A front-tracking algorithm for accurate
representation of surface tension
Stephane Popinet and Stephane Zaleski
Modelisation en Mecanique, CNRS URA 229,
Universite Pierre et Marie Curie, 4 place Jussieu
75005 Paris, France. [email protected]
Abstract
We present a front-tracking algorithm for the solution of the 2D
incompressible Navier-Stokes equations with interfaces and surface
forces. More particularly, we focus our attention on obtaining an
accurate description of the surface-tension terms and the associated
pressure jump. We consider the stationary Laplace solution for a bubble with surface tension. A careful treatment of the pressure gradient
terms at the interface allows us to reduce the spurious currents to machine precision. Good results are obtained for the damped oscillations
of a capillary wave compared with the initial-value linear theory. A
classical test of Rayleigh-Taylor instability is presented.
Keywords: drops and bubbles, surface tracking, front tracking, interfaces,
surface tension, capillary waves.
1 Introduction
A number of methods have been developed in recent years for the solution of
problems involving moving interfaces in multiphase uid ow. These methods
can be divided into two main classes depending on the type of grids used
[20, 10, 33, 32]. In the rst class, the interface is treated as a boundary
between elementary domains. This approach allows a precise representation
of the interfacial jumps conditions, at least in principle. However, it requires
deformable grids in order to follow the motion of the interface [11, 12]. The
second class of methods uses xed grids to describe the velocity eld but
1
requires specic advection schemes in order to preserve the sharpness of
the interfacial front. These advection schemes may in turn be divided into
two groups, either implicitly or explicitly representing the interface. The
implicit methods are sometimes labeled front-capturing in analogy with the
situation in computational gas dynamics, while the explicit methods are often
called front-tracking. Among the implicit methods are Volume Of Fluid and
Level Sets. Modern Volume Of Fluid (VOF) advection schemes yield good
results and ensure an accurate conservation of mass [2, 19, 30]. Level-set
methods are easy to implement and are similar in their properties to volume
of uid methods [34]. Both rely on an implicit description of the interface,
given through phase functions (i.e. volume fraction for the VOF method or
distance function for the level-set method). In the second group, tracking
methods such as interfacial markers uses a more explicit discretization of the
interfacial discontinuity. They are somewhat more complex to implement,
but give the precise location and geometry of the interface [35, 36].
All these methods provide good solutions to the problem of interface
advection; however, accurate representation of surface forces (i.e. surface
tension, membrane eects, . . . ) remains a problem when using xed grids.
A striking feature of these methods (including lattice gases [31]) are the socalled spurious currents shown in gure 1. These numerical artifacts result
Figure 1: Spurious currents around a stationary bubble. The method used
for the interface advection is a VOF scheme.
from inconsistent modeling of the surface tension terms and the associated
pressure jump. In some case they may lead to catastrophic instabilities in
2
interface calculations. More generally, this poses the problem of obtaining
an accurate description of the steep gradients occurring at the interface.
These problems are often compounded by the presence of high density
ratios. Many multiphase ow problems of interest involve uids with very
high ratios of densities and viscosities simply because mixtures of liquids and
gases are the most common uids composing multiphase ows. Thus experiments often involve air bubbles in water, water droplets in air, wave breaking
etc. which are dicult or impossible to address with current multiphase ow
computational techniques. Moreover, with such uids, surface tension eects
are usually large compared to viscous damping. In these cases, the eect of
the unbalanced forces acting on the interface not only reduces the accuracy
but can lead to spurious currents [16] which create oscillations strong enough
to destroy the interface. The spurious currents are therefore more than just
a numerical inaccuracy but are a real limitation of methods used on xed
grids.
In this paper, we present a front-tracking method for the solution of
the two-dimensional incompressible Navier-Stokes equations with interfaces
and surface eects. We will more particularly investigate the accuracy of
the numerical representation of the surface tension and of the associated
pressure jump. In the section which follows, we describe the general scheme
used to solve the Navier-Stokes equations. The second section gives a detailed
description of the front tracking algorithm and in the nal section, some test
cases investigating the numerical accuracy and convergence properties are
presented.
