Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol24.Issue3-4.2001:

Advances in Water Resources 24 (2001) 461±478
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Surface tension e€ects on two-dimensional two-phase
Kelvin±Helmholtz instabilities
Raoyang Zhang, Xiaoyi He, Gary Doolen, Shiyi Chen *,1
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Received 8 February 2000; received in revised form 21 July 2000; accepted 31 August 2000

Abstract
The two-dimensional Kelvin±Helmholtz instability is studied using a lattice Boltzmann multi-phase model in the nearly incompressible limit. This study focuses on the e€ects of surface tension on the evolution of vortex pairing in a two-dimensional
mixing layer. Several types of interface pinch-o€ are observed and the corresponding mechanisms are discussed. The contribution of
surface tension to the ¯ow kinetic energy is mainly negative. Part of this kinetic energy can be transformed to potential energy stored
in the surface tension. The contribution of surface tension to the ¯ow enstrophy is positive and small vortices are generated near
interfaces. Broken interfaces and small vortices near interfaces dominate the late stage of ¯ow ®elds with strong surface
tension. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction
The Kelvin±Helmholtz instability is a ¯ow instability
in which variation of either velocity or density occurs
over a ®nite thickness. An example of this type of ¯ow is

free shear ¯ow, which has served as a generic inhomogeneous model for studying mixing and transport
phenomena in numerous natural and industrial
processes such as chemically reacting ¯ows.
In the past several decades, single-phase free shear
¯ow has been studied extensively by many investigators
using precise experimental measurements, rigorous
analysis, and more recently by direct numerical simulations (DNS). The linear stability of free shear mixing
layer velocity pro®les has been well documented [2,16].
The character of the subharmonic disturbances which
leads to pairing was ®rst explored by Kelly [14] and later
veri®ed numerically by Riley and Metcalfe [21] and by
Pierrehumbert and Widnall [18]. The revolutionary experimental results of Brown and Roshko [1] and Winant
and Browand [30] revealed organized, rollup vortical
structures, i.e. coherent structures, of turbulent planar
mixing layers. These large-scale rollup structures ap*

Corresponding author.
E-mail address: syc@cnls.lanl.gov (S. Chen).
1
Also associated with Department of Mechanical Engineering, The

Johns Hyknis University and National Key Laboratory for Turbulence
Research, Peking University, China.

peared to align primarily with the spanwise direction.
Recently with the rapidly improving computational facilities and algorithms, DNS has become more and more
capable of examining coherent structures and exploring
their detailed nonlinear dynamics. Several of the best
DNS were done by Metcalfe et al. [15], Moser and
Rogers [17] and others. Their results have been used to
complement our understanding of the large-scale structural evolution in mixing layers.
In a single-phase mixing layer, two-dimensional
large-scale rollup vortical structures are a result of the
Kelvin±Helmholtz instability [16,29]. These rollups are
also unstable to subharmonic disturbances [14]. This
instability leads to the pairing of the rollers (i.e., corotation and amalgamation of neighboring rollers) after
an initial linear growth stage. The occurrence of pairing
events depends strongly on the phase di€erence between
the fundamental and subharmonic perturbations. A
smaller phase di€erence leads to faster coalescence of the
pairing vortices. Pairing can occur in both two-dimensional and three-dimensional fully turbulent mixing

layers. The growth of the mixing layer is controlled by
pairings, though this may not be true in strongly turbulent mixing layers [12]. These two-dimensional rollers
are also unstable to three-dimensional perturbations.
This three-dimensional instability leads to arrays of
streamwise, counter-rotating rib structures, which reside
in the braid region between the rollers and connect the
bottom of one roller to the top of the next [5]. As the

0309-1708/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd.
PII: S 0 3 0 9 - 1 7 0 8 ( 0 0 ) 0 0 0 6 7 - 1

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R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

mixing layer becomes increasingly three-dimensional, it
undergoes a transition to turbulence and eventually results in a fully developed turbulent mixing layer.
Although the basic nonlinear behavior of the singlephase mixing layer has been studied extensively
[1,5,15,17], the understanding of immiscible two-phase
mixing layers is still quite limited due to the complexity

of the interfacial dynamics. In its early stage, linear
stability analysis can be applied which describes the
linear growth rate and its dependence on density ratio,
viscosity, surface tension and compressibility. When the
amplitude of a perturbation reaches 10±30% of its
wavelength, the perturbation grows nonlinearly to form
Kelvin±Helmholtz rollups. Linear stability theory fails
in this nonlinear regime and numerical methods become
useful alternatives. However, accurate and ecient
conventional methods, such as the spectral method,
cannot be extended in a straightforward manner. There
are some numerical studies of immiscible two-phase
mixing layers [13,19,20]. Most of them have simpli®ed
the problem and focused on the evolution of surface
tension e€ects in vortex sheets in relatively early stages.
Without taking viscous e€ects into account, several
di€erent boundary integral methods can be applied directly. Weber number dependent interfacial growth and
vorticity concentration along curved interfaces have
been observed. Furthermore, a recent study by Hou et al.
[13] revealed an interfacial pinch-o€ of the vortex sheet

due to surface tension. However, all these results are
limited to the inviscid case. Studies of the e€ects of
surface tension on the evolution of large-scale coherent
structures are still quite rare. The capillary number
(Ca ˆ ul=r) dependent evolution of kinetic energy,
enstrophy and ¯ow patterns in mixing layer have also
not been studied extensively. In vortex sheet simulations, topological singularities, usually associated with
the rapid production of localized circulation, occur and
force numerical simulations to stop whenever interfaces
intersect. In addition, the well-posedness of stable numerical integral methods is still problematic. These
technical problems do not allow the use of vortex integral methods to study the late-stage ¯ow dynamics of
mixing layers with interfacial interactions in which interface pinching and merging happen frequently.
With the recent development of the multi-phase lattice Boltzmann method (LBM) [4,32], DNS of immiscible two-phase mixing layers become increasingly
possible. Because of its kinetic nature, the LBM can
model interfacial dynamics properly by incorporating
molecular interactions [9,10,23,27]. Previous studies of
interface dynamics [31] and Rayleigh±Taylor instabilities [10,11], whose basic mechanism of rollups is the
Kelvin±Helmholtz instability, have shown promising
results for the use of our LBM multi-phase model for
the study of immiscible two-phase mixing layers. Although some conventional methods, such as front

