Moving Interface Problems and Applications in Fluid Dynamics
(8 Jan - 31 Mar 2007)
~ Abstracts ~
Simulations of front evolution in
the filling process of fibre suspension flow using level set
method
Huashu Dou, Temasek Laboratories, NUS, Singapore
The simulation of fibre orientation in dilute suspension
with a front moving is carried out using the projection and
level set methods. The motion of fibres is described using
the Jeffery equation and the contribution of fibres to the
flow is accounted for by the configuration field method. The
dilute suspension of short fibres in Newtonian fluids is
considered. The governing Navier-Stokes equation for the
fluid flow is solved using the projection method with finite
difference scheme, while the fibre-related equations are
directly solved with the Runge-Kutta method. In the present
study for fibres in dilute suspension flow as for injection
molding, the effects of various flow parameters on the fibre
orientation and the velocity distributions as well as the
shapes of the leading flow front are found and discussed.
Our findings indicate that the presence of fibres motion has
little influence on the front shape in the ranges of fibre
parameters studied at the fixed Reynolds number. Influence
of changing fibre parameters only causes variation of front
shape in the region near the wall and the front shape in the
central core area does not vary much with the fibre
parameters. On the other hand, the fibre motion has strong
influence on the distributions of the streamwise and
transverse velocities in the fountain flow. Fibre motion
produces strong normal stress near the wall which leads to
the reduction of transversal velocity as compared to the
Newtonian flow without fibres and in turn the streamwise
velocity near the wall is increased. Thus, the fibre
addition to the flow weakens the strength of the fountain
flow. The Reynolds number has also displayed significant
influence on the distribution of the streamwise velocity
behind the flow front for a given fibre concentration. It is
also found that the fibre orientation is not always along
the direction of velocity vector in the process of mold
filling. In the region of the fountain flow, the fibre near
the centerline is more oriented cross the streamwise
direction comparing to that in the region far behind the
flow front. This leads to that the fibre near the centreline
in the region of fountain flow is more extended along the
transverse direction. Since fibre orientation in the
suspension flow and the shape of the flow front have great
bearing on the quality of the product made from injection
molding, this study has much implications for engineering
applications. These results can also be useful in other
field dealing with fibre suspensions.
Spinning motion of detonation
captured by 3D numerical simulation
Huashu Dou, Temasek Laboratories, NUS, Singapore
The spinning motion of detonation was found in the early
experiments in the 1920’s and also reported in studies in
recent years. This is one kind of unstable motion and it
strongly affects the self-sustenance of the detonation. It
is of importance to analyze the mechanism leading to the
spinning motion in the detonation and its influence on
practical applications. However, the spinning motion of
detonation is three-dimensional and the detonation behaviour
depends on the spatial and temporary evolution of the
detonation variables. Currently, there are only few studies
on three-dimensional simulation for the detonation waves.
Recently, we captured the spinning motion of detonation by
three-dimensional numerical simulation for a narrow square
duct. The governing equation and the numerical method are as
follow. The systems of conservative laws of inviscid fluid
combined with the one-step chemical reaction model are
discretized in a Cartesian coordinates using the fifth-order
WENO (Weighted Essentially NonOscillatory) scheme, and the
final discretized variables are solved with a 3rd order TVD
Rouge-Kutta method. Then, the process of the formation of
the detonation pattern from the premixed gaseous state in 3D
space is observed from the simulation results. The
simulation shows that under an initial disturbance, the
detonation front finally develops to an unsteady
three-dimensional distorted pattern and which translates
between the walls. For a narrow duct, the flow front
displays an unmistakenly spinning motion with a period. But,
for the wide duct, the flow front shows a quasi-steady
“rectangular mode” periodically, no spinning motion occurs.
The mechanism of occurrence of spinning detonation has been
analysed. The simulation result reveals that there are
certainly some differences between the three-dimensional and
two-dimensional detonations. The detonation structure and
the reaction process are more complex in three-dimensional
case. In reality, detonation is always three-dimensional and
the variations of all the flow and reaction variables are in
3D space. Maybe only by careful three-dimensional
simulation, the physical mechanism of detonation can be
further understood.
Applications of shock dynamic and
bubble dynamic research to development of therapeutic
devices
Kazuyoshi Takayama, Tohoku University, Japan
We started our underwater shock wave research in 1975 and
realized underwater shock wave dynamics and bubble dynamics
complement each other. We also struggled to develop
quantitative visualization for flow diagnostics and
eventually reached to work for double exposure holographic
interferometry. My talk contains our early efforts of
cavitation research and micro- explosions. We at first
expanded our understanding to extracorporeal shock wave
lithotripsy (ESWL) in 1982 under collaboration with
urologists in School of Medicine Tohoku University. Later we
became interested in tissue damages associated with ESWL,
which were in principle caused by shock/bubble interaction.
We initiated to damages tissues in a controlled fashion and
expanded it initially to the revascularization of cerebral
embolism. Polishing ideas and improving methodology, we are
now working for development of various therapeutic devices,
for example, to cure pulmonary infarction, and arrhythmia.
With laser induced micro-water jet dissection devices, we
can dissect soft tissues by preserving micro-blood vessels
of over 0.2 mm diameter and deliver dry drug by laser
ablation assisted system. We believe these are one of the
most peaceful interdisciplinary applications of shock wave
research.
Modelling hydrodynamic
interactions between deformable drops
Rogerio Manica, University of Melbourne, Australia
Understanding deformations during interaction of
colloidal or nano particles has important implications in a
wide range of applications such as flotation collection and
emulsion stability. This work developed evolution equations
to model experiments performed with the Atomic Force
Microscope (AFM) in which the forces of interaction between
two deformable drops of radius about 50 microns approaching
each other at velocities of order 10 microns per second were
calculated. The same model was adapted to obtain profiles of
a mercury drop (radius about 2 millimetres) approaching a
flat mica surface and compared with experiments performed
using the Surface Force Apparatus (SFA). The thin films
generated on both problems are on the order of 50 nanometres.
The important feature of the model is the use of matched
asymptotic expansions to derive a new boundary condition for
moving drops to obtain results that are independent of the
size of the computational domain. There is excellent
agreement between both experiments and the theoretical model
which allow us to understand the separate contributions of
hydrodynamic interaction, surface force and drop deformation
in giving rise to observed behavior in such systems.
Bacteria-substrate interaction:
can mathematical models help?
Anil Kishen, National University of Singapore
Adherence of bacteria to a biomaterial surface and
formation of biofilm structure is the major cause of
biomaterial centered infections (BCI). Presence of
biomaterials in close proximity to host-immune system can
increase the susceptibility to such infections. The
adherence of bacteria to biomaterial surface is the first
and the most important step in BCI. This lecture will
address (1) different factors facilitating bacterial
adherence to collagen and dentine substrate and (2)
different stages in the interaction between bacteria and
dentine. Our experiments highlighted the influence of physio-chemical
properties of bacterial cell and substrate on bacterial
adherence to substrate. These experiments emphasized the
need of mathematical modeling to understand
bacteria-substrate interaction and the role of different
factors in promoting BCI.