2 General description of the method
2.1 Basic equations
We seek to solve the incompressible Navier-Stokes equations with varying
density and surface tension. The momentum and mass-balance equations
are
(@tu + u ru) = ?rp + r (2D) + sn + g
(1)
r u = 0:
(2)
The former may be written in conservative form
@u + r (u u) = ?rp + r (2D) + n + g;
(3)
s
@t
where u = (u; v) is the uid velocity, = (x; t) is the uid density,
= (x; t) is the uid viscosity, D is the rate of deformation tensor with
3
components Dij = (@iuj + @j ui)=2. The surface tension term is considered to
be a force concentrated at the interface, is the surface tension coecient, the curvature of the interface, s a distribution concentrated on the interface
and n is the unit normal to the interface. In the case of two immiscible uids,
we may dene a characteristic function which is equal to 1 in phase 1 and 0
in phase 2. Then and n are related by r = ns . In the absence of phase
change simply follows the uid motion and thus satises the advection
equation
@ + u r = 0:
(4)
@t
Density and viscosity are attached to the phases and thus can be expressed
as functions of (5)
= 1 + (1 ? )2
= 1 + (1 ? )2:
(6)
Alternately, the equations of motion may be written in jump condition form.
Jump conditions appear when one investigates eqs. (1) or (3) in the neighborhood of the singular surface S . This leads to the tangential stress condition
[t D n]S = 0;
(7)
where the notation [:]S represents the jump of a quantity across the surface
S . We also obtain from (1) or (3) the normal stress condition
[n (?pI + 2D) n]S = ;
(8)
while the usual assumption of continuity of velocity leads to
[u]S = 0:
(9)
Away from the interface eq. (1) takes the usual form
(@tu + u ru) = ?rp + r2u:
(10)
The normal stress condition (8) leads to the famous Laplace law for a spherical interface of radius R in undeformed ow:
[p]S = 2R :
(11)
4
ρVi,j+1/2
ρUi-1/2,j
Pi,j
ρ Ui+1/2,j
Ci,j
ρVi,j-1/2
Figure 2: MAC discretization of the pressure, volume fraction, momentum
and velocity components
ρVi,j+1/2
Pi,j
ρ Ui-1/2,j
Ωx
ρVi,j+1/2
ρUi+1/2,j
ρUi-1/2,j
Ci,j
Ci,j Pi,j
ρUi+1/2,j
Ωy
ρVi,j-1/2
ρVi,j-1/2
Figure 3: Control volumes for the u and v momentum components
5
2.2 The Navier-Stokes solver
We use a momentum-conserving formulation of the Navier-Stokes equation.
The solution technique is close to the one initially developed for the SURFER
code [16]. It is based on an explicit projection method [23]. The pressure,
volume fraction, momentum and velocity components are discretized on a
uniform Cartesian mesh (x = y = h) using a staggered Marker And Cell
(MAC) [23] distribution (gure 2). If we associate the control volumes x
and y of gure 3, to the components u and v of the momentum, we can
write the integral of (3)
@ Z udx = L(u; ) ? I pdS
(12)
@t @
where
I
I
Z
Z
L(u; ) = ? u u dS + 2D dS + sndx + gdx: (13)
@
@
Actually equation (13) is projected on the x-direction (resp. y) for the x
(resp. y ) control volumes. We denote these control volumes with half
integer indices, so x = i+1=2;j , y = i;j+1=2. (In all these expressions i; j
are integers.) This allows us to dene momentum variables centered on the
relevant cells
Z
1
uxdx
(14)
(ux)i+1=2;j = j
j
i+1=2;j
where j
j = h2 is the measure of the control volume . Similarly control
volumes centered on the pressure nodes have the form ij . The discrete formulation of (12) is then obtained by straightforward approximations of and
u at the center of the faces of the control volumes x and y . Interpolations
are performed as required. The values of and near the interface are obtained by an approximation of the characteristic function by the volume
fraction
Z
Cij = dx
(15)
ij
with i; j integer or half-integer. Then we approximate eqs. (5) and (6) by
ij = Cij 1 + (1 ? Cij )2
(16)
ij = Cij 1 + (1 ? Cij )2:
(17)
R
The discretization of the surface tension term sndx, which is the main
topic addressed in this paper, will be described separately in the next session. First we give an outline of our explicit projection method. It can be
subdivided into four steps:
6
1. The interface is advected using the velocity eld at time tn = n where
is the time step. In our case this is done using a Lagrangian advection
of the markers. This denes the approximation of characteristic function n+1 at time tn+1 . The volume fraction may then be computed by
quadrature.
2. A provisional solution (u)? of the momentum at time (n +1) is built
using an explicit discretization of (12)
(u)? = (u)n + L(un; C n+1)
(18)
where n is the value of the function at time n , L is a discretization
of the operator in eq. (13) and we approximate by C . A detailed
description of this step may be found in [13].
3. We then ensure the divergence-free character of the velocity eld un+1 .
From eq. (13) the eect of pressure on the momentum balance is
I
+1 = (u )?
(ux)ni+1
?