tracking [8] and the level set method [26], have the
ability, in principle, to simulate two-phase Kelvin±
Helmholtz instabilities, no results have been reported.
In this paper, as a ®rst step, we focus on surface
tension e€ects in the evolution of two-dimensional
rollups. To make a direct comparison with previous
spectral results, all simulations have Re ˆ 250, which is
de®ned in the following section. This paper is organized
as follows: in the following section, we give a brief description of the LBM methods used in our study. In
Section 3, we study the single-phase mixing problem.
Section 4 is devoted to the e€ects of surface tension in
multi-phase mixing layers. Finally a summary is given in
Section 5.

2. Numerical method and initial ¯ow setting
2.1. LBM nine-speed single-phase and multi-phase models
To study the e€ects of surface tension in the ¯ow
dynamics in two-phase mixing layers, the following incompressible Navier±Stokes equations must be solved
for traditional methods [6,8]:

r
u ˆ 0;

…1†

oqu
‡ r
quu ˆ ÿrp ‡ q
g ‡ r
l…ru ‡ ruT †
ot
Z
ot
d…x ÿ x0f † dA0 :
‡r
os

…2†

Here u is velocity and divergence free, p the pressure,
r the surface tension coecient, and q and l are the
density and viscosity ®elds, respectively. t is the tangent
vector to the bubble surface, and s is the arclength coordinate. d is a three-dimensional delta function.
Unlike traditional methods, instead of solving these

highly nonlinear Navier±Stokes equations coupled with
highly curved interfaces on the macroscopic scale, we
solve the mesoscopic kinetic Boltzmann equation [3,7].
By solving for particle distribution functions coupled by
molecular interactions, the averaged macroscopic behavior can be simulated. Therefore, the primary tool
used in this study is DNS using the LBM. To test the
accuracy of our simulation results, both the standard
two-dimensional nine-speed single-phase LBM model
[4] and the two-phase LBM model [10] were used to
verify for the zero surface tension case.
Both the basic ideas and the numerical scheme of the
single-phase LBM have been fully discussed in [4]. As a
reminder, we give a brief summary. The density distribution function satis®es the following equation:
fi …x ‡ ei ; t ‡ 1† ˆ fi …x; t† ‡
…i ˆ 0; 1; . . . ; M†:

fi …x; t† ÿ fieq
;
s
…3†

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

Here fi is the particle velocity distribution function
which moves in the ith direction, ei the local particle
velocities in the ith direction, s the particle relaxation
time and fieq is the equilibrium particle distribution
function.
The ¯uid density q and the momentum density qu are
de®ned by
qˆ

M
X

qu ˆ

fi ;

i

M
X

fi ei :

i

The collision term, Xi ˆ …fi …x; t† ÿ fieq †=s, satis®es conservation of total mass and total momentum at each
lattice site
M
X
i

Xi ˆ 0;

M
X
i

Xi ei ˆ 0:

For the two-dimensional nine-speed LBM BGK model
with particle velocities (Fig. 1)
ei
8
0
>
>
>

p
>
>
: 2 cos …iÿ5†p ‡ p ; sin …iÿ5†p ‡ p
i ˆ 5; 6; 7; 8;
4
2
2
2

with w0 ˆ 4=9, wi ˆ 1=9 (i ˆ 1, 2, 3, 4), and wi ˆ 1=36
…i ˆ 5, 6, 7, 8).
The basic strategy of the LBM multi-phase model [10]
is to simulate interfacial dynamics by incorporating
molecular interactions. Unlike the ``standard'' LBM, a
pressure distribution function is proposed to reduce
numerical errors near the interfaces. An index function
is introduced to track the interface between the two
phases. The resulting surface tension is a function of the
density gradient.
Following the 1999 paper of He et al. [10], the distribution functions of the index function and pressure
satisfy the following equations:
fi …x ‡ ei ; t ‡ 1† ÿ fi …x; t†
f…x; t† ÿ fieq …x; t†
ˆÿ i
s
…2s ÿ 1† …ei ÿ u†
rw…/†
Ci …u†;
ÿ
2s
RT

…6†

gi …x ‡ ei ; t ‡ 1† ÿ gi …x; t†

gi …x; t† ÿ gieq …x; t†
s
2s ÿ 1
…ei ÿ u†
‰Ci …u†…Fs ‡ G†
ÿ
2s
ÿ …Ci …u† ÿ Ci …0††rw…q†Š:



ˆÿ

…4†

the equilibrium distribution function can be written to
O…u2 †


9
3
2
fieq ˆ qwi 1 ‡ 3ei
u ‡ …ei
u† ÿ u2 ;
…5†
2
2

463

…7†

Here s is the relaxation time which is related to the
kinematic viscosity by m ˆ …s ÿ 1=2†. fi and gi are newly
de®ned variables for the convenience of discretization.
The use of these two variables guarantees the secondorder accuracy of the scheme both in space and time. fi
and gi are related to the distribution functions of the
index function and pressure, fi and gi , by
…ei ÿ u†
rw…/†
Ci …u†;
fi ˆ fi ÿ
2RT

…8†

1
gi ˆ gi ÿ …ei ÿ u†
‰Ci …u†…Fs ‡ G† ÿ …Ci …u† ÿ Ci …0††rw…q†Š:
2
…9†
fieq

and

gieq

are corresponding equilibrium distributions
#
3ei
u 9…ei
u†2 3u2
eq
ÿ 2 ;
fi ˆ wi / 1 ‡ 2 ‡
…10†
2c4
c
2c
"
!#
2
3ei
u 9…ei
u†
3u2
eq
ÿ 2
‡
:
…11†
gi ˆ wi p ‡ qRT
2c4
2c
c2
"

Here R is the gas constant, T the background temperature and wi is the integral weight, which is the same as wi
in Eq. (5). G is the gravitational force and F is the surface tension written as
Fs ˆ j/rr2 /;
Fig. 1. Nine-speed LBM model.

