An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces

5 MB·

by  Zhouyang Ge

An efficient mass-preserving interface-correction level set/ghost
fluid method for droplet suspensions under depletion forces
Zhouyang Gea,∗, Jean-Christophe Loiseaua,1, Outi Tammisolaa,b, Luca Brandta
aLinn´e Flow Centre and SeRC (Swedish e-Science Research Centre), KTH Mechanics,
S-100 44 Stockholm, Sweden
bFaculty of Engineering, The University of Nottingham, NG72RD Nottingham, UK
Abstract
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1 Aiming for the simulation of colloidal droplets in microfluidic devices, we present here a
0
2 numerical method for two-fluid systems subject to surface tension and depletion forces
among the suspended droplets. The algorithm is based on a fast, second-order-accurate
b
e solver for the incompressible two-phase Navier-Stokes equations, and uses a level set
F
method to capture the fluid interface. The four novel ingredients proposed here are,

1
firstly, an interface-correction level set method (iCLS) that efficiently preserves mass.

] Global mass conservation is achieved by performing an additional advection near the
n
y interface, with a correction velocity obtained by locally solving an algebraic equation,
d which is easy to implement in both 2D and 3D. Secondly, we report a second-order
-
u accurate estimation of the curvature at the interface and, thirdly, the combination of the
l
f ghost fluid method with the fast pressure-correction approach enabling an accurate and
.
s fast computation even for large density contrasts. Finally, we derive a sharp formulation
c
i for the pressure difference across the interface induced by depletion of surfactant micelles
s
y andcombineitwithamultiplelevel-setapproachtostudyshort-rangeinteractionsamong
h
droplets in the presence of attracting forces.
p
[
Keywords: Multiphase flow, Level set method, Ghost fluid method, Colloidal droplet