(19)
x i+1=2;j
=2;j
j
j i+1=2;j pdS:
It is useful at this point to dene the velocity elds in terms of the
momentum elds
+1 =n+1 ; (20)
u?x;i+1=2;j = (ux)?i+1=2;j =n+1 ; unx;i+1+1=2;j = (ux)ni+1
=2;j
with a similar denition for the y elds. Equation (19) can thus be
re-expressed in terms of the velocity elds
I
n
+1
?
u = u ? n+1 j
j pdS
(21)
i+1=2;j
We want to satisfy the continuity equation in the form
r un+1 = 0
(22)
at each point xi;j . This implies
I
pdS) = r u?:
(23)
r ( n+1j
j i+1=2;j
Discretizing as before on the MAC staggered grid, this leads to the
Poisson-like equation for the pressure
(24)
rh ( n+1 rhp(n+1)) = rh u?:
7
where the rh operator is the centered nite-dierence operator on the
staggered grid dened as
rhpx;i+1=2;j = h1 (pi+1;j ?pi;j ); rhpy;i;j+1=2 = h1 (pi;j+1 ?pi;j ): (25)
Although we use eq. (24) in the bulk, near the interface we need a
more accurate discretization of (23) which will be discussed in section
3.6 below. This equation is solved eciently using a multigrid solver
[6, 7, 25, 38].
4. The momentum and velocity elds at time (n + 1) are computed by
equation (19).
2.3 Interface and surface tension
tB
B
ρU
tA
A
Ω
Figure 4: In the \method of tensions" the contribution of the surface tension
to the momentum equation is obtained as the sum of two vectors tangent to
the interface. First the direction of the vectors is determined from a spline
representation of the interface, then the sum of the two tangent vectors times
is added to the momentum balance of the cell.
Given a parametric description (x(s); y(s)) of the location of the interface,
where s is the arc-length, we want to compute the source term due to the
surface tension in the momentum equation (13). If is the control volume
for one of the components of the momentum and AB , the interface segment
inside this control volume (gure 4), the integral source term in (13) is
Z
IB
(26)
sndx = nds;
A
which can be written using the rst Frenet's formula for parametric curves
IB
IB
(27)
nds = dt = (tB ? tA)
A
A
8
where t is the oriented unit tangent to the curve. The integral source term
due to the surface tension is then the sum of the outward unit tangents at
the points where the interface enters or exits the control volume.
3 Front-tracking algorithm
In this section, we present a detailed description of the front-tracking algorithm. The interface is represented using an ordered list of marker particles
(xi; yi), 1 i N . A list of connected polynomials (pxi(s); pyi (s)) is constructed using the marker particles and gives a parametric representation
of the interface, with s an approximation of the arc-length. Both lists are
ordered and thus identify the topology of the interface.
3.1 Advecting the points
The rst step in our algorithm is the advection of the marker particles. We
use a simple bilinear interpolation to nd the velocity inside cell i;j+1=2
u(x; y) = ui?1=2;j (1 ? x ? y + xy) + ui+1=2;j x(1 ? y) +
ui?1=2;j+1y(1 ? x) + ui+1=2;j+1xy;
(28)
where x and y are the coordinates of the point relative to the (i ? 1=2; j )
vertex. (Here and in the remainder of sec. 3 we rescale lengths so that
h = 1.) The marker particles are then advected in a Lagrangian manner
using a straightforward rst-order explicit scheme
xni +1 = xni + u(xni; yin)
yin+1 = yin + v(xni; yin):
(29)
Once the points have been advected we need to reconstruct the parametric
representation of the interface.
3.2 Constructing the polynomials
We have chosen to use connected cubic polynomials with continuous rst and
second-order derivatives. This type of curve is usually known as cubic splines
[37, 15, 24]. The parametric representation is often periodic as the interfaces
are mostly self-connected (drops, bubbles, periodic wave trains . . . ).
We then need to choose a parameter s in order to interpolate the two
sets of points (xi; si) and (yi; si). A simple choice is an approximation of the
9
arclength
si =
i?1 h
i1=2
X
(xj+1 ? xj )2 + (yj+1 ? yj )2 :
j =1
(30)
The connection conditions for the interpolating polynomials lead to two
pseudo tridiagonal systems Ba = c, one for each coordinate of the parametric curve, where B is a N 2 matrix of the form
1
0
b
c
0
0
a
1
1
1
BB a b c
CC
0 C
BB 2 . 2 . 2 .0
.. 0 C
CC
B=B
(31)
BB 0 . . . .
C
@ 0 0 aN ?1 bN ?1 cN ?1 A
cN 0 0 aN bN
where the coecients a1 and cN arise from the periodicity condition [15].