…12†

where j determines the strength of the surface tension.

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R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

The macroscopic variables are calculated using:
X
fi ;
/ˆ
i

pˆ
qRT u ˆ

X
i

X
i

1
gi ÿ u
r…p ÿ qRT †dt ;
2

ei gi ‡

RT
…Fs ‡ G†dt :
2

The density and the kinematic viscosity are calculated
from the index function using
/ ÿ /l
…q ÿ ql †;
/h ÿ /l h
/ ÿ /l
…mh ÿ ml †;
m…/† ˆ ml ‡
/h ÿ /l

q…/† ˆ ql ‡

4 ÿ 2/

3

…1 ÿ /†

ÿ a/2 :

Here the d0w is the vorticity thickness de®ned by
2U
:
…oU =oy† j max

…13†

d0w ˆ

…14†

In addition to the mean velocity, simple perturbations
are added to the mean ¯ow ®eld in order to initiate
Kelvin±Helmholtz rollup. Unlike physical experiments,
unique initial perturbations composed of combinations
of fundamental (harmonic) modes and subharmonic
modes can be used in which the fundamental mode is the
most unstable mode (from linear theory). The initial
two-dimensional velocity ®eld is




p
u1;0 …y†
u1 erf… py=d0w †
exp…iax†
‡ A1;0
uˆ
v1;0 …y†
0




iax
u
…y†
:
…20†
exp
‡ A1=2;0 1=2;0
v1=2;0 …y†
2

where ql and qh are the densities of the light ¯uid and the
heavy ¯uid, respectively, ml and mh are the viscosities of
the light ¯uid and the heavy ¯uid, respectively, and /l
and /h are the minimum and maximum values of the
index function.
The function w in Eqs. (6) and (7) plays an essential
role in phase separation in multi-phase ¯ow simulations.
The following form for w…/† is used in this study:
w…/† ˆ /2 RT

the present study, a temporally growing layer is studied
because periodic boundary conditions can be easily applied in the streamwise direction. Since the ¯ow patterns
are very sensitive to initial conditions, to make a direct
comparison with previous spectral results, the initial
velocity ®eld is exactly the same as that used by Moser
and Rogers [17]. The initial mean ¯ow pro®le in the
streamwise direction x direction) is of the form
p
U ˆ u1 erf… py=d0w †:
…18†

…15†

This relation is associated with the Carnahan±Starling
equation of state for a nonideal gas [10], which has a
supernodal p±V ±T curve when the ¯uid temperature
is below its critical value. It is this supernodal p±V ±T
curve that induces the unstable range of / in which
d…w ‡ /RT †=d/ < 0. This unstable mechanism forces
the index ¯uid into one of its two separate stable states,
causing phase segregation [11]. In this study, we use
a ˆ 12RT , the same as He et al. used in [10] and in [11].
The corresponding macroscopic dynamical equations
are
1 op
‡ r
u ˆ 0;
…16†
qRT ot


ou
q
‡ …u
r†u ˆ ÿrp ‡ r
P ‡ j/rr2 / ‡ G:
ot
…17†

Here P ˆ qm…ru ‡ ur† is the viscous stress tensor. In
the nearly incompressible limit, the time derivative of
the pressure in Eq. (16) is small and the incompressible
condition is approximately satis®ed. Our approach is
close to the pseudo-incompressible technique in classical
CFD methods.
2.2. Initial velocity perturbation
Temporally growing mixing layers and spatially
growing layers have very similar physical phenomena. In

…19†

Here the second term on the right-hand side of the
above equation is the harmonic mode and the last term
is the subharmonic mode. u1;0 …y†, v1;0 …y† and u1=2;0 …y†,
v1=2;0 …y† are complex harmonic and subharmonic eigenfunctions determined by linear instability theory. They
are normalized so that their real parts are equal to one at
y ˆ 0. There is no phase di€erence between the fundamental mode and the subharmonic mode, which assumes that perfect pairing takes place with both modes
being introduced simultaneously. In our simulations,
a ˆ 0:8620 is used. This corresponds to having the most
unstable wave at the wavelength, k ˆ 2p=a ˆ 7:289.
To match the initial ®elds of Moser and Rogers [17],
we set A1;0 ˆ …0:019043166; ÿ0:012901† and A1=2;0 ˆ
…0:019043166; ÿ0:0052838†. The Reynolds number, Re ˆ
u1
d0w =m ˆ 250, is used throughout in this study.
The initial interface is evenly distributed around the
midpoint in vertical direction. Periodic boundary conditions for velocity and density are used in the streamwise direction. Free slip wall boundary conditions are
applied in the vertical direction
ou
j …y ˆ 0; y ˆ Ly †;
oy

v j …y ˆ 0; y ˆ Ly † ˆ 0:

…21†

Since the vortex evolution of the Kelvin±Helmholtz instability is relatively compact, the e€ect of the vertical
boundary condition on the inner ¯ow dynamics is expected to be negligible as long as the vertical domain size
is large enough. As in previous spectral simulations

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

[22,28], a square computational domain is used for most
of this study.

