A level set characteristic Galerkin finite element method for free surface flows

We present a numerical method for free surface flows that couples the incompressible Navier-Stokes equations with the level set method in the finite element framework. The implicit characteristic-Galerkin approximation together with the fractional four-step algorithm is employed to discretize the governing equations. The schemes for solving the level set evolution and reinitialization equations are verified with several benchmark cases, including stationary circle, rotation of a slotted disk and stretching of a circular fluid element. The results are compared with those calculated from the level set finite volume method of Yue et al. , which employed the third-order essentially non-oscillatory (ENO) schemes for advection of the level set function in a generalized curvilinear coordinate system

[1]  T. Tezduyar,et al.  Parallel finite element computation of free-surface flows , 1999 .

[2]  C. W. Hirt,et al.  An Arbitrary Lagrangian-Eulerian Computing Method for All Flow Speeds , 1997 .

[3]  Haecheon Choi,et al.  A fractional four-step finite element formulation of the unsteady incompressible Navier-Stokes equations using SUPG and linear equal-order element methods , 1997 .

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

[5]  W. Rider,et al.  Reconstructing Volume Tracking , 1998 .

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

[7]  Robert L. Street,et al.  A computer study of finite-amplitude water waves , 1970 .

[8]  P. Moin,et al.  Application of a Fractional-Step Method to Incompressible Navier-Stokes Equations , 1984 .

[9]  J. Z. Zhu,et al.  The finite element method , 1977 .

[10]  Richard H. Pletcher,et al.  The Development of a Free Surface Capturing Approach for Multidimensional Free Surface Flows in Closed Containers , 1997 .

[11]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[12]  V. C. Patel,et al.  Numerical simulation of unsteady multidimensional free surface motions by level set method , 2003 .

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

[14]  A. Chorin.  Numerical solution of the Navier-Stokes equations , 1968 .

[15]  Francis H. Harlow,et al.  Numerical Study of Large‐Amplitude Free‐Surface Motions , 1966 .

[16]  W. P. Crowley.  Flag: A free-Lagrange method for numerically simulating hydrodynamic flows in two dimensions , 1971 .

[17]  O. Zienkiewicz,et al.  A general algorithm for compressible and incompressible flow—Part I. the split, characteristic‐based scheme , 1995 .

[18]  R. LeVeque.  High-resolution conservative algorithms for advection in incompressible flow , 1996 .

[19]  M. Manzan,et al.  Analysis of finite element schemes for convection‐type problems , 1995 .

[20]  Balasubramaniam Ramaswamy,et al.  Numerical simulation of unsteady viscous free surface flow , 1990 .

[21]  S. Osher,et al.  Regular Article: A PDE-Based Fast Local Level Set Method , 1999 .

[22]  R. Fedkiw,et al.  A Boundary Condition Capturing Method for Poisson's Equation on Irregular Domains , 2000 .

[23]  G. Tryggvason,et al.  A front-tracking method for viscous, incompressible, multi-fluid flows , 1992 .

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

[25]  J. Yoo,et al.  Time-accurate computation of unsteady free surface flows using an ALE-segregated equal-order FEM , 2000 .

[26]  M. Quecedo,et al.  Application of the level set method to the finite element solution of two-phase flows , 2001 .

[27]  Danesh K. Tafti,et al.  Comparison of some upwind-biased high-order formulations with a second-order central-difference scheme for time integration of the incompressible Navier-Stokes equations , 1996 .

[28]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[29]  M. Rudman.  INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, VOL. 24, 671–691 (1997) VOLUME-TRACKING METHODS FOR INTERFACIAL FLOW CALCULATIONS , 2022 .

[30]  Gordon D. Stubley,et al.  Surface-adaptive finite-volume method for solving free surface flows , 1994 .

[31]  Woo Il Lee,et al.  A new VOF‐based numerical scheme for the simulation of fluid flow with free surface. Part I: New free surface‐tracking algorithm and its verification , 2003 .

[32]  C. K. Thornhill,et al.  Part IV. An experimental study of the collapse of liquid columns on a rigid horizontal plane , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[33]  T. Belytschko,et al.  An Extended Finite Element Method for Two-Phase Fluids , 2003 .

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

[35]  Robert A. Dalrymple,et al.  Using a Three-Dimensional Smoothed Particle Hydrodynamics Method for Wave Impact on a Tall Structure , 2004 .