The solution of this type of system can be reduced to the solution of two
tridiagonal systems which are easily solved using classical techniques [24].
Thus, the construction of the interpolating parametric spline curve from the
set of points (xi) requires the solution of four tridiagonal systems of size N 2.
This can be done in O(N ) operations. All the other operations of our marker
algorithm deal with local computations along the interface. They are thus
also of order N . The ratio between the time spent in the marker algorithm
and in the computations done on the bulk of the uid (the Navier-Stokes
solver) is then of order 1=N . As the domain size increases the proportion
of computational time needed by the marker representation decreases. In
practice, the marker computation accounts for less than 10 % of the time for
a 642 mesh.
3.3 Redistribution
As the interface evolves, the markers drift along the interface following tangential velocities and we may need more markers if the interface is stretched
by the ow. We then need to redistribute the markers in order to ensure
an homogeneous distribution of points along the interface. This is done at
each time step using the interpolating curve (x(s); y(s)). As s is an approximation of the arclength, if we choose a redistribution length l, the
new number of markers is Nnew = sN =l and the points are redistributed
new
as (xnew
i ; yi ) = (x(il); y (il)). l is usually chosen as h, which yields an average number of one marker per computational cell. Decreasing this length
does not apparently improve the accuracy and in some cases leads to instabilities. This is similar to the behavior of boundary integral codes that also use
arclength parameterization, cubic splines and redistribution of nodes [21].
10
3.4 Estimation of the volume fraction
In the case of ows with varying density and/or viscosity between the phases,
we need to estimate the volume fraction C dened in eq. (15). We then have
to create the new volume fraction eld corresponding to the new parametric
interpolation of the interface. This is performed as described in Appendix B.
3.5 Surface tension contribution to the momentum equation
Following the tension formulation (27), if the interface crosses a momentum control-volume boundary, we compute the value of the parameter s
(using the same procedure as above) for the intersection point. We then
add the relevant component of surface tension contribution t where t is
the unit tangent to the curve pointing outward for the considered cell: t =
(x0(s); y0(s))=(x02(s) + y02(s))1=2. As this computation is done once for the
two neighboring cells, the contributions of surface tension to the global momentum cancel to machine accuracy. Thus the sum of surface tension forces
over a droplet or bubble is zero, even if there are errors for the surface tension
in a single cell.
3.6 Pressure gradient correction
For our staggered grid, the xH-component of the integral contribution of the
pressure gradient is given by @
pdS = h(pi;j ? pi?1;j ). For instance, consider
an interface crossing the vertical face BC of the control volume (g. 5).
There is a pressure jump either between pi;j and pi;j?1 (case (a): EB < h=2),
or between pi;j and pi;j+1 (case
(b): EB > h=2). We can have a better
R
C
approximation of the integral B pdS using the nearest pressure nodes
if EB < h=2: RBC pdS = EBpi;j?1 + ECpi;j .
if EB h=2: RBC pdS = EBpi;j + ECpi;j+1 .
This can be viewed as a correction to the standard centered estimate. Dene
if EB < h=2: Ii;j [p] = EB (pi;j?1 ? pi;j ).
if EB h=2: Ii;j [p] = (h ? EB )(pi;j+1 ? pi;j ).
R
Then BC pdS = hpi;j + Ii;j [p]: Ideally, this \pressure-gradient correction"
should be applied when solving the pressure equation (23). This would lead
11
Pi,j+1
C
C
ρUi-1/2,j
ρUi-1/2,j
Pi-1,j
Pi,j
Pi-1,j
E
Pi,j
E
B
B
Pi,j-1
(a)
(b)
Figure 5: The pressure gradient correction for a MAC grid. The location of
the interface relative to the pressure nodes yields two cases: (a): EB < h=2,
(b): EB > h=2.
to a discretization of (23) in the form
rh n+1 rh(p(n+1) + I [pn+1]) = rh u?:
(32)
We have chosen a simpler approach, however, where the pressure gradient
correction is considered as a source term in the right-hand side of the momentum balance equation (13):
rh n+1 rhp(n+1) = ?rh n+1 rhI [p(n)] + rh u?:
(33)
This term is then computed using the pressure eld at time n . This is
done in practice when computing the surface-tension contribution which also
requires the intersection point E . (It is less accurate to use (33) instead of
(32) but justied as the method gives good results for dicult test cases.)
This pressure gradient correction is probably the simplest choice available.
However several other solutions exist and it would be interesting to assess
their accuracy and convergence properties. A simple extension is the choice of
higher order interpolation schemes on both sides of the interface. The mean
square interpolation proposed by Shyy et al. [33] would also be applicable and
could be used to obtain any term with rapid variations across the interface
(e.g. viscous terms for high contrasts of viscosity).