3. Two-dimensional single-phase mixing layer
In spectral shear ¯ow simulations, there are two basic
types: a velocity-based sine±cosine scheme and a vorticity-based Fourier scheme [22,28]. Since far-®eld
vertical boundary conditions are used in both schemes
and because the vorticity-based Fourier scheme is more
compact (reaching the far-®eld boundary condition more
rapidly than the velocity-based scheme), for a given domain size in the y direction, the domain truncation e€ect
is less in¯uential for the vorticity-based scheme than for
the velocity-based scheme. Therefore, results from vorticity-based scheme are relatively more accurate than
results from velocity-based scheme. The accuracy of both
the single-phase and multi-phase LBM can be demonstrated quantitatively by comparison with spectral results of the evolution of the momentum thickness dm . dm
is a measure of the momentum loss due to the presence of
the sheared mixing layer. It is de®ned as

2 !
Z
1 1
u
1ÿ
dy:
…22†
dm ˆ
4 0
u1
The time evolution of the momentum thickness is
shown in Fig. 2. All results have been properly nor-

465

malized. The solid line is the vorticity-based spectral
simulation. The dotted line is the velocity-based sine±
cosine spectral simulation. The vertical domain sizes in
both spectral simulations are set to 2L=k ˆ 1:5. The
dash-dotted line is the single-phase LBM simulation,
and the short-dashed line is the two-phase LBM simulation with no surface tension e€ects. In these two LBM
simulations, the vertical domain sizes are set to
2L=k ˆ 1:0. The grid for the above four results was 1282 .
The ®fth line, the long-dashed line, is a single-phase
LBM simulation with vertical domain size 2L=k ˆ 2:0.
The evolution of the momentum thickness in this stage
appears to be controlled mainly by vortex pairing. All of
the peaks correspond to the vertical vortex pairing.
From the ®gure, we can see that the LBM results match
the spectral results quite well, although we have to note
that the boundary conditions in the y direction are
slightly di€erent. Unlike spectral simulations, a free slip
boundary condition, which requires Eq. (21), is used in
the y direction in our LBM simulations. The peaks of
the LBM results (dash-dotted line and short dashed line)
are slightly slower than the vorticity-based spectral results. This is possibly due to the ®nite vertical domain
size e€ect since the bigger domain simulation (long dashed line) gives about the same peak magnitude. The
peaks of all LBM simulations are also slightly delayed
compared with the spectral cases, which is related to the
pairing vorticity rotating slightly slower in our LBM
simulations. The rotation lag is possibly related to the
compressibility in the LBM simulation. The maximum
local Mach number in our LBM study is close to 0.2.

4. The two-dimensional immiscible multi-phase mixing
layer
4.1. Surface tension e€ects on vorticity ®elds and density
®elds

Fig. 2. Comparison of momentum thickness dm of single-phase mixing
layers from four di€erent numerical methods. The solid line is the result of the spectral method solving vorticity equations. The dotted line
is the result of the spectral method solving velocity equations. The
dash-dotted line is the result of the single-phase LBM. The short
dashed line is the two-phase LBM simulation without surface tension.
The long-dashed line is the the single-phase LBM simulation with twice
the vertical computational domain size, i.e. 2L=k ˆ 2:0. All simulations
have the same initial perturbations and Re ˆ u1 d0w =m ˆ 250. All results
have been nondimensionalized, using the length scale d0w and the
velocity scale u1 .

In Fig. 3, we present the contour plots of the vorticity
®elds for ®ve values of the surface tension at t ˆ 21:5,
when all the momentum thicknesses reach their ®rst
local maximum. The ®rst plot is the result of a singlephase LBM calculation. It matches the ®rst ®gure of
Moser and Rogers [22] very well. The second plot with
zero surface tension is from a result of the two-phase
LBM calculation. There is a very small phase lag compared with the single-phase plot. This delay might be
due to the fact that the nearly incompressible assumption in the LBM multi-phase model is not exactly satis®ed and the pressure derivative in the macroscopic
pressure Eq. (16) might not be always negligible in our
simulations. The remaining four plots are cases with
di€erent surface tensions at time, t ˆ 21:5. The contour
plot with j ˆ 0:01, which corresponds to capillary
number Ca ˆ u0 l=r ˆ 28:8, is quite similar to the zero

466

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

Fig. 3. Vorticity contours with di€erent surface tensions. All contours are plotted at t ˆ 21:5 with Re ˆ 250. The single-phase plot is from the singlephase LBM simulation. The rest are from the multi-phase LBM simulations with di€erent surface tensions. The corresponding capillary numbers are
Ca ˆ 1, Ca ˆ 28:8, Ca ˆ 2:88, Ca ˆ 0:58, and Ca ˆ 0:29 for j ˆ 0, j ˆ 0:01, j ˆ 0:1, j ˆ 0:5, and j ˆ 1:0, respectively. The grid is 512  513 for all
cases.

surface tension plot, except for the occurrence of several
small disturbances. However, when the j increases to
0.1, i.e. Ca ˆ 2:88, the two pairing vortices start to distort and more small vortices appear. When j reaches 0.5
(Ca ˆ 0:58), the two pairing vortex cores almost disappear. Instead, small elongated vortices appear in the
core region. When the surface tension is large enough to

reach j ˆ 1:0 and Ca ˆ 0:29, the vortex pairing cores
collapse and small-scale vortices are activated in the
deformed core region. At this stage, the core regions still
maintain similar outer shapes for all cases.
To compare the density distributions of the zero
surface tension case and the nonzero surface tension
case, the density time evolutions, from t ˆ 4:56 to

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

t ˆ 36:45, are plotted in Fig. 4 for the cases Ca ˆ 1
…j ˆ 0†, Ca ˆ 2:88 …j ˆ 0:1†, and Ca ˆ 0:29 …j ˆ 1:0†,
respectively. For the zero surface tension case, since the
interface does not in¯uence the ¯ow, the evolution of the
interface is purely passive. By following the evolution of
the ¯ow ®eld closely, we observed that the elongated
interface rolls up and forms vorticity pairs. The interface

467

is continuous and smooth. In later stages, many layers
of interfaces are trapped in a relatively circular and
compact core region. For Ca ˆ 2:88 …j ˆ 0:1†, although
the ®rst four plots look quite similar to the zero surface
tension case, starting from the ®fth plot, surface tension
has retarded the leading edge of the interface. The
elongated interface is prevented from being extended.

Fig. 4. Time evolution of the interface in the mixing layer demonstrating the e€ects of increasing surface tension. The ®rst two rows show the results
of the zero surface tension case. The following two rows show the results of the case with Ca ˆ 2:88 …j ˆ 0:1†. The last two rows show the results for
Ca ˆ 0:29, …j ˆ 1:0†. All cases have Re ˆ 250 and the same grid size, 512  513.

468

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

Unlike the sharp ends of the stretched interface in the
zero surface tension case, the interfacial ends in this case
are blunt due to the smoothing e€ects of surface tension.
In the last two plots of this case, interfacial pinching and
breaking occur and small droplets appear. In this late
stage, the layer thickness and the length of the stretched
interfaces are less than those of the zero surface tension
case. Though the interfaces appear broken, a circular
compact trapped region is still formed. When surface
tension becomes much stronger, i.e., Ca ˆ 0:29
…j ˆ 1:0†, even in the early stages, quantitative di€erences emerge. The ¯ow rotation seems still to be able to
keep up with the zero surface tension case. However, the
interface evolution is signi®cantly delayed. The tips of
the ®ngers created by strong surface tension cannot be
stretched easily by the surrounding ¯ow. Instead, the
tips broaden and roll with the ¯ow, as can be seen
clearly from plots 3 through 6. In plot 6, a pinch-o€ is
about to occur for the upper tips. Then an isolated big
droplet is formed in plot 7. The interface pattern is no
longer symmetric. In the last plot of this case, the
compact core no longer exists. Stretched thicker interfacial layers form a ¯at elliptical core.
There are several possible mechanisms to explain the
interfacial pinch-o€ associated with surface tension. The
small droplets that appeared in the last two plots of
Ca ˆ 2:88 …j ˆ 0:1† case fall o€ from the elongated interface. This is similar to the ``end-pinching'' observed
by Stone and Leal in 1989 [25], and Song and
Tryggvason in 1999 [24], when they studied relaxation
and breakup of an initial extended drop in Stokes ¯ow.
Due to the surrounding pressure acting on the neck,
end-pinching could happen and the bulbous end separates from the stretched drop. The pinch-o€ that
occurred for Ca ˆ 0:29 …j ˆ 1:0† seems to require a
di€erent explanation. Hou et al. observed a similar
phenomenon in their study of interface e€ects on a
vortex sheet in 1997 [13]. They believed their pinch-o€
was the result of ``self-intersecting''. They o€ered the
following explanation: since the ®nger lengthening is
associated with ¯owing into the ®nger, a jet can form
when the neck becomes narrow. Any irregularity due to
capillary waves can induce a local Kelvin±Helmholtz
instability, causing growth of the irregularity and
eventually causing pinch-o€ of the neck. From our
simulation, we believe that another explanation for selfintersecting is also valid. For the zero surface tension
case, since the interface always follows streamlines, selfintersecting is impossible. However, for the nonzero
surface tension case, because of the resistance of surface
tension, the interface cannot keep up with the streamlines. Therefore, self-intersecting becomes quite possible.
From plots 5 and 6 of Ca ˆ 0:29 …j ˆ 1:0†, we can
clearly see that although the ``T-shaped'' black tip rolls
with the ¯ow, an inertial lag due to surface tension
occurs. In plot 6, the left end of the T-shaped black tip

almost hits the upper black layer and the white neck is
nearly broken due to inertial lag. In plot 7, a large
elongated droplet has already been formed. Thus, we
believe that in a convective ¯ow ®eld, as long as surface
tension provides enough resistance, self-intersecting of
the interface will occur, causing interfacial pinching and
breaking.
Another point we note here is the symmetry of the
¯ow structure. For the zero surface tension case, our
simulations always show symmetrical structures for the
two di€erent ¯uids, even in the very late stages, which do
not have adequate resolution. However, for nonzero
surface tension, our simulations show that some asymmetry of the ¯ow structures appears when interface
pinching and breaking occur. Then the asymmetry
spreads, changing the ¯ow ®eld eventually. The stronger
the surface tension, the earlier the asymmetries appear.
We are not sure whether this asymmetry is due to surface tension or due to our numerical approximation.
Fig. 5 shows a comparison of vorticity evolution from
t ˆ 4:56 to t ˆ 36:45, with the same initial conditions as
the density evolution in Fig. 4, for three di€erent values
of the surface tension. For zero surface tension case, it is
seen that high vorticity concentrates in the cores of the
vortex pairs. The vorticity pairing goes quite smoothly.
The vortex structures match the corresponding density
structures very well. When surface tension is included,
the vorticity ®eld becomes disturbed. Vorticity concentrations appear on the interfaces. Plots 2, 3, and 4 for
both the Ca ˆ 2:88 …j ˆ 0:1† and the Ca ˆ 0:29
…j ˆ 1:0† cases show large vorticity concentrations at
the ®nger tips. Similar observations have been reported
by previous studies [13,20]. Later, increasingly small
vortices occur on the interface at locations with relatively sharp curvature, i.e. at the tips of ®ngers and near
broken droplets. Small vortices also show up in the
necks of the density jets. The vortex structures become
more and more fragmented. The vortex core region
becomes increasingly stretched and ¯at.
4.2. Surface tension e€ects on momentum and density
mixing
To further quantify the e€ects of surface tension, we
study the evolution of momentum thickness for increasing values of surface tension. In Fig. 6, the momentum thickness is plotted as a function of time for
Ca ˆ 1 (solid line), Ca ˆ 2:88 (dotted line), Ca ˆ 0:58
(dash-dotted line), and Ca ˆ 0:29 (dashed line). For zero
surface tension, dm increases slowly and oscillates. When
the surface tension increases, the oscillation of dm is
retarded, indicating that surface tension resists the ¯ow
rotation. When the surface tension becomes even
stronger, dm increases faster. Therefore, momentum loss
is faster and momentum dissipation is increased compared with the nonzero surface tension case. There are

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

469

Fig. 5. Time evolution of vortices in the mixing layer as a function of increasing surface tension. The ®rst two rows are for the zero surface tension
case. The following two rows are for Ca ˆ 2:88 …j ˆ 0:1†. The last two rows are for Ca ˆ 0:29 …j ˆ 1:0†. All cases have Re ˆ 250 and the same grid,
512  513.