A coupled immersed
interface-boundary element method for the simulation of cell
deformation in single-cell traps
Le Duc Vinh, Singapore-MIT Alliance, Singapore
We present a coupled Immersed Interface Method - Boundary
Element Method (IIM-BEM) numerical technique that can
explore the behavior of target cells in detail, and describe
the cell deformation and motion under the effects of both
the electric and the flow fields. We apply this technique to
the analysis of the characteristics of a hybrid
electrical-mechanical trap for single-cell trapping. The
Immersed Interface Method provides the means of calculating
hydrodynamic effects and fluid-structure interaction effects
such as cell deformation, and the Boundary Element Method is
used to calculate the electric fields and their effects on
the particle. We report on the effect of different
combinations of electrode positions and mechanical
properties of the trap on the maximum loading and unloading
Reynolds numbers, and show the effect that cells moving with
the flow have on cells which have been already trapped.
Modeling and simulations of nasal
obstruction
Heow Pueh Lee, Institute of High Performance Computing,
Singapore
The nose is a natural and physiological respiratory
passageway with approximately 10,000-20,000 liters of air
moving through it daily on their way to the lungs. It has
important physiological functions of conditioning and
filtering inspired air, as well as immune defence function
as the nasal mucosa is the first site of interaction between
the host tissue and foreign invaders (i.e., bacteria,
allergens and etc). On the other hand, nasal obstruction,
one of the most common symptoms, is difficult to quantify by
clinical examination with the available techniques in
clinical practice. Moreover, consequence or impairment of
human airway physiology caused by nasal obstruction remains
unclear. The aim of this study is to establish a
computational model of human nose from MRI/CT scans and to
investigate the effect of geometrical configuration of human
nose in health and disease, and their impact on nasal
airflow and its related physiologic functions.
Free-Lagrange simulations of
planar shock and lithotripter shock interactions with an air
bubble
Cary Turangan, Institute of High Performance Computing,
Singapore
A free-Lagrange hydrocode, Vucalm, is used to simulate
the axisymmetric jetting collapse of air bubbles in water.
The simulations are performed for both planar step shock-
and lithotripter shock-induced collapses of initially stable
air bubbles. For planar step shock case, simulations are
conducted for a bubble of initial radius 1.0 mm located near
an aluminium layer and a shock strength of P = {528 MPa, 264
MPa, 132 MPa}. The simulations suggest that the interaction
deforms the bubble to create a high speed jet that pierces
the bubble and the collapse is violent. The jet eventually
impacts the downstream wall of the bubble and the surface of
the aluminium emitting an intense blast wave. The bubble
continues to contract, collapses and is fragmented by the
violent collapse, and the jet impact on the aluminium as
well as the blast wave emission cause plastic deformation on
the aluminium. For the shock-induced collapse, a
lithotripter shock, consisting of 56 MPa compressive and -10
MPa tensile waves, interacts with a bubble of initial radius
0.04 mm located in a free-field (case 1) and near a rigid
boundary (case 2). The interaction of the shock with the
bubble causes it to involute and forms a liquid jet that
achieves a velocity exceeding 1.2km/s for case 1 and 2.6
km/s for case 2. The impact of the jet on the downstream
wall of the bubble generates a blast wave with peak
overpressure exceeding 1 GPa and 1.75 GPa for case 1 and 2,
respectively. The results show that the simulation technique
retains sharply resolved gas/liquid interfaces regardless of
the degree of geometric deformation, and reveal details of
the dynamics of bubble collapse. The effects of
compressibility are included for both liquid and gas phases,
while stress distributions can be predicted within
elastic-plastic solid surfaces (aluminium) in proximity to
cavitation events.
Experimental study of single
cell capillary obstruction arising from human diseases
Chwee Teck Lim, National University of Singapore
It has been known that any deviation in the structural
and mechanical properties of a living cell can not only
result in the breakdown of its physiological functions, but
may also lead to diseases. For example, red blood cells
(RBC) transport oxygen to the various parts of the human
body by squeezing their way through narrow capillaries.
However, these cells are also highly coveted by the
protozoan Plasmodium falciparum, the single-cell parasites
that cause malaria. The parasite invades the RBC and
releases proteins that interact with and induced changes on
the membrane skeleton. These changes drastically reduce the
RBC deformability and cause the cells to be sticky. This not
only impairs blood flow but can also lead to coma and even
death. This talk focuses on the use of experimental
techniques to probe the progressive stiffening of the
malaria infected RBCs and their effects on the flow
behaviour of infected RBCs through blood capillaries. These
include techniques such as laser traps or optical tweezers,
micropipette aspiration and microfluidics.
On bubble-acoustic interaction
for biological applications
Evert Klaseboer, Institute of High Performance Computing,
Singapore
An overview will be given of the developments concerning
bubble dynamics for biological applications using acoustics.
The term ‘acoustic’ here not only refers to weak acoustic
waves, but also strong acoustic perturbations, such as shock
waves. The Institute of High Performance Computing (IHPC)/
NUS has developed a numerical code for more than 10 years
that can simulate oscillating bubbles efficiently, ranging
from very large bubbles (~10 meter) to microscopically small
bubbles. The method is based on a numerical technique called
the boundary integral method. The advantage of this method
is that only a mesh is needed on the surfaces of the
problem, thus reducing enormously the amount of
computational time needed.
Initially developed for underwater explosions, where a large
bubble with diameter of about 10 m occurs, which generates a
high-speed jet when it collapses. The developed numerical
code can also simulate bubbles on a much smaller scale (mm
to micrometer), since the physics for large and small
bubbles are similar. An example in a bio-related area is the
recent discovery of ‘particle shooting’ where cavitation on
a small (micrometer sized) particle hit by a shock wave
causes this particle to accelerate at great speed. If the
particle contains some kind of drug, it could be used in
drug delivery into a nearby tissue.
If (micro) bubbles are ‘radiated’ with ultrasound, they
start to oscillate. It has recently been discovered that
depending on the nature of a nearby material (or tissue), a
jet can develop in the bubble towards or away from the
material. The understanding of the physics involved is of
direct importance to certain medical treatments (for example
if the jet is supposed to be used to destroy any nearby
tissue).
Another bio-related example is the appearance of a jet
inside a bubble when it is hit by a shock wave. Extremely
high jet velocities have been observed of several km/s. If a
(micro-) bubble is guided towards a cancer cell by coating
the bubble with certain receptors and a shockwave is
released, it might be able to destroy the cancer cell.
Similar techniques are also used in shockwave lithotripsy
(the removal of kidney stones).
A last example is the ‘bubble pump’, where a (small)
collapsing bubble produces a jet towards a plate with a
small hole in it. Instead of hitting the plate (as in an
underwater explosion), the jet will ‘pump’ liquid from one
end of the plate towards the other. The Institute of High
Performance Computing has patented this technology (together
with NUS) and is searching for applications, in either bio
or other areas such as inkjet printing.
Analysis of dielectrophoretic
cell traps using the boundary element method
Carlos Rosales Fernandez, Institute of High Performance
Computing, Singapore
In this talk I will describe the advantages of the
Boundary Element Method as a numerical tool for the analysis
of dielectrophoretic traps. In the analysis of
dielectrophoretic traps, as in other moving boundary
problems, the Boundary Element Method has the great
advantage that it does not require re-meshing, but rather a
simple displacement of the mesh at every time step. This
greatly simplifies the study of complex geometries and makes
possible detailed analyses of the performance of different
traps under different operating conditions before a
particular design is chosen for production. I will present
calculations done with the Boundary Element Method that
predict the movement of a rigid particle inside 2D
dielectrophoretic traps of different designs and show how
these calculation can be used to choose an optimized trap
design.
Shock wave lithotripsy -
mechanisms and potential for improvement
Pei Zhong, Duke University, USA
Shock wave lithotripsy (SWL) has revolutionized the
treatment of kidney stones since its introduction in the
early 1980s. In this talk I will briefly review the history
and current status of SWL. I will present a series of
studies that help to elucidate the fundamental mechanisms of
stone comminution and tissue injury in SWL. Based on this
knowledge, we have developed methods to selectively suppress
cavitation activity in renal tissues while enhancing the
collapse of cavitation bubbles near the target stone. It has
been demonstrated in both in vitro and in vivo models that
these technical improvements can significantly increase
stone fragmentation while reducing concomitantly the
collateral injury in SWL. The synergistic interactions
between lithotripter-generated stress waves and cavitation
in stone comminution have also been investigated, which may
help to better design the clinical treatment strategies for
kidney stone patients.
Fluid dynamic models of
flagellar and ciliary beating
Robert Dillon, Washington State University, USA
The motility of sperm flagella and cilia are based on a
common axonemal structure. In this article, we describe a
fluid-mechanical model for the ciliary and sperm axoneme.
This fluid-mechanical model, based on the immersed boundary
method, couples the internal force generation of dynein
molecular motors through the passive elastic axonemal
structure with the external fluid mechanics governed by the
Navier-Stokes equations. We show recent numerical simulation
results for sperm motility and multiciliary interaction.
An introduction to the immersed
boundary/interface methods
Robert Dillon, Washington State University, USA and Zhilin Li, North Carolina State University, USA
The immersed boundary method was developed by Peskin for
to model blood flow in the heart. It provides a framework
for coupling elastic dynamics of flexible boundaries with a
surrounding viscous, incompressible fluid. While
traditionally used in the context of biological
applications, the immersed boundary method is now considered
a classical method in computational fluid dynamics and has
impacted non-biological applications. In this tutorial, we
will give an overview of the method and provide an practical
introduction on using the immersed boundary method.
The Immersed Interface Method (IIM) was motivated by the
Peskin's Immersed Boundary Method. The IIM shares many
characteristics of the IB method. Both methods use simple
grid structure. The original motivation of IIM is to improve
the accuracy of the IB method from first order to second
order. This has been achieved by incorporating the jump
conditions into numerical schemes near or on the interface.
In this talk, I will explain the IIM through some simple
examples in 1D and 2D. I will explain how the IIM can be
coupled with evolution schemes such as front tracking and
level set method for moving interface and free boundary
problems. I will also present some recent progress of the
immersed interface method including the finite element
formulation and augmented IIM.
Cavitation-cell interaction with
adherent and suspension cells
Claus-Dieter Ohl, University of Twente, The Netherlands
I will report on several experimental approaches to
elucidate the physical mechanisms of drug uptake into cells
by bubbles. First we observed that adherent cells when being
exposed to a single shock wave give rise to peculiar
patterns: The uptake of drug occurs on annular rings with
detached cells in the center, bordered by dead cells, and
surrounded by positive cells. We could correlate these
patterns with single cavitation bubbles collapsing
close to the interface, in particular with a jet flow
impacting and spreading on the surface. In contrast to this
rather poorly controlled bubble nucleation experiment we
designed a second experiment to study the effect of a now a
single bubble. Therefore, the bubble is created with a laser
at variable distances from the boundary covered with
adherent cells. Here, high-speed recordings and wall shear
stress measurements reveal that the cells are expose to high
levels of shear stress. It is concluded that drug delivery
is a hydrodynamic boundary-layer interaction induced by the
non-spherical bubble dynamics. In contrast, experiments with
cells in suspension are much more
difficult to conduct. Here, the 3-dimensionality of the flow
makes is nearly impossible to track a single cell after its
interaction with a cavitation bubble. Yet, it is feasible in
planar structures which are used for lab-on-a-chip devices.
I will report on recent experiments in these systems which
show viable and permanent membrane poration of
suspension cells in this type of environments. The bubble is
created with a laser technique and can be controlled in
space, time, and size. This new technique allows also to
study the effect of jetting flow vs. radial flow and to
determine the minimum distance between the cell and bubble
needed for a viable interaction.
An application of the boundary
integral method to modelling cell motility in a viscous
fluid
Rafael P. Saldaña, Ateneo de Manila University,
Philippines
In this study, we used the boundary integral method to
elucidate the fluid dynamics of amoeboid motion which may
have some relevance to the surface motions of mobile cells
in vertebrates such as T-cells, microphages, and
fibroblasts.
As a first approximation, we considered the motion of a
single bubble or drop in an incompressible Newtonian fluid
which undergoes some undisturbed motion far from the bubble
or drop. Starting with the Navier-Stokes equations and using
the boundary integral method, we derived the integral
equations that are necessary for the numerical simulations
of amoeboid motion.
Higher-Order, Cartesian grid
based finite difference schemes for elliptic equations and
Stokes Systems wit interface
Kazufumi Ito, North Carolina State University, USA
Compact fourth order Cartesian grid based finite
difference methods are developed for variable elliptic
equations and elliptic interface problems. The resulting
system has the $M-$matrix property and thus satisfies the
discrete maximum principle. Our methods are based on the
continuation of solution across the interface using
multi-variable Taylor's expansion of the solution about
selected interface points and a set of appropriate interface
conditions. The interface conditions are derived from the
continuity of the solution and its flux and equating the
equation and its derivatives. The method is also
successfully extended to the heat equation and Stokes
systems. The validity and effectiveness of the proposed
methods are demonstrated through our numerical results.
Computation of laryngeal flow and
sound
Young J. Moon, Korea University, Korea
The human larynx is computationally modeled as axi-symmetric
and its flow and sound are computed by solving the
incompressible Navier-Stokes equations and the linearized
perturbed compressible equations (LPCE) in a coupled manner.
The governing equations are discretized with a sixth-order
compact scheme and a four-stage Runge-Kutta method, with the
moving grid system employed. This study investigates
transient characteristics of the glottal flow related to the
glottis motion as well as to the sound generation process.
It is found that the laryngeal flow is characterized by a
pulsating air jet within the glottis with periodically
changing local Reynolds number and flow separation point and
these flow features are closely related to the physics of
phonation aeroacoustics. The acoustic fields computed by the
linearized perturbed compressible equations (LPCE) show that
(i) the main phonation process (i.e. periodic changes of
compression and rarefaction waves in the sub-and
supra-glottal regions) is generated mainly by periodic
motions of the glottis (translation) and pulsating air jet
within the glottis, via a dipole sound generation mechanism,
(ii) the sound wave strength in the subglottal region is
much weaker than that in the supraglottal region, (iii) a
difference in fundamental frequency for male and female
determines the voice quality and it is associated with the
difference in local Reynolds number of the air jet within
the glottis, and (iv) a rotational motion of the glottis
controls the glottal impedance by changing the flow
separation point between the leading- and trailing-edge of
the vocal folds and this increases the mechanical efficiency
of the glottis as a sound generator in phonation process.
Nonlinear modeling of solid
tumor growth
John Lowengrub, University of California, Irvine, USA
I. Introduction and Basic Models
II. Effect of the Microenvironment
III. Angiogenesis
IV. Vascular Growth
Multiscale models of solid tumor
growth and angiogenesis: the effect of the microenvironment
John Lowengrub, University of California, Irvine, USA
We present and investigate models for solid tumor growth
that incorporate features of the tumor microenvironment
including tumor-induced angiogenesis. Using analysis and
nonlinear simulations, ects of the interaction between the
genetic characteristics of the tumor and the tumor
microenvironment on the resulting tumor progression and
morphology.
We find that the qualitative behavior of the tumor
morphologies is similar across a broad range of parameters
that govern the tumor genetic characteristics. Our findings
demonstrate the importance of the impact of microenvironment
on tumor growth and morphology and are consistent with
recent experiments. We discuss the implications for cancer
therapy protocols.
Finite element modelling of the
human brain and applications in neurosurgical procedures
Eng Hock Tay, National University of Singapore
This paper presents the development of a biomechanical model
of the human brain to investigate brain deformation under
surgical conditions. A physics-based brain atlas was
constructed to connect the finite element framework with the
atlas assisted analysis. This atlas with associated meshes
consists of 44 brain structures to account for the salient
anatomical features. Typical element types of the atlas were
generated in a multiple-object fashion to address the
general requirements. The mechanical model was characterized
by generalizing existing experimental data on brain tissue
tests and integrated into this physical atlas. A
hyperviscoelastic material model was adopted particularly to
suit the loading conditions consistent with surgical
settings. To validate the utility of this model in desired
applications, finite element simulations of two typical
brain deformation cases were performed with various strain
rates. Incorporated with the specifically characterized
model, the physical atlas reproduces complex brain
deformation phenomena induced by craniotomy and
ventriculoperitoneal shunting. This modelling approach
enables a more realistic and informative representation of
brain deformation than conventional models by providing
insights into the subcortical brain behaviours. With a
further understanding of the brain physical properties, the
application of the proposed model and approaches may extend
to various computer assisted surgery analysis areas.
Quasineutral limit of
drift-diffusion models for semiconductors
Shu Wang, Beijing University of Technology, China
In this talk I will talk about quasineutrality problems in
the applied sciences such as semiconductors and plasma. Some
interesting phenomena like depletion (vacuum) regimes,
concentration and oscillatory behaviors, shock layer,
interface problem, and sheath boundary layer will be
discussed by performing quasineutral limit of
drift-diffusion models for semiconductors. Some new ideas
and results will also be given.
Level set methods for watershed
segmentation and some PDE techniques for noise removal
Xue-Cheng Tai, Nanyang Technological University,
Singapore
In this talk, we will present some of the recent work we
have done in using level set methods for analyze some
medical imaging problems. The first application is to use
level set methods to segment real 3D MR images. We will show
advantages and weak points in using the methods for real
clinical applications. The second application is to use
level methods for watershed segmentation. The task is to
automatically detect human cell boundary from some
biomedical imaging techniques. For real imaging data, the
noise is huge and traditional watershed methods give high
rate of false detection. It is the first time that level set
idea has been used for this kind of application. We also use
the Four-Color theorem to design a method that only needs
two level set functions to detect arbitrary number of
objects.
In the second part of the talk, I will present some PDE
method to do image inpainting and denoising. This part is
based on some joint works with S.Osher, R. Holm and R. Talal.
We propose a two-step algorithm. Observing that the isophote
directions of an image correspond to an incompressible
velocity field, we impose the constraint of zero divergence
on the tangential field. Combined with an energy
minimization problem corresponding to the smoothing of
tangential vectors, this constraint gives rise to a
nonlinear Stokes equation where the nonlinearity is in the
viscosity function. Once the isophote directions are found,
an image is reconstructed that fits those directions by
solving another nonlinear partial differential equation. In
both steps, we use finite difference schemes to solve. We
present several numerical examples to show the effectiveness
of our approach.
A simple implementation of the
immersed interface method for the stokes flow with singular
forces
Ming-Chih Lai, National Chiao Tung University, Taiwan
In this talk, we shall introduce a simple version of
immersed interface method (IIM) for the Stokes flow with
singular forces along an interface. The numerical method is
based on applying the Taylor's expansions along the normal
direction and incorporating the solution and its normal
derivative jumps into the finite difference approximations.
The fluid variables are solved in the staggered grid, and a
new accurate interpolating scheme for the non-smooth
velocity has been developed. The numerical results show that
the scheme is second-order accurate.
Gauge methods for variable density
and multi-phase flows
Shuo Jia, Murex Southeast Asia Pte Ltd
In this talk, we will present the variable density gauge
formulation for solving variable density incompressible
Navier-Stokes equations. This formulation is an extension
from the conventional gauge formulation for constant density
Navier-Stokes equations. We develop a finite difference
scheme on uniform non-staggered grids using the variable
density gauge formulation to obtain fully second-order
accuracy in velocity, pressure and density. When the surface
tension is taken into account, we couple the variable
density gauge formulation with the level set method to
numerically simulate the motion of two immiscible viscous
fluids separated by a free moving interface. The numerical
performance of the variable density gauge formulation is
illustrated by several numerical results.
Computational fluid simulation using
meshless finite difference
Choon Seng Chew, Institute of High Performance
Computing, Singapore
A scheme using the mesh-free generalized finite differencing
(GFD) on flows past moving bodies is proposed. The aim is to
devise a method to simulate flow past an immersed moving
body that avoids the intensive re-meshing of the
computational domain and minimizes data interpolation, as
such procedures are time consuming and are a significant
source of error in flow simulation. In the present scheme,
the moving body is embedded and enveloped by a cloud of
mesh-free nodes, which convects with the body motion against
a background of Cartesian nodes. GFD with weighted least
squares (WLS) approximation is used to discretize the
two-dimensional viscous incompressible Navier-Stokes
equations at the mesh-free nodes, while standard FD are
applied elsewhere. The convecting mesh-free nodes are
treated by an Arbitrary Lagrangian-Eulerian (ALE)
formulation of the flow equations. The proposed numerical
scheme was tested on a number of problems including the
decaying-vortex flow, external flows past.
Simulation of flows with interfaces
using discontinuous Galerkin method
Nguyen Vinh Tan, National University of Singapore
The discontinuous Galerkin method is presented to solve the
hyperbolic equations of conservation laws involving
interfaces which can be found in many engineering problems.
A front tracking technique is introduced to treat the
discontinuities in the solutions due to the presence of the
interfaces. It considers the discontinuities as the interior
boundaries coupled to the discontinuous Galerkin
discretization. The technique also uses lower dimensional
grids called fronts to represent the discontinuities in the
numerical solutions. It therefore avoids smearing the jumps
across the interface and leads to improvement of accuracy.
Motion of the front is obtained by solving Riemann problems
at element interfaces. The presented method can track the
interface sharply while still preserving the conservation of
the solution over the whole domain. The method is applied to
solve for the several typical interface problems including
shock-interface interaction, interface with surface tension.
Coupled adaptive moving mesh and
phase field method for the incompressible mixture flows
Zhijun Tan, National University of Singapore
A phase field model which describes the motion of mixtures of
two incompressible fluids is presented by Liu and Shen [Phys.
D 179 (2003) 211–228]. The model is based on an energetic
variational formulation. In this work, we develop an efficient
adaptive mesh method for solving a phase field model for the
mixture flow of two incompressible fluids. It is a coupled
nonlinear system of Navier-Stokes equations and Allen-Cahn
phase equation (phase field equation) through an extra stress
term and the transport term. The numerical strategy is based
on the approach proposed by Li et al. [J. Comput. Phys. 170
(2001) 562–588] to separate the mesh-moving and PDE
evolution. In the PDE evolution part, the phase-field
equation is numerically solved by a conservative scheme with
a Lagrange multiplier, and the coupled incompressible Navier-Stokes
equations are solved by the incremental pressure-correction
projection scheme based on the semi-staggered grid method.
In the mesh-moving part, the mesh points are iteratively
redistributed by solving the Euler-Lagrange equations with a
parameter-free monitor function. In each iteration, the
pressure and the phase are updated on the resulting new grid
by a conservative-interpolation formula, while the velocity
is re-mapped in a non-conservative approach. A simple method
for preserving divergence-free is obtained by projecting the
velocity onto the divergence-free space after generating the
new mesh at the last iterative step. Numerical experiments
are presented to demonstrate the effectiveness of the
proposed method for solving the incompressible mixture flows.
Performance prediction of a long
narrow reverse osmosis filtration channel
Lianfa Song, National University of Singapore
Reverse osmosis membrane process is playing the key role in
the reclamation of high quality water from non-conventional
sources, such as wastewater, brackish water and seawater.
The common configuration of reverse osmosis filtration
process is characterized by a long narrow channel (typically
several meters in length but a fraction of millimeter in
height). The feed water is fed in one end and the
concentrate exits from the other end of the membrane
channel. Because of the substantial variations in the
operating parameters and water properties along the channel,
the membrane filtration system presents strong nonlinear
behaviors that cannot be adequately explained within the
framework of the classic membrane filtration theories.
Mathematical model for the heterogeneous membrane system is
first developed. The model is then solved numerically and
analytically for a better understanding of the long narrow
reverse osmosis filtration channel. Interesting results of
paramount importance to the reverse osmosis processes have
been obtained and will be reported in this talk, such as the
role of concentration polarization in the spiral wound
membrane modules, controlling mechanisms of the performance
of the long narrow reverse osmosis channel, and
characteristics of membrane fouling.
Mass transfer across the air-water
interface
Zhifeng Xu, National University of Singapore
The phenomena of mass transfer across the air-water
interface widely exist in various natural industrial
processes. Because of its wide application, a general model
capable of predicting the mass transfer rate across the
air-water interface under various flow conditions is highly
desirable.
In the past decade, a new kind of model named as ‘surface
divergence model’ attracted many researchers’ attention. It
is generally accepted that the gradient of the vertical
fluctuating velocity with respect to the interface plays an
important role in mass transfer process. This new developed
model tries to correlate this parameter to the mass transfer
velocity across the air-water interface. This parameter is
acquired from the turbulence structure in the very vicinity
of the interface. This feature guarantees it can be used in
various flow conditions.
We have developed a reliable method to measure this key
parameter in a circular wind wave channel. In this method,
two cameras are use to observe the flow field from two
different view angles. One from above to capture the clear
view of the water interface, and the other one form below to
get the detailed information under the water interface.
These two different images from the two views are correlated
through careful calibration before the experiment.
The mass transfer experiments were carried out under gas
absorption and/or gas evasion conditions in a sealed
circular channel, where the flow experiment is also taken.
For gas evasion experiments, carbon dioxide was used to
flush the headspace of the experimental set-up. It yields a
known zero oxygen surface concentration. For gas absorption
experiment, pure oxygen gas was introduced such that the
total gaseous volume above the interface was considered to
be saturated. Under such condition, the water surface is
maintained at the saturation of oxygen in the water side at
that temperature. By measuring the initial and final
dissolved oxygen concentration, the mass transfer velocity
can be obtained. Following figure gives out the comparison
of the mass transfer results between gas absorption and gas
evasion. It is noted that the oxygen sensor, used to measure
the dissolved oxygen concentration in this work, use two
known oxygen concentration standards for calibration. These
two standards are a zero concentration standard and an
environmental saturated concentration standard. So the
oxygen sensor works in extrapolated region of calibration
for the gas absorption experiments. Larger experimental
measurement error exists for high level oxygen
concentration. The pure gas was input just for 10minutes to
make sure the work range of the instrument is not very far
away from the calibration region.
The random projection method for
stiff detonation capturing
Weizhu Bao, National University of Singapore
In this talk, I review the random projection method for
underresolved numerical simulation of stiff detonation waves
in chemically reaction flows. In the problem, the chemical
time scales may be orders of magnitude faster than the fluid
dynamical time scales, making the problem numerically stiff.
A classical spurious numerical phenomenon, the incorrect
propagation speeds of discontinuities, occurs in
underresolved numerical solutions.
We introduce a random project method for the reaction term
by replacing the ignition temperature with a uniformly
distributed random variable. The statistical average of this
method corrects the spurious shock speed, as will be proved
with a scalar model problem and demonstrated by a wide range
of numerical examples in inviscid detonation waves in both
one and two space dimensions as well as for detonation with
multispecies.
A general moving mesh framework in 3D
and its application for simulating the mixture of
multi-phase flows
Tang Tao, Hong Kong Baptist University, Hong Kong
In this talk, we present an adaptive moving mesh algorithm
for meshes of unstructured tetrahedra in three space
dimensions. The algorithm automatically adjusts the size of
the elements with time and position in the physical domain
to resolve the relevant scales in multiscale physical
systems while minimizing computational costs.
Since the mesh redistribution procedure normally requires to
solve large size matrix equations (arising from discretizing
the Euler-Lagrange equations or a minimization problem), we
will describe a procedure to decouple the matrix equation to
a much simpler block-tridiagonal type which can be solved by
multi-grid methods efficiently. To demonstrate the
performance of the proposed 3D moving mesh strategy, the
algorithm is implemented in finite element simulations of
fluid-fluid interface interactions in multiphase flows. In
this talk, we will propose a general framework on how to
design an adaptive grid method useful for this kind of
simulations. To demonstrate the main ideas, we consider the
formation of drops by using an energetic variational phase
field model which describes the motion of mixtures of two
incompressible fluids. The phase field model consists of a
Navier-Stokes system coupled with volume preserving
Allen-Cahn type phase equations. Numerical results on two-
and three-dimensional simulations will be presented.
From immersed boundary method to
immersed continuum method
X. Sheldon Wang New Jersey Institute of Technology, USA
The objective of this talk is to present an overview of the
newly proposed immersed continuum method in conjunction with
the traditional treatment of fluid-structure interaction
problems, the immersed boundary method, and the fictitious
domain method. In particular, the key aspects of the
immersed continuum method in comparison with the immersed
boundary method are discussed. The immersed continuum method
retains the same strategies employed in the extended
immersed boundary method and the immersed finite element
method, namely, the independent solid mesh moves on top of a
fixed or prescribed background fluid mesh, and employs fully
implicit time integration with a matrix-free combination of
Newton-Raphson and GMRES iterative solution procedures. The
immersed continuum method is capable of handling
compressible fluid interacting with compressible solid.