2
v
5
8
3 1. Introduction
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0 In the field of colloidal science, much progress has been made on the synthesis of
.
1 complex building blocks mimicking molecular structures to elaborate innovative mate-
0
rials, e.g. materials with complete three dimensional band gaps [1, 2, 3, 4]. The basic
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1 elements of such colloidal molecules are particles or droplets less than one millimeter
:
v in size, and their self-assembly relies on either lengthy brownian motion or careful mi-
i crofludic designs, on top of typical colloidal interactions, e.g. depletion attraction and
X
electrostatic repulsion [5, 6, 7]. Regardless of the approach, however, questions remain
r
a
why the colloidal particles/droplets undergo certain path to organize themselves and how
∗Corresponding author.
Email addresses: [email protected] (Zhouyang Ge), [email protected]
(Jean-Christophe Loiseau), [email protected] (Outi Tammisola), [email protected] (Luca Brandt)
URL: https://github.com/GeZhouyang (Zhouyang Ge)
1Presentaddress: LaboratoireDynFluid,ArtsetM´etiersParisTech,151boulevarddel’hˆopital,75013
Paris, France
Preprint submitted to Journal of Computational Physics February 2, 2017
such process can be controlled and optimized. Since full data are not yet accurately ac-
cessible from experiments in such miniature systems, computer simulations will be useful
to provide supplemental information.
Scaling down to microscale appears first to be a convenience for the numerical simu-
lations of multicomponent and multiphase systems as the non-linear Navier-Stokes (NS)
equations can be reduced to the linear Stokes equations. This allows the use of boundary
integral methods (BIM) [8], e.g. most recently the GGEM-based BIM [9, 10] solving the
Stokes equations in general geometries. However, it is also possible to use the conven-
tional unsteady, fractional-step/projection-method NS solver at low Reynolds number,
combined with an interface description method [11, 12]. The latter approach is more
versatile, probably less difficult to implement, and enjoys a rich literature of standard
numerical techniques. Here, in view of a rich range of possible applications and consider-
ing also the rapid development of inertial microfluidics (where inertial effects are used to
better control the flow behavior) we take the approach of simulating the incompressible,
two-fluid NS as outlined in [13]. The splitting procedure proposed in [13] enables the use
of fast solvers for the pressure Poisson equation also for large density and viscosity con-
trasts. The remaining choice then is to be made among the available interface-description
methods.
Generally, there are two catagories of methods to resolve an interface in the frame of
a NS solver, i.e. front-tracking methods and front-capturing methods. An example of the
front-tracking method is the immersed boundary method (IBM) [14]. It is a Lagrangian-
based method, hence it offers a sharp description of the interface. However, for deforming
interfaces, these approaches requires frequent mesh rearrangement. Furthermore, topol-
ogy changes are to be handled manually, making such methods typically more expensive
and of limited interest for droplet simulations.
Front-capturingmethods,ontheotherhand,areEulerianandhandletopologychanges
automatically; theyarethereforetypicallyeasiertoparallelizetoachievehigherefficiency.
One of such methods is the volume-of-fluid (VOF) method [15], which defines different
fluids with a discontinuous color function. The main advantage of VOF is its intrinsic
mass conservation property. It suffers however from inaccurate computations of the inter-
face properties, e.g. normals and curvatures. This makes it less favorable for simulations
of microfluidic systems where surface tension becomes dominant.
Another popular front-capturing method is the level set (LS) method [16, 17]. Con-
trary to VOF, LS prescribes the interface through a smooth level set function which
usually takes the form of the signed distance to the interface. Due to its smoothness,
normals and curvatures of the interface can be readily and accurately computed. The
problem, nevertheless, is that mass loss/gain may occur and accumulate because of nu-
merical diffusion in the solution of the LS transport and because of the errors introduced
in a procedure called reinitialization, commonly required to reshape the level set into a
distance function. Therefore, additional measures have to be taken to fix this problem.
Many different approaches have been proposed to make LS conservative, which can
be classified into the following three methodologies. The first is to improve the level set
discretization and reinitialization, so that numerical errors are reduced. In practice, one
can increase the order of level set fluxes [18], minimize the displacement of the zero level
setduringreinitialization[19,18],oremploylocalmeshrefinement[20]. Bydoingso,mass
loss can be greatly reduced, although the formulation is still inherently non-conservative.
The second remedy couples the LS with the VOF, e.g. the so-called CLSVOF methods
2
[21], MCLS [22], or the most recent curvature-based mass-redistribution method [23].
With varying level of coupling, these methods can usually preserve mass equally well
as VOF; yet the complexity and some inaccuracy of VOF will be imported, making
it less elegant. The third approach to guarantee mass conservation is to modify the
the formulation of the LS, or the LS or NS equations, so that LS becomes globally
mass-conserving. Examples of this kind include the hyperbolic-tangent level set [24], the
extension-velocitylevelset[25], andthemass-preservingNSprojectionmethod[26]. Such
approaches are conceptually simple and can lead to good mass conservation; however,
they may also result in unphysical numerical artifacts or have limited applications.
With the physical background of colloidal droplets in mind, and partially inspired by
the mass-preserving NS projection method [26], we propose a simple interface-correction
level set (iCLS) method. The idea is to obtain a normal velocity field at the droplet
interface and use it to perform an additional advection of the LS function to compensate
for mass loss, in a way similar to inflating a balloon. Because no coupling with VOF is
required, the simplicity and high accuracy of the original LS method is preserved, yet the
extra computational cost of this procedure is negligible.
Based on the newly proposed iCLS method, we extend the single level set (SLS)
function to multiple level sets (MLS). MLS has the benefits that each droplet can be
treated individually, is allowed to interact with the other droplets and is guarded from
its own mass loss. MLS also prevents numerical coalescence of droplets when they get
close. The computational cost of MLS is proportional to the number of droplets and the
number of total cells, i.e. (lN2) in 2D or (lN3) in 3D, with l being the number of
O O
droplets and N being the number of cells in each dimension. With a simple narrow band