two reasons for this. First, because of surface tension, interfacial pinching and breaking occur and
droplets appear. These a€ect the ¯ow structures and
possibly promote momentum exchange. Just as seen
in Fig. 5 with Ca ˆ 0:29 at t ˆ 36:45, the whole
vortex core region is no longer as compact as the
zero surface tension case. Surface tension appears to
be able to break and spread vortices. Second, the

kinetic energy loss leads to momentum loss. The
deformed droplet with surface tension could transfer
part of the kinetic energy into ``potential energy'' and
store energy in the strained surface, dissipating the
¯ow momentum. In addition, large viscous dissipations occur on the interface due to the small vortices
induced by the surface tension. Therefore, ¯ow kinetic energy decreases faster and causes a loss of ¯ow

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R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

interaction between the surface tension and the ¯ow
®eld spreads the mixing layer. In the very late stages, the
dm for nonzero surface tension cases do not change
much. Because of the compactness of the original mean
¯ow, the density thicknesses for all cases are bounded.
The mean velocity pro®les at two di€erent times,
t ˆ 36:45 and t ˆ 145:8, are plotted in Fig. 8. Because of
momentum di€usion and dissipation, all streamwise
mean velocity pro®les become ¯atter than the initial
deep pro®le. The larger the surface tension, the faster
the mean pro®le decays.
4.3. Surface tension e€ects on the evolution of the kinetic
energy and the evolution of the enstrophy

Fig. 6. Time evolutions of the momentum thickness for di€erent surface tensions. The solid line is for the zero surface tension case. The
dotted line is for Ca ˆ 2:88 …j ˆ 0:1†. The dot-dashed line is for
Ca ˆ 0:59 …j ˆ 0:5†. The dashed line is for Ca ˆ 0:29 …j ˆ 1:0†. All
results have been properly nondimensionalized. The resolution is
512  513, and Re ˆ 250.

momentum. We will discuss this again in the following section.
Fig. 7 presents the time evolution of the mixing layer
density thickness for the above four cases. The density
thickness dd is de®ned as the vertical width of the region
in which two di€erent ¯uids interact with each other.
The results show a behavior of dd that is like that of the
momentum thickness dm , since both evolutions are directly related to the vortex evolution. Without surface
tension, dd oscillates in a small range, for 2/3 of the
whole computational spatial domain. When surface
tension is included, dm becomes wider. The nonlinear

Fig. 7. Time evolution of density thickness dd for di€erent surface
tensions. The solid line is the zero surface tension case. The dotted line
is Ca ˆ 2:88 …j ˆ 0:1†. The dot-dashed line is Ca ˆ 0:59 …j ˆ 0:5†, and
the dashed line is Ca ˆ 0:29 …j ˆ 1:0†:

To study the details of the e€ects of surface tension
on the mixing layer, it is important to examine the
evolution of the kinetic energy. Neglecting the body
force in the macroscopic equation (17), it is easy to derive the following equation for the ¯uid kinetic energy:
1 du2
u
rp
/
ˆÿ
‡ mu
r2 u ‡ j u
rr2 /:
2 dt
q
q

…23†

Then, the total kinetic energy equation can be written by
integrating Eq. (23)
Z
Z
Z
dE
u
rp
/
2
ˆÿ
dv ‡ m u
r udv ‡ j
u
rr2 /dv
dt
q
v
v
v q
Z
Z
/
2
u
rr2 /dv:
…24†
ˆ ÿ m …ru† dv ‡ j
q
v
v
Here periodic boundary conditions were used in the
x direction and slip boundary conditions were used in
the y direction. In addition, we assume that all integrations on boundaries are small. Therefore, for the zero
surface tension case, the total kinetic energy should always decrease due to dissipation from the viscous term.
The monotonically decreasing solid line in Fig. 9 agrees
well with this description. The other three lines represent
the results for Ca ˆ 2:88 (dotted line), Ca ˆ 0:58 (dotted±dashed line), and Ca ˆ 0:29 (dashed line), respectively. Although we cannot tell analytically whether the
net contribution from the surface tension term in
Eq. (24) is positive or negative, the simulation results
imply that surface tension mainly dissipates the kinetic
energy as well as viscous term in this case.
Fig. 10 presents the contributions of dissipation from
both the viscous term and the surface tension term. The
solid lines are the results for zero surface tension. For
this purely viscous case, it is well known that the viscous
dissipation always decreases the energy. In the ®gure,
the negative viscous dissipation causes the curve to
move monotonically toward zero. When surface tension
becomes more important, the negative viscous dissipation no longer monotonically decays, and the contribution from the surface tension becomes quite
interesting. When the surface tension is strong enough

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

471

Fig. 8. Mean velocity pro®les for four di€erent cases with j ˆ 0, j ˆ 0:1, j ˆ 0:5, and j ˆ 1:0, respectively. The solid lines are the initial mean
velocity pro®le. The dotted lines are for Ca ˆ 1 …j ˆ 0†. The dashed-dotted lines are for Ca ˆ 2:88 …j ˆ 0:1†. Short dashed line are for Ca ˆ 0:58
…j ˆ 0:5†. The long dashed lines are Ca ˆ 1:0 …j ˆ 1:0†. The upper two plots are the pro®les at t ˆ 36:45 and the bottom two are the pro®les at
t ˆ 145:8.

(Ca ˆ 0:29, dashed line in both plots), the evolution for
both viscous dissipation and surface tension become
irregular and large ¯uctuations appear. Most of the
contributions to the kinetic energy from surface tension
are negative. Sometimes the surface tension contribution to the kinetic energy becomes positive. The following is a physical explanation. Because of the strong
shear e€ect, the interface is elongated and distorted. Part
of the kinetic energy can be transformed to a kind of
``surface potential energy'' which is stored in the elastic
interface due to the surface tension. This type of energy
transform accelerates the kinetic energy decay rate in
Fig. 9. On the other hand, the interface cannot be
stretched inde®nitely. Pinching and breaking occur.
During the droplet break-o€, the surface tension rapidly
smoothes the sharp breaking interface. The deformed
droplet relaxes from the breaking point and releases

some of the surface potential energy. This part of the
energy is transformed back into kinetic energy, making
the contribution from surface tension positive. From
previous ®gures, we have noted that with strong surface
tension, vortices seem to be concentrated near the interface. This local concentration of vortices leads to
large local viscous dissipation near the interface. During
pinch-o€, the induced vortices are usually much larger
than normal, causing much larger viscous dissipation.
The two peaks in Fig. 10 exactly correspond to the
breaking of the interface.
Similar to the kinetic energy equation, the two-dimensional vorticity equation (x ˆ r  u) can be derived
by taking the curl of Eq. (17)
 
dx
rq  rp
/
2
…25†
ˆÿ
‡ mr x ‡ jr
 rr2 /:
dt
q2
q

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R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