Several numerical examples are also presented to demonstrate
that the proposed immersed continuum method is a good
candidate for multi-scale and multi-physics modeling
platform.
Vector extension velocity Level Set
methods
Weijun Tang, Institute of Applied Physics and
Computational Mathematics, China
A vector extension velocity method is introduced in this
paper. The new equations for vector extension velocity can
be used to advance the Level Set function. It is pointed out
that this method differs with extension velocity in normal
direction proposed by Adalsteinsson and Sethian. The later
has a disadvantage of much numerical error for treating
those interfaces with corners and numerical examples confirm
this. Furthermore, a new iteration model equation is
proposed to solve the extension velocity. This new model
differs from the model with sign function proposed by
Sussman, and also the model with smooth function put forward
by Peng et al. Analysis for the first-order upwind scheme to
those three iteration models mentioned above are given. It
shows the reason why the interface excursion takes place
during the iteration procedure for other two models,
contrary, the new model this paper proposes keeps the
interface stable.
Modeling diffuse interfaces and
transformation fronts
Richard Saurel, Polytech Marseille, France and
University Institute of France, France
The numerical resolution of interfaces separating
compressible fluids involves artificial mixing zones
resulting of numerical diffusion. In such artificial
mixtures it is difficult to determine the correct
thermodynamical state. By considering such zones as physical
multiphase mixtures reliable modelling is possible.
The single velocity and single pressure but
multi-temperatures flow model of Kapila et al. (2001) is
examined for the modelling of diffuse interfaces separating
compressible fluids. This non-conservative hyperbolic model
with five partial differential equations requires shock
relations for its closure. The shock relations derived in
Saurel et al. (2007a) are presented and validated. With
these relations, the Riemann problem is solved, and a
relaxation-projection method is developed to solve interface
problems and shock propagation into mixtures (Saurel,
2007b). The method is conservative, oscillation free and
able to deal with very large density and pressure ratios.
Then, extensions to extra physics are considered.
First, capillary effects at interfaces separating
compressible fluids are introduced. A conservative momentum
and energy formulation is obtained (Perigaud and Saurel
(2005)).
Second, heat and mass transfer at interfaces are introduced
in order to deal with phase transition fronts (Saurel et
al., 2007c). When the interface structure is numerically
solved evaporation jump conditions are recovered. In
particular, the kinetic CJ relation proposed by Simoes
Moreira and Shepherd (1999) for evaporation fronts in
metastable liquids is recovered. Examples for
multidimensionnal cavitating flows are presented.
Last, multiphase detonation modelling is addressed. The
mechanical equilibrium multiphase flow model with heat and
mass transfer is examined in the context of exothermic
decomposition. Generalized CJ conditions for multiphase
explosives are obtained. They involve specific mixture speed
of sound, chemical decomposition and heat transfer rates.
Conventional CJ and WK conditions are recovered as limit
cases. Multidimensional detonation computations in the
presence of material interfaces are presented.
With this approach, interfaces, shocks, detonations,
capillary fluids, evaporation fronts are solved with the
same equations and the same numerical method.
Kapila, A.K., Menikoff, R., Bdzil, J.B., Son, S.F. and
Stewart, D.S. (2001) Two-phase modeling of deflagration to
detonation transition in granular materials: Reduced
equations. Physics of Fluids 13(10), pp 3002-3024
Perigaud, G and Saurel, R. (2005) A compressible flow model
with capillary effects. Journal of Computational Physics,
Vol. 209, pp 139-178
Saurel, R., Le Metayer, O., Massoni, J. and Gavrilyuk, S.
(2007a) Shock jump relations for multiphase mixtures with
stiff mechanical relaxation. Shock Waves, Vol. 16 (3), pp
209- 232
Saurel, R., Franquet, E., Daniel, E. and Le Metayer, O.
(2007b) A relaxation-projection method for compressible
flows. Part I : The numerical equation of state for the
Euler equations. Journal of Computational Physics, in press
Saurel, R., Petitpas, F. and Abgrall, R. (2007c) A
hyperbolic non-equilibrium model for cavitating flows.
Journal of Fluid Mechanics, submitted
Simoes-Moreira, J.R. and Shepherd, J.E. (1999) Evaporation
waves in superheated dodecane. Journal of Fluid Mechanics
382: 63-86
Drop dynamics in a liquid
crystalline medium
James Feng, University of British Columbia, Canada
Owing to their molecular orientation, nematic liquid
crystals exhibit complex rheological behavior. In
particular, their tendency to anchor on interfaces at a
fixed angle leads to unusual interfacial dynamics for drops
and bubbles. In this talk, I describe numerical simulations
of rising drops in a quiescent nematic medium with a
vertical far-field orientation. The moving interface is
computed in a diffuse-interface framework, and the
anisotropic rheology of the liquid crystal is represented by
the Leslie-Ericksen theory, regularized to permit
topological defects. Results reveal interesting coupling
between the flow field and the orientational field
surrounding the drop, especially the defect configuration.
The flow sweeps a point or ring defect downstream, and may
transform the ring defect into a point defect. The nematic
orientation affects the flow field in return, and modifies
the rise velocity and drag on the drop. The drop takes on
unusual shapes when the anchoring energy is high enough to
compete with the interfacial tension.
Efficient kinetic schemes for steady
and unsteady flow simulations on unstructured meshes
Song Jiang, Institute of Applied Physics and
Computational Mathematics, China
This talk presents efficient discontinuous and continuous
2nd-order kinetic schemes on unstructured meshes for
compressible unsteady and incompressible steady flows,
respectively. For compressible unsteady flows, we use the
time-dependent gas distribution function to get the exact
values of the flow variables at cell interfaces to evaluate
efficiently the numerical flux at the cell interfaces of
unstructured mesh, resulting in a 2nd-order gas kinetic BGK
scheme. For incompressible steady flows, a continuous
2nd-order gas kinetic BGK type scheme is presented, for
which the time-dependent gas distribution function with
continuous particle velocity is constructed and used in the
evaluation of the numerical flux across the cell interfaces
of unstructured meshes. Numerical examples are presented
which are compared to the benchmark solutions and the
experimental measurements.
Extrinsic meshfree method for
interfacial discontinuity
Do Wan Kim , Han Yang University, Korea
As a generalization of smooth
approximations in meshfree methods, a sharp meshfree
approximation for derivative discontinuities across
arbitrary interfaces is proposed. The interface can be
arbitrarily located in a domain in which nodes are
distributed uniformly or irregularly. The proposed meshfree
approximations explicitly consist of two parts,
singular(local step, scissor, and wedge functions) and
regular(smooth meshfree apprxomation). By using the fast
moving least square meshfree approximation, the derivatives
of these approximations are readily calculated at arbitrary
point in the considered domain, so that the point
collocation method enables us to calculate the accurate
numerical solution with discontinuities along the interface.
