AN ADAPTIVE FINITE ELEMENT METHOD FOR INCOMPRESSIBLE FLOWS WITH INTERFACES

An adaptive finite element method for solving incompressible steady state free surface flow problems is presented. Although the method is primarily aimed at material forming processes, such as coextrusion, it is applicable to many other multifluid flow problems. An Eulerian stabilized finite element formulation is proposed to solve the coupled system of partial differential equations. Adaptive remeshing is applied to improve the accuracy of the computed solution. The methodology is verified by solving a problem with a know closed form solution. The methodology is then applied to the coextrusion of polymers in an industrial annular die. Steady state drop dynamics simulations complete the paper.

[1]  John W. Dold,et al.  An efficient surface-integral algorithm applied to unsteady gravity waves , 1992 .

[2]  A. Jameson,et al.  A finite volume method for transonic potential flow calculations , 1977 .

[3]  J. Agassant,et al.  Prediction of stationary interfaces in coextrusion flows , 1994 .

[4]  Dominique Pelletier,et al.  AN ADAPTIVE FINITE ELEMENT METHOD FOR A TWO- EQUATION TURBULENCE MODEL IN WALL-BOUNDED FLOWS , 1997 .

[5]  Dominique Pelletier,et al.  Adaptive remeshing for convective heat transfer with variable fluid properties , 1994 .

[6]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 1: The recovery technique , 1992 .

[7]  Rainald Löhner.  Adaptive remeshing for transient problems , 1989 .

[8]  Gareth P. Williams.  Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow , 1969, Journal of Fluid Mechanics.

[9]  O. C. Zienkiewicz,et al.  Adaptive remeshing for compressible flow computations , 1987 .

[10]  S. Lo.  A NEW MESH GENERATION SCHEME FOR ARBITRARY PLANAR DOMAINS , 1985 .

[11]  Charles L. Lawson,et al.  Transforming triangulations , 1972, Discret. Math..

[12]  J. Z. Zhu,et al.  The superconvergent patch recovery and a posteriori error estimates. Part 2: Error estimates and adaptivity , 1992 .

[13]  J. Brackbill,et al.  A continuum method for modeling surface tension , 1992 .

[14]  Dominique Pelletier,et al.  ADAPTIVE REMESHING FOR HYPERBOLIC TRANSPORT PROBLEMS , 1994 .

[15]  C. Han,et al.  Multiphase flow in polymer processing , 1981 .

[16]  Roland W. Lewis,et al.  Finite element modelling of surface tension effects using a Lagrangian-Eulerian kinematic description , 1997 .

[17]  Dominique Pelletier,et al.  Fast, Adaptive Finite Element Scheme for Viscous Incompressible Flows , 1992 .

[18]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[19]  Andrew N. Hrymak,et al.  Transient free‐surface flows in injection mold filling , 1988 .

[20]  Yves Demay,et al.  A Two-dimensional Numerical Method for the Deformation of Drops with Surface Tension , 1998 .

[21]  S. Rebay.  Efficient Unstructured Mesh Generation by Means of Delaunay Triangulation and Bowyer-Watson Algorithm , 1993 .

[22]  J. T. Oden,et al.  A posteriori error estimation of finite element approximations in fluid mechanics , 1990 .

[23]  A. Hrymak,et al.  Three-dimensional studies on bicomponent extrusion , 1990 .

[24]  Rainald Löhner,et al.  An adaptive finite element solver for transient problems with moving bodies , 1988 .

[25]  Steven A. Orszag,et al.  Generalized vortex methods for free-surface flow problems , 1982, Journal of Fluid Mechanics.

[26]  A. Jameson.  Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings , 1991 .

[27]  L. Quartapelle.  The incompressible Navier—Stokes equations , 1993 .

[28]  M. Crochet,et al.  Stratified flows of Newtonian and viscoelastic fluids , 1989 .

[29]  C. W. Hirt,et al.  Volume of fluid (VOF) method for the dynamics of free boundaries , 1981 .

[30]  J. Peraire,et al.  Finite Element Flux-Corrected Transport (FEM-FCT) for the Euler and Navier-Stokes equations , 1987 .

[31]  L. Franca,et al.  Stabilized finite element methods. II: The incompressible Navier-Stokes equations , 1992 .

[32]  Andrew N. Hrymak,et al.  Finite‐element simulation of stratified multiphase flows , 1987 .

[33]  Howard A. Stone,et al.  Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid , 1989, Journal of Fluid Mechanics.

[34]  T. Vinje,et al.  Numerical simulation of breaking waves , 1981 .

[35]  Arthur Rizzi,et al.  Computation of inviscid incompressible flow with rotation , 1985, Journal of Fluid Mechanics.

[36]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[37]  Erik G. Thompson,et al.  Use of pseudo‐concentrations to follow creeping viscous flows during transient analysis , 1986 .

[38]  Howard A. Stone,et al.  Dynamics of Drop Deformation and Breakup in Viscous Fluids , 1994 .

[39]  M. Fortin,et al.  A generalization of Uzawa's algorithm for the solution of the Navier-Stokes equations , 1985 .

[40]  Antony Jameson,et al.  A new implicit algorithm with multigrid for unsteady incompressible flow calculations , 1995 .

[41]  Paresh Parikh,et al.  Generation of three-dimensional unstructured grids by the advancing-front method , 1988 .

[42]  S. Orszag,et al.  Approximation of radiation boundary conditions , 1981 .

[43]  Pierre J. Carreau,et al.  Rheology of Polymeric Systems: Principles and Applications , 1997, Engineering Plastics.

[44]  Michael Selwyn Longuet-Higgins,et al.  The deformation of steep surface waves on water - I. A numerical method of computation , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[45]  Jerzy M. Floryan,et al.  Numerical Methods for Viscous Flows With Moving Boundaries , 1989 .

[46]  Yves Demay,et al.  Numerical simulation of coextrusion and film casting , 1995 .

[47]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[48]  Dick K. P. Yue,et al.  Deep-water plunging breakers: a comparison between potential theory and experiments , 1988 .

[49]  Antony Jameson,et al.  Multigrid algorithms for compressible flow calculations , 1986 .

[50]  M. Crochet,et al.  Newtonian stratified flow through an abrupt expansion , 1987 .

[51]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .