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Computer Aided Geometric Design
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Computer Aided Geometric Design
www.elsevier.com/locate/cagd
Topology analysis of time-dependent multi-fluid data using the Reeb
graph
Fang Chen
a
,
, Harald Obermaier
b
, Hans Hagen
a
, Bernd Hamann
b
, Julien Tierny
c
,
Valerio Pascucci
c
a
Department of Computer Science, University of Kaiserslautern, Kaiserslautern 67663, Germany
b
Institute for Data Analysis and Visualization, Department of Computer Science, University of California, Davis, One Shields Avenue, Davis, CA 9561
6, USA
c
72 South, Central Campus Drive, Salt Lake City, UT 84112, USA
article info
abstract
Article history:
Available online xxxx
Keywords:
Multi-phase fluid
Level set
Topology method
Point-based multi-fluid simulation
Liquid–liquid extraction is a typical multi-fluid problem in chemical engineering where
two types of immiscible fluids are mixed together. Mixing of two-phase fluids results in
a time-varying fluid density distribution, quantitatively indicating the presence of liquid
phases. For engineers who design extraction devices, it is crucial to understand the density
distribution of each fluid, particularly flow regions that have a high concentration of the
dispersed phase. The propagation of regions of high density can be studied by examining
the topology of isosurfaces of the density data. We present a topology-based approach to
track the splitting and merging events of these regions using the Reeb graphs. Time is used
as the third dimension in addition to two-dimensional (2D) point-based simulation data.
Due to low time resolution of the input data set, a physics-based interpolation scheme is
required in order to improve the accuracy of the proposed topology tracking method. The
model used for interpolation produces a smooth time-dependent density field by applying
Lagrangian-based advection to the given simulated point cloud data, conforming to the
physical laws of flow evolution. Using the Reeb graph, the spatial and temporal locations
of bifurcation and merging events can be readily identified supporting in-depth analysis of
the extraction process.
©
2012 Elsevier B.V. All rights reserved.
1. Introduction
In chemical engineering, liquid–liquid extraction is a widely used method, where compounds of one liquid are separated
by mixing it with a finely dispersed solvent (
Drumm et al., 2008
). Numerical simulations model this process as a multi-fluid
flow with two liquid phases. These multi-fluid simulations can be used to improve the design parameters of extraction
devices in order to optimize the extraction process.
The desired output of these simulations is the predicted evolution of the density distribution for each of the phases,
which indicates how well the two liquids are mixed. The density of a phase in a given space denotes the fraction of
the space occupied by the fluid. In liquid–liquid extraction, a finely dispersed solvent results in low density values for the
second phase in large regions of the data set. Examples for Computational Fluid Dynamics (CFD) codes capable of simulating
This research was supported by the International Research and Training Group at the University of Kaiserslautern (IRTG) and Deutsche
Forschungsgemeinschaft (DFG, German research foundation).
*
Corresponding author.
E-mail addresses:
chen@cs.uni-kl.de
(F. Chen),
harald.obermaier@itwm.fhg.de
(H. Obermaier),
hagen@cs.uni-kl.de
(H. Hagen),
hamann@cs.ucdavis.edu
(B. Hamann),
jtierny@sci.utah.edu
(J. Tierny),
pascucci@sci.utah.edu
(V. Pascucci).
0167-8396/$ – see front matter
©
2012 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.cagd.2012.03.019
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two-phase flows are Fluent (commercial), OpenFoam (open source) and finite point set method (FPM) (
Kuhnert and Tiwari,
2003
). The latter is used throughout this work and produces scattered point sets, carrying velocity and density information.
Typically, there are three types of multi-phase fluid models tackling different flow regimes:
volume of fluid
(VOF) modeling
which focuses on tracking the interface of the two fluids for slug and surface flow; Eulerian multi-phase modeling which
deals with heat and momentum transfer between the phases; and a discrete phase modeling of the mixture. The last type
of modeling is widely used for simulating bubbly flow and slurry flow, such as fluid mixture in a bubble column reactor
for liquid–liquid extraction. However, traditional approaches of capturing fluid material boundaries cannot handle the case
where multiple regions within a cell are occupied by different fluids, as is the case in finely dispersed liquids. Thus a higher
level visualization technique is required for the understanding of these flow density distributions.