In this method, interface conditions are directly
discretized on the given interface model. The interface
model can be constructed by employing the FE-approximation
on the interface. However, it is only for the purpose of
interface function(normal flux) approximation and is
independent of the background nodes. Therefore, discretizing
the interface condition on the interface model is performed
independently of doing the governing equation inside the
domain. This is another advantage of our method when
treating with problems involving interfacial
discontinuities.
The approximations for discontinuities are applied in a
meshfree point collocation method to obtain solutions of the
Poisson problem with a layer delta source on the interface
and second order elliptic problems with discontinuous
coefficients and/or the singular layer sources along the
interface. Although they are not moving interface problems,
our method can be extended to such problems as well. The
numerical calculations show that this method has good
performance even on irregular node models.
Runge-Kutta Discontinuous Galerkin
methods for compressible two-medium flow simulations:
one-dimensional case
Jianxian Qiu, Nanjing University, China
The Runge-Kutta discontinuous Galerkin method for solving
hyperbolic conservation laws is a high order finite element
method, which utilizes the useful features from high
resolution finite volume schemes, such as the exact or
approximate Riemann solvers, TVD Runge-Kutta time
discretizations, and limiters. In the presentation we will
describe our recent work on using Runge-Kutta discontinuous
Galerkin finite element methods for multi-medium flow
simulations in one dimension where the moving material
interfaces by is effected by a conservative method.
Numerical results for both gas-gas and gas-water flows in
one dimension are provided to demonstrate the characteristic
behavior of this approach.
Fluid transport models for a
two-phase core-annular flow
Long Lee, University of Wyoming, USA
We consider a two-phase core annular flow in a cylindrical
pipe in this talk. The inner core is assumed to be a
pressure driven gas flow. The other phase is highly viscous
fluid lining the inner wall of the pipe. Several models are
presented, including the classic Poiseuille solution for
two-phase flows, to predict the mean thickness of the liquid
layer in the experiment by Kim et al. , where a fixed flow
rate of gas drags the liquid upward in a vertical pipe, in
which the liquid is injected into the pipe at a fixed feed
rate.
We derive a nonlinear evolution equation based on the
lubrication approximation for the interface. The equation
incorporates the strong pressure-driven gas flow as a
forcing term into the equation for the liquid, with an
effective viscosity for turbulent flow replacing the
molecular viscosity of the gas. We study numerically the
interface evolution of an initially axisymmetric disturbance
of the annular film of viscous liquid. The mean height of
the liquid layer in the experiment can be accurately
predicted using this model, and the existence of the
ring-like waves reported in the experiments is confirmed by
the interfacial dynamics of the model.
Biofilms and microbial mats:
Multiphase physical-biological systems
Isaac Klapper, Montana State University, USA
Microbial colonies, in the forms of biofilms and microbial
mats, are ubiquitous biological ecosystems consisting of
large numbers and, generally, large diversity of microbial
organisms embedded in self-secreted polymeric matrices.
These colonies are typically to be found in wet, flowing
environments where fluid-colony multiphase mechanical
interactions are of central importance.
Colony spatial structure and physical properties play
important roles in community function, protection, and
ecology so that it is important to determine physical
properties of biofilms and mats regarded as macroscale
materials. Measurements indicate that characterization as
viscoelastic fluids is appropriate. Observations demonstrate
the tendency for material failure via sloughing of large
chunks. In this talk I will discuss some efforts to
understand physical properties of biofilms and microbial
mats as materials and to use this knowledge to construct
models capable of describing their long-time behavior.
Numerical modelling of cavitation
inception in high-speed cavitating flow:
One-fluid/Free-Lagrange model
Boo Cheong Khoo, National University of Singapore
High speed objects traveling underwater may initiate
formation of cavitation near the body of the object. This
cavitation region may enclose the object that the only
significant contact of the object and the water is at the
objects’ head and the hydrodynamic drag is therefore reduced
significantly. Consequently, high subsonic or even
supersonic speed underwater is possible. Here, we present a
model based on One-fluid model coupled with the
Free-Lagrange method for simulations of high speed
cavitating flow. The One-fluid model can be used to model
creation, evolution and collapse of unsteady cavitation by
assuming that the cavitating flow is a homogeneous mixture
of isentropic gas and liquid components and the flow is
compressible. Using the Free-Lagrange method we have
simulated a 12.5 mm radius cylinder with a hemispherical
head traveling underwater with 254 m/s velocity and 0° angle
of attack. The simulation shows that a cavity region is
created along the surface of the cylinder’s body when the
pressure of water in its vicinity drops below the water
saturated vapour pressure, as well as the growth of the
cavity downstream.
An adaptive ghost fluid finite
volume method for compressible gas-water simulations
Chunwu Wang, Nanjing University of Aeronautics &
Astronautics, China
An adaptive ghost fluid finite volume method is developed
for one- and two-dimensional compressible multi-medium flows
in this work. It couples the real ghost fluid method (GFM)
[SIAM J. Sci. Comput. 28(2006) 278] and the adaptive moving
mesh method [SIAM J. Numer. Anal. 41(2003) 487; J. Comput.
Phys. 188(2003)543], and thus retains their advantages. This
work shows that the local mesh clustering in the vicinity of
the material interface can effectively reduce both numerical
and conservative errors caused by the GFM around the
material interface and other discontinuities. Besides the
improvement of flow field resolution, the adaptive ghost
fluid method also largely increases the computational
efficiency. Several numerical experiments are conducted to
demonstrate robustness and efficiency of the current method.
They include several 1D and 2D gas-water flow problems,
involving the large density gradient at the material
interface and strong shock-interface interactions. The
results show that our algorithm can capture the shock waves
and the material interface accurately, and is stable and
robust even for large density and pressure.
Moving interfaces in solids
Deborah Sulsky, University of New Mexico, USA
The material-point method is applied to study the motion of
granular material and fracture in solids. These problems are
characterized by the interaction of multiple, deformable
bodies in contact. The changing contact surfaces presents a
challenge to numerical simulation methods.
The material-point method (MPM) is a versatile method for
solving problems in continuum mechanics. For example, it is
useful for problems containing multiple materials, with
bodies in contact (including friction), with moving
interfaces, and with complicated geometry.
The flexibility of the method has been achieved by combining
two discretizations of the material.One is a Lagrangian
description based on representing the continuum by a set of
material points that are followed throughout the
calculation. The second is a background grid that is used to
solve the momentum equation efficiently.
Transitions of liquid crystals
and critical value of elastic coefficients
Xingbin Pan, East China Normal University, China
P. G. de Gennes predicted analogies
between the effect of elastic coefficients to liquid
crystals and the effect of applied magnetic fields to
superconductors, and predicted that all elastic coefficients
diverge to infinity at smectic-C to nematic transition.
One would expect quantitative comparison in the analogies.
In the case of equal elastic coefficients, we define the
critical value K^c and make comparison of it with the upper
critical magnetic field for type II superconductors. We
classify smectic liquid crystals into subcritical, critical
and supercritical cases according to the Ginzburg-Landau
parameter, the wave number and the boundary value of the
director at surface.
We show that in the subcritical case the liquid crystal does
not undergo phase transition; and in the supercritical case
both phase transition and hysteresis occur. The prediction
of De Gennes is true only in the critical case, where a
necessary relation between the wave number and Ginzburg-Landau
parameter must hold true.
