A Preconditioning Mass Matrix to Avoid the Ill-Posed Two-Fluid Model
Abstract
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 non-hyperbolic and
non-conservative 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. Lots of efforts have 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 the 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 non-hyperbolic 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 non-hyperbolic 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. [To appear in Journal of Applied Mechanics]
processes in two-phase homogenized systems. Even though this physical
model has been widely accepted, an inherently non-hyperbolic and
non-conservative 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. Lots of efforts have 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 the 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 non-hyperbolic 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 non-hyperbolic 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. [To appear in Journal of Applied Mechanics]