12
4 Results
In this section, we present some simple test cases intended to illustrate the
ability of our method to cope with high surface-tension ows without loss
of accuracy. All the computations have been done on an IBM RS6000/370
workstation. Typically one time step on a 1282 grid requires 0.7 second, the
most time-consuming procedure being the multigrid solver.
4.1 Stationary bubble and spurious currents
An interesting test case is the verication of the stationary Laplace solution
for a circular bubble or droplet. As shown previously, in principle a stationary
solution exists for our method and consequently Laplace's law should be
veried exactly. However, one cannot interpolate a circle (which is C 1)
exactly using a parametric spline curve (C 2 in our case) and small dierences
will subsist. Two questions follow, the stability of this approximate solution
and its numerical convergence to the theoretical solution.
In the absence of externally imposed velocity, the relevant dimensionless
numbers are the Ohnesorge number Oh = =(D)1=2, (or the less-often used
Laplace number La = 1=Oh 2) and the ratios 1=2 and 1=2 . When numerical simulations are performed, spurious currents of amplitude U are observed. In [16] it was conjectured that the amplitude of the spurious currents
must be proportional to =. This is equivalent to having an approximately
constant value of the capillary number Ca s = U=.
The Reynolds number UD= of the spurious currents is then proportional to La = D=2 . When La is large, computations become dicult
in our previous scheme [16] because the spurious currents develop a kind
of turbulence, shaking the droplet in a kind of Zitterbewegung. However,
in our new scheme with pressure correction, we observe two interesting features. First there is remarkable constancy of Ca s with Oh , indicating that
spurious currents are very predictable. Second we can reach high values of
1=Oh 2 which was not previously possible. The nal bonus is that increasing
the resolution at xed Oh makes the spurious currents decrease to machine
accuracy.
Table 1.a illustrates the constant character of Ca over a broad range of
Ohnesorge numbers. We used a 322 Cartesian mesh with periodic boundary
conditions in the x direction and reecting conditions on the horizontal walls,
the density and viscosity ratios are one, the ratio between the diameter and
the box width is 0.4. We measured the maximum amplitude of the spurious
currents after 250 characteristic time scales (t = tphys=(D) = 250). A
second series of tests was performed with the same geometry, for a value
13
1=Oh2
1.2
12
120
1200
12000
jU j=
8.58e-6
6.76e-6
5.71e-6
5.99e-6
8.76e-6
(a)
Grid size Diameter in grid points
162
6.4
322
12.8
642
25.6
1282
51.2
2
256
102.4
(b)
jU j=
3.76e-5
6.68e-6
10.7e-7
11.5e-8
17.4e-9
Table 1: Amplitude of the spurious currents around a circular bubble. (a)
Independence of the non-dimensional maximum velocity with respect to the
Ohnesorge number on a 322 mesh. (b) Convergence to zero of the nondimensional maximum velocity with spatial resolution.
of 1=Oh 2 of 12,000 and increasing spatial resolutions. Table 1.b shows a
convergence of the method to the theoretical solution which is faster than
second order in the spatial resolution (the exponent is approximately 2.75).
For the maximum resolution of 2562 the absolute error is of the order of the
machine precision.
The same test was done using the Piecewise Linear Interface Calculation
(PLIC/VOF) code [19]. Figure 6 shows that without the correction to the
pressure gradient, the amplitude of the spurious currents is independent of
the spatial resolution for PLIC/VOF, which results in spurious currents 105
times stronger than with the markers method for the 2562 case.
4.2 Capillary wave
Another simple but important test is the solution of the damped oscillations
of a capillary wave. The linear theory for the small amplitude oscillation of
an interface between two inviscid uids of equal density in an unbounded
domain, gives the dispersion relation
3
(34)
!02 = k
2 ;
where k is the wavenumber [18].
4.2.1 Normal-mode analysis
The normal-mode analysis for an interface between two viscous uid may
be found for instance in Lamb [17]. For small amplitude perturbations, and
14
−2
10
−3
10
|U|mu/sigma
−4
10
PLIC
Markers
−5
10
−6
10
−7
10
−8
10
0.0
20.0
40.0
60.0
80.0
Diameter in grid points
100.0
Figure 6: Amplitude of the spurious currents versus spatial resolution for the
PLIC method without pressure correction and the current method.