Numerical simulation of
multiphase/interfacial flows using front-tracking method
Hua Jinsong, Institute of High Performance Computing,
Singapore
Multiphase/interfacial flows are found
in many common natural phenomena like drops, bubbly flows
and wave breaking, as well as in industrial processes such
as combustion, particle transport, petroleum refining and
boiling. Due to the difficulties encountered in performing
experimental and theoretical studies, numerical simulation
has become a powerful tool to investigate such complex
flows. A front tracking method has been developed to
simulate the complex multiphase/interfacial flows. The front
tracking method is well known for its capability of
capturing sharp interfaces in the multi-fluid systems. The
multi-fluid phases are treated as a single fluid with
variable material properties, and one single set of
governing equations for the whole computational domain is
solved. The interface properties, such as surface tension,
are computed on the front and distributed to the background
grid through a distribution function for solving the flow
field. The advection of fluid properties, such as density
and viscosity, is achieved by following the motion of the
front. In addition, a number of new features will be
introduced to extend the model’s capability in handling the
multiphase flow problem with large density ratio, e.g. in
gas bubble-liquid system; parallelization and mesh
adaptation to deal with large scale problem; moving
reference frame to tackle the long simulation of moving
interface; and systematic validation against the
experiments. A brief introduction will also be presented on
the various applications of the current method to
investigate the bubble dynamics in bubble rising,
bubble-bubble interaction, gas bubble pinch-off in liquid,
and droplet generation in micro-channels.
The Modified Ghost Fluid Method
and its applications
Tiegang Liu, Institute of High Performance Computing,
Singapore
The Ghost Fluid Method (GFM) is a very
latest technique developed for treating moving material
interfaces, and has quickly gained attention in practice.
The GFM possesses the features of simplicity, easy extension
to multi-dimensions, maintenance of a sharp interface and
applicability to fluids of vastly different properties even
in different coordinate systems. However, it was found by us
that the original Ghost Fluid Method (GFM) can provide
incorrect results or even fails to work in some situations.
A modified Ghost Fluid Method (MGFM) has been developed by
us to overcome the difficulties encountered by the original
GFM. In this talk, I shall introduce the original GFM and
its variants, and the MGFM. The focus is on mathematically
analyzing the accuracy and conservation errors of various
existing GFMs. I show that the original GFM indeed has no
accuracy when applying to shock or jet impacting on a
material interface, while the MGFM is second order accurate.
Finally, I will give various applications of the MGFM.
An Energy-law Preserving C^0 Finite
Element Methods for Liquid Crystal Flows and Phase Field
Models
Lin Ping, National University of Singapore
Liquid crystal flow model is a coupling
between orientation (director field) of liquid crystal
molecules and
a flow field. The model is the same as a phase field model
of multiphase flows if the orientation variable is changed
to phase function. It is important to preserve the energy
law of the model. We shall use a C^0 finite element method
which is simpler than existing C^1 element methods and mixed
element formulation. Through a reformulation the energy law
can be achieved by the C^0 finite element method. A discrete
energy law is achieved by a modified midpoint time discretization with a special treatment of the nonlinear
phase change term. Apparently the discrete energy law is an
approximation of the continous energy law. A
fixed point iterations automatically separates the flow and director (or phase field) equations and thus reduces the
size of the stiffness matrix and at the same time keeps the
stiffness matrix time independent. The latter avoids solving
a linear system at every time step and largely reduces the
computational time, especially when direct linear system
solvers are used. A number of examples are computed to
demonstrate the algorithm.
DNS of bubbly channel flows
Gretar Tryggvason, Worcester Polytechnic Institute,
USA
Recent results from direct numerical
simulations of bubbly flows in vertical channels are
presented. The computations are done using a
front-tracking/finite-volume method that allows full
resolution of the flow around several bubbles moving in both
laminar and turbulent flows. For nearly spherical buoyant
bubbles, the lift force result in a lateral migration of the
bubbles, leading to a bubble-rich wall-layer for upflow and
a wall-layer void of bubbles for downflow. The implications
of the results for modeling are address. The effect of the
void fraction and bubbles size, as well as the changes in
the flow structure as the bubbles become more deformable,
are discussed.
A meshfree method for fluid
flow simulation
Shengyin Wang, National University of Singapore
The Radial Basis Functions (RBFs)
originally developed for curve fitting and function
approximation have become increasingly popular recently. In
this study, an integration-free meshfree multiquadric RBF
collocation method is presented as an accurate and efficient
approach for fluid flow simulation. The infinitely smooth
multiquadric RBFs are used to achieve a high level of
smoothness and accuracy. A polynomial is adopted in the RBF
discretization to guarantee the nonsingularity of the
resulting system. The collocation method is used to
discretize the governing equations and the boundary
conditions. The weak formulation is not involved and thus
the present method can be integration-free and truly
meshless. The existence of a unique RBF solution may be
obtained with mild constrictions on the RBF knots
distribution. The resulting system belongs to the popular
saddle point system and iterative methods may become more
effective for solving moderately large problems. The
benefits of the present meshfree method are higher-order
accuracy and a faster convergence speed than the finite
element method due to the global support basis functions and
the existence of a free shape parameter. It is illustrated
that spectral convergence can be obtained by using either
the h-scheme or the c-scheme and the c-scheme can be more
effective due to the ease of implementation. To make full
use of the c-scheme, an unconstrained optimization procedure
is developed to obtain the optimal shape parameter based on
the direct pattern search method. It is obtained that the
numerical errors may decrease significantly with the
increase of the shape parameter until breakdown due to the
ill-conditioning and round-off errors. Numerical results
show that the present method can achieve significantly
better performance in accuracy and convergence over a
higher-order finite element method for fluid flow
simulation. It is suggested that a meshfree RBF approach for
fluid flow simulation can be an appealing alternative to the
popular solution methods.
Nonlinear diffusions as limit of
BGK-type kinetic equations
Peter Markowich, University of Vienna, Austria
We consider nonlinear BGK-type kinetic
equations with one free parameter which is to be determined
by local mass conservation. In the diffusive scaling limit
we obtain nonlinear diffusion equations whose diffusivities
depend on the global Gibbs state used to set up the BGK
equation.
A SQP-Semismooth Newton-type
algorithm for control of the instationary Navier-Stokes
system subject to control constraints
Michael Hintermueller, University of Graz Heinrichstr,
Austria
Sequential quadratic programming (SQP)
methods for the optimal control of the instationary Navier-Stokes
equations with pointwise constraints on the control are
considered. Due to the presence of the constraints the
quadratic subproblems (QP) of SQP require a more
sophisticated solver compared to the unconstrained case. In
this talk, a semismooth Newton method is proposed for
efficiently solving the QPs. The convergence analysis, which
is performed in an appropriate function space setting,
relies on the concept of Newton differentiability for
proving locally superlinear convergence of the QP-solver.
For the analysis of the outer SQP-iteration a generalized
equations approach is utilized. Sufficient conditions for
guaranteeing strong regularity of the generalized equation
are established which, in turn, allows to argue a quadratic
rate of convergence of the SQP-method. The talk ends with a
report on numerical results supporting the theoretical
findings.