approach, this cost can be roughly reduced to (lN) and (lN2), respectively.
O O
One difficulty of the simulation of multiphase flow is the presence of surface tension,
which manifests itself as a pressure jump across an infinitesimally thin interface. In
microfluidic systems, particularly, surface tension becomes more prominent since it scales
linearly with the dimension of the system (hence diminishes slower than volumetric forces
such as gravity). To handle this discontinuity, the most widely-adopted approach is the
continuumsurfaceforce(CSF)[27], whichsmearsoutthedensitycontrastandmodelsthe
surface tension as a smoothed forcing term near the interface. CSF effectively introduces
an artificial spreading of the interface and is prone to errors when two interfaces are
within its smoothing width.
Another interesting technique, in the context of finite-difference discretizations of
the NS equations, is the ghost fluid method (GFM). It was originally devised to keep
a sharp density profile in the compressible Euler equations [28], but has later been ex-
tended to treat the normal stress discontinuities in incompressible NS equations [? ]. As
shown by Desjardins et al [30], using GFM for the pressure jump due to surface tension
and neglecting the normal jump due to viscosity difference result in a second-order ac-
curate pressure field, while preserving the discontinuity for both pressure and density.
Their original algorithm is framed for solving the pressure Poisson equation iteratively,
e.g. through a multi-grid solver. Here, we first propose a second-order accurate, grid-
converging interface curvature estimation, and then adapt it to a fast pressure-correction
method (FastP*) [13] that benefits from an efficient, FFT-based, direct solver. The only
additional requirement from [13] is that the computational domain is periodic in at least
two directions.
Finally, an unique challenge to the simulation of colloidal droplets is the modeling of
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near-field interactions. It is known that two or more colloids can interact via dispersion,
surface, depletion, and hydrodynamic forces [5]. Apart from the hydrodynamic forces,
which can be determined from the NS solver directly, and the dispersion forces, which
arise from quantum mechanical effects, the depletion and surface forces have to be mod-
eled. These forces can be either attraction or repulsion and are typically calculated from
the gradient of a potential. Hence, MLS is needed to determine the interaction forces.
Combining with GFM, we propose a novel method to compute the near field interactions
in our NS solver. The GFM makes the interaction force sharp and the MLS makes it
possible for two droplets to be closer than one grid cell without automatic merging.
The paper is organized as follows. In Sec. 2, the governing equations for the incom-
pressible, two-phase flow are briefly presented. We then describe in Sec. 3, the classical
signed-distance level set methodology and the numerical methods adopted. In sec. 4, we
propose an interface-correction level set method that is mass-conserving. In Sec. 5, we
derive an interface curvature estimation and demonstrate its second-order convergence.
Our coupled flow solver is outlined in Sec. 6 along with three validation cases. Finally,
we propose a MLS/GFM-based method for the modeling of near-field depletion potential
in Sec. 7.
2. Governing equations for interfacial two-phase flow
The dynamics of the incompressible flow of two immiscible fluids is governed by the
Navier-Stokes equations, written in the non-dimensional form
u = 0, (1a)
∇·
(cid:18) (cid:19)
∂u 1 1 (cid:2) (cid:3) 1
+u u = p+ µ ( u+ uT) + g, (1b)
i
∂t ·∇ ρ −∇ Re∇· ∇ ∇ Fr
i
where u = u(x,t) is the velocity field, p = p(x,t) is the pressure field, and g is a unit
vector aligned with gravity or buoyancy. ρ and µ are the density and dynamic viscosity
i i
ratios of fluid i (i = 1 or 2) and the reference fluid. These properties are subject to a
jump across the interface, which we denote as [ρ] = ρ ρ for density and [µ] = µ µ
Γ 2 1 Γ 2 1
− −
for viscosity. For viscous flows, the velocity and its tangential derivatives are continuous
on the interface [31]. However, the pressure is discontinuous due to the surface tension
and the viscosity jump, i.e.
1 2
[p] = κ+ [µ] nT u n, (2)
Γ Γ
We Re ·∇ ·
where κ is the interface curvature, and n is the normal to the interface. If the surface
tension coefficient, σ˜, varies on the interface the tangential stress is also discontinuous.
In this paper, we assume constant and uniform σ˜. In Eqs. (1b) and (2), Re, We, and Fr
are, respectively, the Reynolds, Weber, and Froude numbers, defined as
ρ˜U˜L˜ ρ˜U˜2L˜ U˜2
1 1
Re = , We = , Fr = , (3)
µ˜1 σ˜ g˜L˜
˜ ˜
where U, L, ρ˜, µ˜, and g˜ denote the reference dimensional velocity, length, density,
1 1
dynamic viscosity, and gravitational acceleration. Note that ρ = 1 and µ = 1 (i.e. we
1 1
define fluid 1 as the reference fluid).
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3. Classical level set methodology
In the level set framework, the interface Γ is defined implicitly as the zero value of
a scalar function φ(x,t), i.e. Γ = x φ(x,t) = 0 . Mathematically, φ(x,t) can be any
{ | }
smooth or non-smooth function; but it is classically shaped as the signed distance to the
interface [32, 17], viz.
φ(x,t) = sgn(x) x x , (4)
Γ
| − |
where x denotes the closest point on the interface from nodal point x, and sgn(x) is
Γ
a sign function equal to 1 or 1 depending on which side of the interface it lies. For
−
two-fluid problems with single level set, sgn(x) provides a natural “color function” for
phase indication. Furthermore, with this definition, geometric properties such as the unit
normal vector, n, and the local mean curvature, κ, can be conveniently computed as
φ
n = ∇ , (5)
φ
|∇ |
κ = n. (6)
−∇·
3.1. Advection
The motion of a fluid interface is governed by the following hyperbolic equation
∂φ
+u φ = 0, (7)
∂t ·∇
where u is the flow velocity field. Despite of its simple form, obtaining an accurate and
robust solution to Eq. (7) is challenging. For two-fluid problems, state-of-the-art level
set transport schemes include the high-order upstream-central (HOUC) scheme [18], the
weighted essentially non-oscillatory (WENO) scheme [31], the semi-Lagrangian scheme
[33], or the semi-jet scheme [34]. Quantitative comparisons of these schemes in various
test cases can be found in [18, 34]. We note that the choice of the scheme is case-
dependent, i.e. depending on the smoothness of the overall level set field or the stiffness
of Eq. (7). For flows involving moderate deformations, HOUC is usually sufficient and
most efficient. For more complex flows, WENO or semi-Lagragian/jet schemes combined
with subcell resolution might be pursed. In the present study, we use either HOUC5 or
WENO5 (5 denotes fifth-order accuracy) to evaluate φ.
∇
For the temporal discretization of Eq. (7), we use a three-stage total-variation-
diminishing (TVD) third-order Runge-Kutta scheme [35]. At each time step ∆t, it up-
dates φ from time level n to n+1 in three sub-steps