Fig. 9. Time evolution of the total kinetic energy Ek . The solid line is
for the zero surface tension. The dotted line is for Ca ˆ 2:88 …j ˆ 0:1†.
The dotted-dashed line is for Ca ˆ 0:59 …j ˆ 0:5†. The dashed line is
for Ca ˆ 0:29 …j ˆ 1:0†. All Ek have been nondimensionalized by u21 .

The equation for x2 can be shown to be
 
1 dx2
xrq  rp
/
2
ˆÿ
 rr2 /:
‡
mxr
x
‡
jxr
2 dt
q2
q

…26†

Since our study is nearly incompressible, barotropic, and both phases have a same density, the pressure
terms can be neglected. Thus, the equation for total
enstrophy, X ˆ hx2 i, satis®es the following equation
(which is similar to the total kinetic energy equation,
Eq. (24)):
 
Z
Z
1 dX
/
2
ˆ ÿm …rx† dv ‡ j xr
 rr2 / dv:
2 dt
q
v
v
…27†

Here boundary conditions have been applied and all
contributions from the boundaries have been neglected.
Fig. 11 shows the enstrophy evolution for the four
cases considered before. Again, the solid line is for zero
surface tension, the dotted line is for Ca ˆ 2:88, the dotdashed line is for Ca ˆ 0:58, and the dashed line is for
Ca ˆ 0:29. For the zero surface tension case, like total
kinetic energy E, the total enstrophy X decays monotonically in time. This occurs because the total enstrophy
dissipation is always negative for the two-dimensional
case. However, when the contribution from the surface
tension is not negligible, the enstrophy evolution shows
signi®cant di€erences. The enstrophy does not continuously decrease. When the surface tension is large enough,
the enstrophy oscillates and tends to increase. For short
times, the enstrophy ¯uctuations can even reach quite
large values. The two peaks for Ca ˆ 0:29 (dashed line) in
the ®gure appear at exactly the same time as the peaks in
Fig. 10, (t ˆ 45:6 and t ˆ 59:2). As in the discussion for
energy dissipation, these two peaks correspond to the
pinch-o€ of the interface.
To show additional e€ects of surface tension on
enstrophy evolution, we calculate the contributions
from both surface tension and viscosity to the enstrophy
equation (27). From Fig. 12, it is interesting to see that
the contribution from surface tension is always positive.
This positive contribution generates vortices near the
interface. When a droplet pinches o€, before being
smoothed, the breaking interface is often very sharp.
Since rr2 / in Eq. (27) is related to the local curvature,
the surface tension term usually becomes large. Therefore, the surface tension contributes signi®cantly and
vorticity is generated. Meanwhile, the contributions
from viscous dissipation also become larger than that in
the zero surface tension case. Altogether, the net contributions from the right-hand side of the enstrophy

Fig. 10. Time evolution of kinetic energy dissipation from viscosity and surface tension e€ect. ev is the viscous dissipation, and es is the dissipation
due to surface tension. The solid line is zero surface tension case. The dotted line is Ca ˆ 2:88 …j ˆ 0:1†. The dot-dashed line is Ca ˆ 0:59 …j ˆ 0:5†.
The dashed line is Ca ˆ 0:29 …j ˆ 1:0†.

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

473

Fig. 11. Time evolution of the total enstrophy X . The solid line is zero surface tension. The dotted line is Ca ˆ 2:88 …j ˆ 0:1†. The dot-dashed line is
Ca ˆ 0:59 …j ˆ 0:5†, and the dashed line is Ca ˆ 0:29 …j ˆ 1:0†.

Fig. 12. Time evolution of enstrophy dissipation due to viscosity and surface tension. The top plot shows the contributions for both the viscosity
(negative) and surface tension (positive) for the four cases previously discussed. The bottom plot shows the total contribution from viscosity and
surface tension. The solid line is for zero surface tension. The dotted line is Ca ˆ 2:88 …j ˆ 0:1†. The dot-dashed line is Ca ˆ 0:59 (j ˆ 0:5†. The
dashed line is Ca ˆ 0:29 …j ˆ 1:0†.

Fig. 13. Interface distribution for Ca ˆ 0:29 …j ˆ 0:1† at two times, t ˆ 54:72 and t ˆ 59:28. Re ˆ 250 and grid size, 512  513.

474

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

equation are positive for suciently large surface tension during interface break-up.
Fig. 13 shows the density distribution for Ca ˆ 0:29
at two times, t ˆ 54:72 and t ˆ 59:28. The latter time
corresponds to the time of the second peaks in Figs. 10
and 11. In Fig. 13, interfacial pinching and breaking are
about to occur at t ˆ 54:72 and some formed droplets
appear at t ˆ 59:28. There are more small droplets at
t ˆ 59:28 than at t ˆ 54:72. This implies that the inter-

facial pinching and the relaxation of broken droplets
occur between these two times and should be responsible
for the peaks in the previous ®gures. Fig. 14 shows the
distributions at the same two times as in the previous
®gure. The top plot is the vorticity distribution. The
second one is the distribution of the viscous energy
dissipation. The third one is the distribution of the viscous vorticity dissipation. The bottom one is the distribution of the viscous dissipation for x2 . All these

Fig. 14. Distributions of vorticity, dissipation of E, dissipation of x, and dissipation of x2 (from top to bottom, respectively) at two times, t ˆ 54:72
and t ˆ 59:28. Ca ˆ 0:29; Re ˆ 250.