15
when the viscosities and densities of both uids are the same we have the
dispersion relation
3
2
!2 = k
(35)
2 (1 ? k=q) = !0 (1 ? k=q);
where
q
q = k2 ? i!=:
(36)
which gives the viscous correction to the inviscid case (34) and the damping
ratio of the amplitude. Eliminating ! between (35) and (36) one nds that
q is a root of the quartic equation
3
k
2
q(q ? k)(q + k) = ? 22 ;
(37)
which can be solved numerically.
4.2.2 Initial-value problem
As will be shown from the numerical simulations, this normal-mode analysis
is not suitable for the study of the related initial-value problem. As shown
by Prosperetti et al. [8, 26], an analytical solution (given in Appendix B)
exists for the initial-value problem in the case of the small-amplitude waves
on the interface between two superposed viscous uids, provided the uids
have the same dynamic viscosity.
4.2.3 Comparison with the numerical simulations
The test case for the capillary wave is a square box divided in two equal
parts by a sinusoidal perturbation. The wave length is equal to the box
width. The boundary conditions are free slip on the top and bottom walls,
periodic along the horizontal axis. The ratio between the initialp interface
perturbation H0 and the box height is 0.01. We let Oh = 1= 3000 the
non dimensional viscosity = k2=!0 6:472 10?2 , and the two uids
densities are the same. Figure 7 shows the evolution of the amplitude with
time for the numerical solution and both the normal-mode and initial-value
analytical solutions. The normal-mode curve is obtained using H0ei!t =!0
where ! is given by eq. (35).
In order to study the spatial convergence of the method, we have repeated
this test for resolutions of 82, 162, 322 and 642 . Table 2 summarizes the
results. We obtain a good convergence and a relative error of order 10?2 from
0
16
Relative error
0.10
0.05
0.00
−0.05
−0.10
0.010
Front tracking
Initial value
Normal mode
Relative amplitude
0.005
0.000
−0.005
−0.010
0.0
10.0
Non dimensional time
20.0
Figure 7: Time evolution of the amplitude of a capillary wave in the viscous case. Our front-tracking method is used on a 1282 square grid. The
theoretical curves are obtained from a normal-mode analysis and from the
exact solution to the initial-value problem in the linearized viscous case (see
Appendix B).
17
0.010
Relative amplitude
Front tracking
Initial value
Normal mode
0.000
−0.010
0.0
2.0
Non dimensional time
4.0
Figure 8: Close up of the previous gure illustrating the dierence between
the normal-mode and initial-value solutions.
Grid size Error/Initial value Error/Normal mode
82
0.2972
0.3077
162
0.0778
0.0959
2
32
0.0131
0.0332
642
0.0098
0.0307
2
128
0.0065
0.0280
Table 2: Evolution of the relative error between numerical computations and
the normal-mode and initial-value analytical solutions. The relative error is
the r.m.s. of the dierences between the solutions divided by the amplitude
of the initial perturbation. We compare the solutions for the rst 25 nondimensional
p time units. 1 = 2, non-dimensional viscosity 6:472 10?2 ,
Oh = 1= 3000.
18
Grid size Error/Initial value
82
0.3593
162
0.1397
2
32
0.0566
642
0.0264
2
128
0.0148
Table 3: Evolution of the relative error between numerical computations and
thepinitial-value analytical
solution. 1 = 102 , 4:799 10?2 , Oh 1 =
p
1= 3000, Oh 2 = 1= 30000.
a spatial resolution of 322 compared to the initial-value analytical solution
of Prosperetti. Moreover, we treat here the case of a bounded domain of
aspect ratio h= = 1=2 where h is the half height of the computational
box and = 2=k the wavelength. The analytical solution is given in the
innite depth case and should be corrected with a term of order cotanh(kh) =
cotanh 1:0037.
The previous test was performed with a density ratio of one. Table 3
illustrates the results for a density ratio of ten.
4.3 Rayleigh-Taylor instability
In order to illustrate the capability of our method to deal with more complex
cases, we present here a classical test. The original version of this computation has been performed by Puckett et al. [29, 30] using a VOF type method.
A one-meter wide, four-meters high rectangular domain is discretized using
a 64 256 grid. The uid densities are 1.225 and 0.1694 kg=m3. The uid
viscosities are 0.00313 kg:m?1:s?1. The interface between the uids is an initially sinusoidal perturbation of amplitude 0:05 meters. Figure 9 shows the
evolution of the interface at times 0, 0.7, 0.8 and 0.9 seconds. The maximum
mass uctuation of our method is approximately 0:14%, which is larger than
the observed variation for a VOF-type method (about 0:01%). The interface
evolution compares well with the results of Puckett et al. [29] and with the
simulation done using the VOF/PLIC algorithm of ref. [19] on a 128 512
grid (gure 10).