We present a topology-based approach for studying the volume fraction field given by an arbitrarily distributed numer-
ically computed point set. The simulation data we are looking at is a two-dimensional, time-varying fluid field discretized
by particles with associated density and velocity values. The low time resolution of this reference data set complicates
the tracking of material boundaries, as a non-physically-based interpolation scheme can result in wrong topology. For this
matter, we make use of a physically-based interpolation scheme that improves correspondences between time steps of the
simulation and allows for more robust feature extraction.
The major goal of this paper is to develop plausible and practical interpolation schemes for point-based multi-fluid
density data, and to characterize fluid interface behavior with Reeb graphs. To identify interesting time intervals where
regions that are densely occupied by a certain phase of fluid split or merge, we first define these regions by a certain
level
set
of the density field. Using a physically-based interpolation scheme, we compute a time-continuous density field at a
re-sample grid points. Finally we carry out a topological analysis of the extracted time-varying level sets using the Reeb
graph. The major contributions of this work are as follows:
It proposes an interpolation scheme for point-based time-dependent density data sets which have no connectivity
information. The proposed interpolation scheme is capable of handling sparse data with large time intervals, preserving
both the physical properties as well as topology of the flow.
It introduces a framework to extract and analyze fluid interface topology. The framework is practical for the analysis
point-based multi-fluid data sets. It offers novel views and tools for domain experts for further analysis and estimation
of solvent efficiency.
The paper is organized as follows: In Section
1.1
, we introduce related work about particle-based fluid simulation and
topology-based feature tracking techniques. In Section
2
, suitable interpolation schemes are applied separately for time
and spatial direction in order to obtain a topological-clean extraction of the level sets. Section
3
contains the example and
topology analysis of our result. The final section highlights some areas of possible improvements as well as future work
concerning our method.
1.1. Related work
A
level set
of a scalar field is given by the set of points with identical scalar value. Our idea of studying the level sets
of the density field was inspired by existing research which focuses on material boundary and fluid interface tracking,
including the
front tracking
(
FT
)
method
(
Unverdi and Tryggvason, 1992
),
level set method
(LSM) (
Osher and Sethian, 1988
),
and
volume of fluid method
(
VOF
)(
Hirt and Nichols, 1981
).
The FT method (
Unverdi and Tryggvason, 1992; Terashima and Tryggvason, 2009; Gloth et al., 2003
)advectsthemarked
interface from an initial configuration and keeps the topology of the interface constant during the simulation. Therefore,
this method is limited to topological changes in multi-phase fluids, such as merging or breaking of droplets.
The LSM was introduced by Osher (
1988
) in 1988. The material boundary or interface is defined as the zero set (
Osher
and Fedkiw, 2001
,
2003
;
Sethian, 1985
) of the given scalar field. Sethian (
2003
) and Lakehal (
2002
) applied the concept to
fluid simulation. In 2002, Enright et al. (
2002
) combined Lagrangian marker particles with LSM to obtain and maintain a
smooth geometrical description of the fluid interface. However, it has been pointed out by Müller (
2009
) and Garimella et
al. (
2005
) that material volume is not well-preserved in the level set method, which is a main drawback of this approach.
The VOF method (
Hirt and Nichols, 1981
) is one of the best established interface volume tracking methods currently
in use (
Marek et al., 2008; Sussman and Puckett, 2000
). It employs the idea of using mass conservation for the volume of
each fluid. Apart from VOF, other interface tracking algorithms include
simple line interface
(SLIC) (
Noh and Woodward, 1976
)
and piecewise linear interface construction (PLIC) (
Rider and Kothe, 1998
). In the SLIC method, the interface is taken to be
perpendicular or parallel to coordinate axes directions, while in PLIC, the interface is given by a piecewise linear function
with arbitrary orientation.
However, the interface tracking algorithms mentioned above do not fully apply to liquid–liquid extraction simulation
as one phase of fluids is fully dispersed and no continuous interface exists between the two fluids. Therefore, a material
interface is not traceable in the case of slurry flow. Instead, topology-based techniques for the analysis of level sets are
more suitable and useful in this context because of their ability to capture the absolute and relative behavior of small-scale
features like dispersed bubbles directly (
Tierny et al., 2009
).