φ1 = φn +∆t f(φn)

 ·
φ2 = 3φn + 1φ1 + 1∆t f(φ1) (8)
 4 4 4 ·
 φn+1 = 1φn + 2φ2 + 2∆t f(φ2)
3 3 3 ·
where f(φ) = u φ.
− ·∇
5
(a) 50 50 (b) 100 100 (c) 200 200
× × ×
Figure 1: Comparison of the initial interface and its shape after one full rotation for
different mesh resolutions. Solid lines depict the initial interface. Two different schemes
have been used to evaluate the gradients, namely HOUC5 (dashed lines) and WENO5
(dash-dotted line).
Zalesak’s disk. The Zalesak’s disk [36], i.e. a slotted disc undergoing solid body rotation,
is a standard benchmark to validate level set solvers. The difficulty of this test lies in
the transport of the sharp corners and the thin slot, especially in under-resolved cases.
The initial shape should not deform under solid body rotation. Hence, by comparing the
initial level set field and that after one full rotation one can characterise the degree of
accuracy and mass conservation of a numerical solver. Here, the parameters are chosen
so that a disk of radius 0.15, slot width of 0.05 is centered at (x,y) = (0,0.25) of a
[ 0.5,0.5] [ 0.5,0.5] box. The constant velocity field is given as
− × −
u = 2πy, v = 2πx. (9)
−
Three different mesh resolutions have been considered, namely 50 50, 100 100 and
× ×
200 200. Fig. 1 depicts the shape of the interface after one full rotation of the disk for
×
the different meshes. Along with the results of the HOUC5 scheme (red dashed line), the
shape of the interface obtained using the WENO5 scheme (green dash-dotted line) is also
reported on this figure. Both schemes yield good results on fine grids, but the HOUC5
scheme clearly outperforms the WENO5 scheme on the coarsest mesh considered here.
3.2. Reinitialization
Although the level set function is initialized to be a signed-distance function, it may
lose this property as time evolves. This can cause numerical issues, particularly in the
evaluation of the normals n and curvature κ [17]. In order to circumvent these problems,
an additional treatment has to be introduced to reshape φ into a distance function, as in
Eq. (4). There are mainly two approaches to enforce Eq. (4), either with a direct, fast
marching method (FMM) [16], or converting it into a time-dependent Hamilton-Jacobi
equation [17]
∂φ
+S(φ )( φ 1) = 0 (10)
0
∂τ |∇ |−
6
5
0
5
− 5 0 5 5 0 5 5 0 5
− − −
(a) Initial condition (b) After 25 steps (c) After 100 steps
Figure 2: Illustration of the reinitialization procedure. The shape of the ellipsoid is
depicted as the thick solid line. The dashed lines then depict iso-contours of φ(x,y)
ranging from 1 to 1 by increments of 0.25.
−
where τ is a pseudo-time, and S(φ ) is a mollified sign function of the original level set,
0
defined as

−1 if φ0 < −∆x
S(φ0) = 1 if φ0 > ∆x (11)
 φ0 otherwise.