19
t=0.0
t=0.7
t=0.8
t=0.9
Figure 9: Rayleigh-Taylor instability on a 64 256 grid using the fronttracking algorithm.
20
t=0.0
t=0.7
t=0.8
t=0.9
Figure 10: Rayleigh-Taylor instability on a 128 512 grid using the
VOF/PLIC algorithm.
21
5 Conclusions
In summary, we have presented a front-tracking numerical algorithm able to
deal accurately with surface tension. For the case of a stationary bubble, the
method shows a convergence to the theoretical solution which is faster than
second order in the spatial resolution. For reasonable mesh sizes, the spurious
currents usually present when using xed grids can then be reduced to machine accuracy. Good results are obtained for the test case of a capillary wave
with viscous damping. We obtain very good agreement with Prosperetti's
solution to the linearized initial-value problem. The relative error for the
time evolution of the amplitude is smaller than 10?2 when using 32 or more
points per wavelength. The method is robust and remains accurate even for
very small Ohnesorge numbers (10?3 or less). The implementation of the
method leads relatively easily to a fast code: as we use a xed regular grid,
the Navier-Stokes algorithm is ecient. Moreover, due to its one-dimensional
nature, the surface-tracking part is negligible in terms of computational time.
The marker representation gives a precise location of the interface and is
therefore well-suited for the study of small scale eects.
We note however, that we have not yet implemented a reconnection mechanism to deal with topology changes. Consequently, our method conserves
the small lamentary structures even for low resolutions. We are currently
studying a reconnection mechanism based on a physical representation of
the short-range interactions between interfaces. Whereas VOF-type methods introduce an arbitrary reconnection length (usually the mesh spacing),
this method should provide a physical criterion for the reconnections. This
advantage of explicit front-tracking in providing a potentially better answer
to the reconnection problem was recognized in ref. [36].
The main idea of this paper is to take into account the fact that the
interface is a sharp discontinuity when building the surface tension representation. This idea could be generalized to include the other singularities of
the equation, of which the most prominent example is the jump in density.
It remains to be seen whether such an improvement will be as easy to obtain
as in the case of surface tension.
An interesting comment on our approach is that it runs counter to several
attempts to deal with singularities in the equations by smoothing them, i.e.
distributing them over neighboring mesh nodes through appropriate smoothing kernels [36, 5, 1].
The ideas developed in this paper can be generalized to dierent types of
Navier-Stokes solvers and are not limited to projection methods on Cartesian
grids. In particular, we emphasized the necessity of a precise and consistent
discretization of the surface tension and of the associated pressure jump.
22
This requires the precise knowledge of the interface position relative to the
underlying grid that front tracking usually provides.
It is important to note that this approach could be and should be applied
to the discretization of all the gradient terms with rapid variations across the
interface, in particular the viscous stress tensor for high viscosity contrasts.
Numerous applications require an accurate representation of the surface
eects: motion of drops of water in air [39, 28], wave breaking [3] or sonoluminescence [27]. Moreover, the exact description of the interface position
allows an easy implementation of the membrane models found in biological
mechanics [4, 14] and of the eect of surfactants [22]. An interesting study
could also be a detailed sub-grid modeling of the reconnections between interfaces [9].
We are currently working on an axisymmetric version of the code. In the
future, we plan to develop a fully three-dimensional front-tracking algorithm.
References
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Tahoe, CA, 1995. May be obtained from the author's web page at
http://math.math.ucdavis.edu/ puckett/.
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23
[9] Jens Eggers. Universal pinching of 3D axisymmetric free-surface ow.
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Comput. Phys., 113:134{147, 1994.
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1995.
[20] James Mc Hyman. Numerical methods for tracking interfaces. Physica
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[21] Ali Nadim. private communication.
24
[22] Y. Pawar and K. J. Stebe. Marangoni eects on drop deformation in an
extensional ow: The role of surfactant physical chemistry. I. Insoluble
surfactants. Phys. Fluids, 8:1738{1751, 1996.
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Mechanics, 25:577{602, 1993.
[29] E. G. Puckett, A. S. Almgren, J. B. Bell, D. L. Marcus, and W. J.
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25
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26
Appendix A: estimation of the volume fraction
In the case of ows with varying density and/or viscosity between the phases,
we need to estimate the volume fraction C dened in eq. (15). We then have
to create the new volume fraction eld corresponding to the new parametric
interpolation of the interface.
