A Preconditioning Mass Matrix to Avoid the Ill-Posed Two-Fluid Model

Two-fluid models are central to the simulation of transport processes in two-phase homogenized systems. Even though this physical model has been widely accepted, an inherently nonhyperbolic and nonconservative ill-posed problem arises from the mathematical point of view. It has been demonstrated that this drawback occurs even for a very simplified model, i.e., an inviscid model with no interfacial terms. Much effort has been made to remedy this anomaly and in the literature two different types of approaches can be found. On one hand, extra terms with physical origin are added to model the interphase interaction, but even though this methodology seems to be realistic, several extra parameters arise from each added term with the associated difficulty in their estimation. On the other hand, mathematical based-work has been done to find a way to remove the complex eigenvalues obtained with two- fluid model equations. Preconditioned systems, characterized as a projection of the complex eigenvalues over the real axis, may be one of the choices. The aim of this paper is to introduce a simple and novel mathematical strategy based on the application of a preconditioning mass matrix that circumvents the drawback caused by the nonhyperbolic behavior of the original model. Although the mass and momentum conservation equations are modified, the target of this methodology is to present another way to reach a steady-state solution (using a time marching scheme), greatly valued by researchers in industrial process design. Attaining this goal is possible because only the temporal term is affected by the preconditioned The obtained matrix has two parameters that correct the nonhyperbolic behavior of the model: the first one modifies the eigenvalues removing their imaginary part and the second one recovers the real part of the original eigenvalues. Besides the theoretical development of the preconditioning matrix, several numerical results are presented to show the validity of the method.

[1]  Keun-Shik Chang,et al.  Pressure wave speeds from the characteristics of two fluids, two-phase hyperbolic equation system , 1998 .

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

[3]  Andrea Prosperetti,et al.  Ensemble phase‐averaged equations for bubbly flows , 1994 .

[4]  J. Brock,et al.  Volume tracking of interfaces having surface tension in two and three dimensions , 1996 .

[5]  D. Jacqmin.  Regular Article: Calculation of Two-Phase Navier–Stokes Flows Using Phase-Field Modeling , 1999 .

[6]  J. A. Trapp,et al.  A nearly-implicit hydrodynamic numerical scheme for two-phase flows , 1986 .

[7]  V. H. Ransom.  RELAP5/MOD2 code manual , 1985 .

[8]  Frédéric Coquel,et al.  A Numerical Method Using Upwind Schemes for the Resolution of Two-Phase Flows , 1997 .

[9]  D. M. Anderson,et al.  DIFFUSE-INTERFACE METHODS IN FLUID MECHANICS , 1997 .

[10]  Drag and non-drag force influences in numerical simulations of metallurgical ladles , 2005 .

[11]  E. Turkel,et al.  PRECONDITIONING TECHNIQUES IN COMPUTATIONAL FLUID DYNAMICS , 1999 .

[12]  D. B. Kothe,et al.  RIPPLE: A NEW MODEL FOR INCOMPRESSIBLE FLOWS WITH FREE SURFACES , 1991 .

[13]  Jr.,et al.  Fixing Einstein's equations , 1999, gr-qc/9901021.

[14]  D. Bestion,et al.  The physical closure laws in the CATHARE code , 1990 .

[15]  S. Osher,et al.  A Non-oscillatory Eulerian Approach to Interfaces in Multimaterial Flows (the Ghost Fluid Method) , 1999 .

[16]  V. H. Ransom,et al.  Regularization of Two-Phase Flow Models , 1997 .

[17]  J. Saut,et al.  Short-wave instabilities and ill-posed initial-value problems , 1990 .

[18]  A. Chorin.  A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[19]  Peter Smereka,et al.  Axisymmetric free boundary problems , 1997, Journal of Fluid Mechanics.

[20]  D. Drew,et al.  The three-dimensional time and volume averaged conservation equations of two-phase flow , 1988 .

[21]  Jean-Marc Hérard,et al.  Closure laws for a two-fluid two-pressure model , 2002 .

[22]  N. Zuber,et al.  Drag coefficient and relative velocity in bubbly, droplet or particulate flows , 1979 .

[23]  S. Osher,et al.  Level set methods: an overview and some recent results , 2001 .

[24]  D. Drew,et al.  Theory of Multicomponent Fluids , 1998 .

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

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

[27]  H. Städtke,et al.  Numerical simulation of multi-dimensional two-phase flow based on flux vector splitting , 1997 .

[28]  J. Lowengrub,et al.  Quasi–incompressible Cahn–Hilliard fluids and topological transitions , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[29]  Keun-Shik Chang,et al.  Numerical solution of hyperbolic two-fluid two-phase flow model with non-reflecting boundary conditions , 2002 .

[30]  Oscar Reula.  STRONGLY HYPERBOLIC SYSTEMS IN GENERAL RELATIVITY , 2004 .

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

[32]  D. Juric,et al.  Computations of Boiling Flows , 1998 .

[33]  Yang-Yao Niu,et al.  Advection upwinding splitting method to solve a compressible two‐fluid model , 2001 .

[34]  Abraham M. Lenhoff,et al.  Drop formation in liquid-liquid systems before and after jetting , 1995 .

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

[36]  C. Gundlach,et al.  Hyperbolicity of second order in space systems of evolution equations , 2005, gr-qc/0506037.

[37]  Andrea Prosperetti,et al.  ON THE SUITABILITY OF FIRST-ORDER DIFFERENTIAL MODELS FOR TWO-PHASE FLOW PREDICTION , 1985 .