√φ2+∆x2
0
The first approach is more complicated to implement and is typically at best second-order
accurate. Here, we take the second approach. Specifically, we use the SSP-RK3 scheme
for time integration, and WENO5 to evaluate the spatial derivatives.
Distorted elliptic field. Inordertoillustratetheredistancingprocedure, atestcasesimilar
to the one by Russo and Smereka [37] is considered. Let us define the initial level set
field as
(cid:32)(cid:115) (cid:33)
(cid:18) (cid:19)
x2 y2
φ(x,y,0) = f(x,y) + 1
4 16 −
with f(x,y) a distortion function that leaves only the location of the interface unchanged.
The initial condition is displayed in Fig. 2(a), where the shape of the ellipse is depicted
as the thick blue line. The red dashed lines depict iso-contours of φ(x,y) ranging from -1
to 1. Clearly, this initial condition is far from being a distance function, the iso-contours
not being equidistant. As the function φ is evolved under Eq. (10), it slowly converges
towards a signed-distance function as can be assessed from Fig. 2(b) and (c). From a
practical point of view, it has to be noted that, provided the initial condition φ(x,y,0) is
closeenoughtoadistancefunction, thenveryfewiterationsofthereinitializationEq.(10)
are required for φ to be transformed back into a signed-distance function. Typically, one
iteration every ten to a hundred time steps is sufficient to reshape φ, see e.g. discussions
in [38]. Therefore, the computational cost of the reinitialization step is usually negligible.
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3.3. Narrow band approach
The level set methodology introduced so far applies to the whole computational do-
main. In fact, for multiphase-flow problems, only the local level set values adjacent the
interface are important. This has motivated the development of the so called narrow
band methods, which can reduce the operation counts from (N2) (2D) or (N3) (3D)
O O
to roughly (N) or (N2), with N being the number of points in one dimension; see
O O
e.g. [39, 40]. Here, we implement a similar narrow band approach as in [40], in which
the level set advection and reinitialization are solved within a short tube surrounding
the interface. In the original algorithm, a cut-off function c(φ) is applied to the advec-
tion equation φ +c(φ)u φ = 0 to avoid numerical oscillations propagating from the
t
·∇
boundary of the narrow band. The cut-off function, based on ENO or WENO schemes,
includes a smooth transition from the inner tube to the outer tube. For the HOUC5
scheme, among all the tests we have done, we find that a simple step function will work,
i.e.
(cid:40)
1 if φ < γ
c(φ) = (12)
0 otherwise
where γ is the distance from the edge of the narrow band to the interface. Note that γ
needs to be sufficiently large that the stencil of spatial discretization scheme lies totally
within it. For both HOUC5 and WENO5, the minimal value of γ is 3∆x. In the current
work, γ = 8∆x is used to provide the additional information required to model droplet
interactions (Sec. 7).
4. Interface-correction level set (iCLS) method
It is known that classical level set methods lead to mass loss when applied to multi-
phase flows, partially because there is no underlying mass conservation in the level set
formalism, partially because of the reinitialization procedure. Such errors can be re-
duced or even removed by using elaborated schemes [20] or coupling with VOF [21, 23].
However, doing so often makes the level set schemes complicated to implement and less
efficient. To maintain the simplicity of the original level set method, we propose the
alternative approach to conserve mass by performing small corrections near the interface.
Because such corrections are done by directly solving a PDE (same as Eq. (7)), the pro-
posed method is straightforward to implement in both 2D and 3D. Meanwhile, because
the correction does not need to be performed at every time step, the additional cost is
negligible. Below, we first present the derivation of the correction, then we demonstrate
the mass conservation with an example.
Let Γ divide a domain into two disjoint subsets Ω (e.g. a droplet) and Ω (e.g. the
1 2
ambient fluid), and V denote the volume of Ω . Without loss of generality, we let φ < 0
1
in Ω , and φ > 0 in Ω . The rate of change of V can be written as the integral of a
1 2
normal velocity, u , defined on Γ [26], i.e.
c
(cid:90)
δV
n u dΓ = , (13)
c
· δt
Γ
where n is the outward-pointing normal from the interface Γ. If δV/δt corresponds to
−
the mass loss over an arbitrary period of time (it does not have to be the time step of
the level set advection), then u can be thought as a surface velocity that corrects the
c
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volume by an amount δV/δt, hence compensating the mass loss. In other words, if u is
c
known, then the following PDE can be solved,
∂φ
+u φ = 0, (14)
c
∂t ·∇
after which the mass loss accumulated over δt is removed.
To obtain such a surface correction-velocity u , we introduce a speed function, f , an
c s
auxiliary variable, p , and express the rate of change of u as
c c
du
c
= f p . (15)
s c
dt − ∇
Here,p canbeseenasanon-dimensionalcorrection-pressureinΩ . Iff = 1,thephysical
c 1 s
interpretation of Eq. (15) is to inflate or deflate a balloon by δV with pressure p over ∆t
c
(see Fig. 3). This is more evident when we recast it in the form of the impulse-momentum
theorem (per unit “mass” of the interface)
(cid:90) ∆t
u = p dt, (16)
c c
− ∇
0
in which the correction-velocity is zero at t = 0, and we require f 1.
s
≡
For a general choice of f , we may substitute Eq. (16) into Eq. (13) which results in
s
(cid:90) ∆t (cid:90) δV
dt n ( f p )dΓ = . (17)
s c
· − ∇ δt
0 Γ
In order for p to be compatible with u , p has to be differentiated at the interface.
c c c
∇
This turns out to be easy in the level set framework, as we can use a 1D regularized
Heaviside function of φ, such as