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

distributions are strongly related to the interface distributions. Large vorticity and dissipations appear near the
interface, especially near the places where interfacial
pinching and droplet relaxation occur. The observations
in Figs. 13 and 14 support our previous physical explanations.
4.4. Mesh convergence
Grid convergence of the solutions is one of the most
important issues in numerical simulations of immiscible
two-phase ¯ows. Fig. 15 shows the grid dependence of
one of our solutions for Ca ˆ 2:88. In this ®gure, we
plotted density distributions at three times, t ˆ 27:33,
t ˆ 31:89, and t ˆ 36:45, on two di€erent meshes,

475

512  513 and 1024  1025. At t ˆ 27:33, all the
structures look same except for the long thin ®lament
between the pair. On the ®ne grid (1024  1025), the
thickness of the thin ®lament is 6±8 grid spacings.
Since the interface requires 3±4 grid spacings in our
scheme, it is reasonable that the coarse grid cannot
resolve this thin ®lament well. The basic structures at
the later time, t ˆ 31:89 and t ˆ 36:45, are well converged, although the small structures show some differences. The rotations of ¯ow ®eld show little
dependence on grid size. Interfacial pinching and
breaking occur at similar locations on two di€erent
grids but the shapes of the small droplets are di€erent.
How the grid convergence depends on capillary number requires more study.

Fig. 15. Grid convergence of numerical simulations for Ca ˆ 2:88 …j ˆ 0:1†. Plots of density distributions are for three times, t ˆ 27:33, t ˆ 31:89
and t ˆ 36:45.

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R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

Fig. 16. Density correlation for three surface tensions, Ca ˆ 1 …j ˆ 0†, Ca ˆ 2:88 …j ˆ 0:1†, and Ca ˆ 0:29 …j ˆ 1:0†, at t ˆ 145:8. The solid line is
for Ca ˆ 1 , the dot±dashed line is for Ca ˆ 2:88, and the dashed line is for Ca ˆ 0:29.

4.5. Very late stage ¯ow structures
In the very late stage, because of the interface
pinching and breaking, the ¯ow ®elds with ®nite surface
tension di€er from the zero surface tension case. To
show the dependence of interface distribution and vorticity distributions on surface tension in late stages, we
examined the correlation functions of density and vorticity. The correlation functions of density and vorticity
are de®ned as follows:
R
/…r ‡ x†/…x† dx
;
…28†
Cq …r† ˆ
h/2 i
R
x…r ‡ x†x…x† dx
;
…29†
Cx …r† ˆ
hx2 i
where / is the index function of the two phases and x is
the vorticity.
Fig. 16 shows the density correlations along the x
direction on the middle line of the vertical direction. For
small separations, r, the correlations of the zero surface
tension (solid line) decay rapidly. The correlation of
Ca ˆ 2:88 (dot-dashed line) decay slower than the zero
surface tension case. The correlations of Ca ˆ 0:29
(dashed-line) decay the slowest. This implies that the
characteristic size of the density distribution increases
with increasing Ca number in late stages. This observation is reasonable. For the zero surface tension case,
the interfaces can be stretched inde®nitely and become
very thin. When surface tension is included, the tension
prevents the interfaces from being stretched and causes
interface pinching. The larger the surface tension, the

Fig. 17. Vorticity correlation for three surface tensions, Ca ˆ 1
…j ˆ 0†; Ca ˆ 2:88 …j ˆ 0:1†, and Ca ˆ 0:29 …j ˆ 1:0†, at t ˆ 145:8.
The solid line is for Ca ˆ 1 , the dot±dashed line is for Ca ˆ 2:88, and
the dashed line is for Ca ˆ 0:29.

larger the size of droplets. For large separations, r, the
correlations of the zero surface tension oscillate rapidly.
The correlations of Ca ˆ 2:88 (dot±dashed line) oscillate
slower. The correlations of Ca ˆ 0:29 (dashed line) oscillate much slower. Since the interface for zero surface
tension is stretched to form an organized dense spiral,
the correlation is a regular oscillation. For strong surface tension, the organized ¯ow pattern is broken, the
interface distribution becomes irregular and the correlations oscillate much slower.

R. Zhang et al. / Advances in Water Resources 24 (2001) 461±478

The vorticity correlations are shown in Fig. 17. The
correlations of the zero surface tension (solid line) decay
slowly and smoothly over the whole separation. This
occurs because the large stable vortex core dominates in
very late stages. All small-scale vorticities are depressed.
For nonzero surface tensions, the correlations show
large di€erences. For small separations, r, the correlations of Ca ˆ 2:88 decay rapidly. The correlation of
Ca ˆ 0:29 decays even more rapidly. This implies that
for strong surface tension, it is the small-scale vortices,
instead of the large stable vortex cores, that dominate in
the very late stages.

477

be observed near the interface. Eventually broken
droplets and small vortices near interfaces dominate
the ¯ow ®elds in the late stages.
In the future, high resolution simulations are needed
to model more accurately the late stage ¯ow. A detailed
study of the two-dimensional interface pinch-o€ mechanism would also be interesting. Certainly, the most
important and interesting study is of the three-dimensional immiscible multi-phase Kelvin±Helmholtz instability. Finally, a direct comparison with experiments is
very much needed.
Acknowledgements

5. Conclusions
In this paper, we have used a recently proposed LBM
multi-phase model to study the immiscible two-phase
Kelvin±Helmholtz instabilities. In this model, an index
¯uid is used to track the two phases and the interface
between them. A pressure distribution function is introduced to describe the ¯ow dynamics. Surface tension
is implemented in the model by incorporating molecular
interactions. Interfaces in this model are maintained
automatically.
Using a combination of harmonic and subharmonic
perturbations as initial velocity conditions, we have
carried out simulations of Kelvin±Helmholtz instabilities for four di€erent values of surface tension. The
nonlinear behavior of the two-phase mixing shows
signi®cant changes with increasing surface tension.
For the zero surface tension case, our numerical results match previous spectral results quantitatively.
When the surface tension becomes more important
(Ca becomes smaller), the surface tension and ¯ow
®eld in