerian computational mesh is not subject to mesh distortion, and tracking eliminates spurious numerical diffusion at interfaces and the need for mixed-material computational cells. We have developed such an approach within the framework of the front tracking method, as implemented in the FronTier code. Our two-dimensional solid dynamics module is based on a fully conservative formulation of the governing equations for large-strain deformation, a hyperelastic equation of state that allows for large volume change, and a rate-dependent plasticity model. It features conservative finite differencing, use of a Riemann solver to enforce the Rankine-Hugoniot conditions at material interfaces, and an implicit method for integrating the plastic source terms. This paper presents an overview of solid dynamics in FronTier and some preliminary applications to high-velocity expanding ring and shock-accelerated interface problems. 1.