1 if φ > (cid:15)


H (φ) = 1(cid:2)1+ φ + 1 sin(πφ)(cid:3) if φ (cid:54) (cid:15) (18)
(cid:15) 2 (cid:15) π (cid:15) | |
0 otherwise,
with (cid:15) = 1.5∆x being the half smoothing width. This way, the correction-pressure and
its gradient in Eq. (17) can be conveniently written as
(cid:0) (cid:1)
p = 1 H (φ) p , (19)
c (cid:15) 0
−
and
(cid:90) (cid:90)
p = δ (φ) φp , (20)
c (cid:15) 0
∇ − ∇
where δ (φ) is the derivative of H (φ), and p is a constant. Noting that n φ = φ =
(cid:15) (cid:15) 0
(cid:82) ·∇ |∇ |
1 for a distance level set function, and denoting f δ (φ)dΓ = A , we arrive at a simple,
Γ s (cid:15) f
explicit expression for the constant pressure
δV 1
p = , (21)
0
δt A ∆t
f
9
u
c
Ω
2
Ω
1
Figure 3: 2D illustration of the mass correction. The solid line represents the interface.
The arrows indicate the normal correction-velocity located at cell centers of the grid.
in which we have made a first-order approximation for the time integration in Eq. (16),
(cid:82)∆t
i.e. A dt = A ∆t. Finally, from Eqs. (15) (20) and (21), the correction-velocity can
0 f f
be written as
δV f δ (φ)
s (cid:15)
u (φ) = φ, (22)
c
δt A ∇
f
or
δV f
s
u (φ) = H (φ). (23)
c (cid:15)
δt A ∇
f
Eq. (22) or (23) gives an expression for the surface correction-velocity in Eq. (14).
Here, we have required a bounded support for u , i.e. u = 0 for φ (cid:62) (cid:15). There are two
c c
| |
benefits of spreading the surface velocity. First, it allows an easy handling of the interface
location, as u only depends on a 1D Dirac delta function of the level set. The choice of
c
δ (φ) can also be different from the trigonometric form in Eq. (18), e.g. one can use the
(cid:15)
linear hat in [41]. The important point is that we spread the correction-velocity rather
than the interface. The interface remains sharp, as it is implicitly represented by the
level set function. The second benefit of spreading u is that it greatly reduces the risk of
c
numerical instability. As u is supported on a 2(cid:15) band around the interface, the maximal
c
nodal value of u scales with 1/(cid:15). In our tests, we have never found its non-dimensional
c
value to exceed 1. Therefore, the CFL conditions imposed by Eq. (14) is always satisfied
as long as we use the same temporal scheme (e.g. RK3) for solving Eq. (7) and Eq. (14).
In Eq. (22) or (23), u is written as a function of φ only. The speed function, f ,
c s
introduced in Eq. (15), acts as a pre-factor for u . Specifically, we can define it in two
c
10