Consider a cell crossed by the interface. Phase 1 occupies a subset (gure 11). We wish to compute the volume fraction Cij = jj=j
j. This
problem can be reduced to the computation of a circulation along a parametric curve as illustrated below. Following Stokes theorem, if P and Q are
B
A
C
Ω
δΩ
Cij
D
Figure 11: Computation of the volume fraction
two functions of the space coordinates (x; y) we can write
Z @Q @P !
I
Pdx + Qdy =
? @y dxdy;
@x
@
if we choose P = 0, Q = x we get
Z
I
xdy = dxdy;
@
(38)
(39)
which can be written using the parametric description of @ . We redene
x(s); y(s) to follow the sum of the interface S and the \wet" boundary of for s = s1 to s2.
Z s2
x(s)y0(s)ds = jj:
(40)
s1
For our third order polynomial parametric function (x(s); y(s)),
x(s) = axs3 + bxs2 + cxs + dx
y(s) = ay s3 + by s2 + cy s + dy ;
27
(41)
thus the contribution of the interface S to the integral is given by
Z sD
x(s)y0(s)ds = 12 axay s6 + 51 (3bxay + 2axby )s5 +
sA
1 (a c + 2b b + 3c a )s4 + 1 (b c + 2c b + 3d a )s3 +
x y
x y
x y
x y
4 xy
3 xy
1 (c c + 2d b )s2 + d c s:
(42)
x y
x y
2 xy
where the points sA and sB correspond to the intersections with @ as on
the gure. We must then add the circulation along the faces of the cell. In
the case of our Cartesian grid, three useful observations can be made:
R
1. As for the horizontal faces xdy = 0, only the vertical faces contribute
to the circulation.
R
2. If Ra vertical face is not crossed by the interface it contributes xdy =
x dy = x, which is an integer number.
3. We impose a maximum CFL number of 0.5. In this case the variation
of the volume fraction during any time step can not be larger than 0.5
in any cell. Consequently, we can write
cn ? 0:5 cn+1 cn + 0:5
(43)
If we compute only the fractional part of the circulation, c~n+1, then the
sought result is given by cn+1 = c~n+1 + i, where i is an unknown integer.
However i is uniquely dened by (43) and can take only four distinct values
f?1; 0; 1; 2g.
It is now easy to compute the volume fraction, we just follow the interface, as dened by the list of polynomials. We test whether the interface
intersects the horizontal and/or vertical grid lines. The point of intersection
is computed using a combination of bisection and Newton-Raphson methods
[24]. If there is no intersection (e.g. segment BC in gure 11), we just add
the circulation along the whole polynomial segment, computed using (42),
to the cell volume fraction c~. If there are any intersections, we compute the
circulation for each part of the polynomial segment and add the circulation
to the corresponding cell (segments AB and CD). When the intersection is
on a vertical segment, we add the vertical contribution to the circulation to
one cell and subtract it from the neighboring cell (gure 12). Once all the
interfaces contributions have been added, we use (43) to compute the value
of the integer constant i to be added to get cn+1 . With this algorithm, the
28
Ω
δΩ
δΩ
Ω
(a)
(b)
Figure 12: Intersection of an interface with a vertical face. Two cases arise
depending on the topology of the domain.
cases where an interface (or several interfaces) crosses more than once the
same cell are treated implicitly.
Cells which were crossed by the interface at the previous time step have
their volume fraction eld updated to zero or one according to whether they
now lie outside or inside the domain respectively. The inequality (43) gives
the correct value.
Appendix B: initial value solution
We describe the solution to the initial-value problem [8, 26] which we use for
comparisons. We consider here the case of an initial sinusoidal perturbation
of amplitude H0, the two uids being at rest. If we take as fundamental time
!0?1, with !0 given by !02 = k3=(1 + 2), and as fundamental length k?1,
we can set
(t0) = !0t; = k2=!0:
(44)
Using these non-dimensional time and viscosity, the analytical solution for
the non-dimensional amplitude a = H=H0 is given in compact form by
2
a((t0)) = 8(14(1? ?44)2)+ 1 erfc(1=2(t0)1=2)
4
2
X
+ Zzi z2 ?!0!
0
i=1 i i
2
exp[(zi ? !0)(t0)=!0]erfc(zi(t0)1=2=!01=2);
29
(45)
where the zi's are the four roots of the algebraic equation
z4 ? 4 (!0)1=2z3 + 2(1 ? 6 )!0z2 +
4(1 ? 3 )(!0)3=2z + (1 ? 4 )(!0)2 + !02 = 0;
(46)
and Z1 = (z2 ? z1)(z3 ? z1)(z4 ? z1) with Z2, Z3, Z4 obtained by circular
permutation of the indices. The dimensionless parameter is given by =
12=(1 + 2)2.
30