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TECHNICAL PAPERS

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

[+] Author and Article Information
Angel L. Zanotti

International Center for Computational Methods in Engineering (CIMEC), INTEC-Universidad Nacional del Litoral-CONICET, S3000GLN, Güemes 3450, Santa Fe, Argentinaazanotti@intec.unl.edu.ar

Carlos G. Méndez, Norberto M. Nigro, Mario Storti

International Center for Computational Methods in Engineering (CIMEC), INTEC-Universidad Nacional del Litoral-CONICET, S3000GLN, Güemes 3450, Santa Fe, Argentina

J. Appl. Mech 74(4), 732-740 (Oct 10, 2006) (9 pages) doi:10.1115/1.2711224 History: Received June 27, 2005; Revised October 10, 2006

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 preconditioner. 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.

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Copyright © 2007 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic representation of two-fluid model

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Figure 2

Eigenvalues for a sweeping in alpha (0.01:0.01:0.99) and velocity relations (1:5:100), without preconditioning

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Figure 3

Eigenvalues for a sweeping in alpha (0.01:0.01:0.99) and velocity relations (1:5:100), with preconditioning and γ=1. All eigenvalues are real, but are different from the real part of eigenvalues of Fig. 2.

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Figure 4

Eigenvalue for a sweeping in alpha (0.01:0.01:0.99) and velocity relations (1:5:100) with preconditioning and γ≠1. All eigenvalues are real and are equal to the real part of eigenvalues of Fig. 2.

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Figure 5

The water faucet problem. Void fraction with preconditioning at steady state, for a mesh of 320 uniform lineal elements. Comparison between numerical and analytical solutions.

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Figure 6

The water faucet problem. Liquid velocity with preconditioning at steady state, for a mesh of 320 uniform lineal elements. Comparison between numerical and analytical solutions.

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Figure 7

The water faucet problem. Void fraction with preconditioning at t=0.4s, for six different meshes of 40, 80, 160, 320, 640, and 1280 uniform lineal elements.

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Figure 8

The water faucet problem. Liquid velocity with preconditioning at t=0.4s, for six different meshes of 40, 80, 160, 320, 640, and 1280 uniform lineal elements.

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Figure 9

The wave traveling problem. Void fraction without preconditioning at four time steps of 0.001s.

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Figure 10

The wave traveling problem. Gas velocity without preconditioning at four time steps of 0.001s.

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Figure 11

The wave traveling problem. Void fraction with preconditioning at five time steps of 0.001s.

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Figure 12

The wave traveling problem. Gas velocity with preconditioning at five time steps of 0.001s.

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Figure 13

The wave traveling problem. Void fraction with preconditioning at 25 time steps of 0.001s.

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Figure 14

The wave traveling problem. Gas velocity with preconditioning at 25 time steps of 0.001s.

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