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

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 in-compressible 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 inhomo-geneous 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 simula-tions (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 ex-perimental 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-peared to align primarily with the spanwise direction. Recently with the rapidly improving computational fa-cilities 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 struc-tural 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., co-rotation 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-dimen-sional and three-dimentwo-dimen-sional fully turbulent mixing layers. The growth of the mixing layer is controlled by pairings, though this may not be true in strongly tur-bulent 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

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*

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.

mixing layer becomes increasingly three-dimensional, it undergoes a transition to turbulence and eventually re-sults in a fully developed turbulent mixing layer.

Although the basic nonlinear behavior of the single-phase 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 di-rectly. 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

(Caˆul=r) dependent evolution of kinetic energy,

enstrophy and ¯ow patterns in mixing layer have also not been studied extensively. In vortex sheet simula-tions, 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 nu-merical integral methods is still problematic. These technical problems do not allow the use of vortex inte-gral methods to study the late-stage ¯ow dynamics of mixing layers with interfacial interactions in which in-terface pinching and merging happen frequently.

With the recent development of the multi-phase lat-tice Boltzmann method (LBM) [4,32], DNS of immis-cible 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 instabili-ties [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. Al-though 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 Reˆ250, which is de®ned in the following section. This paper is organized as follows: in the following section, we give a brief de-scription 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 in-compressible Navier±Stokes equations must be solved for traditional methods [6,8]:

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.tis the tangent vector to the bubble surface, and sis the arclength co-ordinate.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 be-havior 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 distri-bution function satis®es the following equation:

fi…x‡ei;t‡1† ˆfi…x;t† ‡

fi…x;t† ÿfieq

s ;

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 densityqand the momentum densityquare de®ned by

servation of total mass and total momentum at each lattice site

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

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

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 dis-tribution functions of the index function and pressure satisfy the following equations:

Here s is the relaxation time which is related to the kinematic viscosity bymˆ …sÿ1=2†.fiandgiare newly

de®ned variables for the convenience of discretization. The use of these two variables guarantees the second-order accuracy of the scheme both in space and time.fi

and gi are related to the distribution functions of the

index function and pressure,fiandgi, by

i are corresponding equilibrium distributions

fieqˆwi/ 1

Here Ris the gas constant,T the background tempera-ture andwiis the integral weight, which is the same aswi

in Eq. (5).Gis the gravitational force andFis the sur-face tension written as

Fsˆj/rr2/; …12†

wherej determines the strength of the surface tension.

The macroscopic variables are calculated using:

The density and the kinematic viscosity are calculated from the index function using

q/_{† ˆ}ql‡

whereqlandqhare the densities of the light ¯uid and the

heavy ¯uid, respectively,mland 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 functionwin Eqs. (6) and (7) plays an essential role in phase separation in multi-phase ¯ow simulations. The following form forw…/†is used in this study:

w…/† ˆ/2RT 4ÿ2/

…1_ÿ/†3ÿa/

2_:

…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 dw‡/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 aˆ12RT, the same as He et al. used in [10] and in [11]. The corresponding macroscopic dynamical equations are 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

the present study, a temporally growing layer is studied because periodic boundary conditions can be easily ap-plied 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 directionxdirection) is of the form

U _ˆu₁erfp_p

y=d0_w†: …18_†

Here thed0_w is the vorticity thickness de®ned by

d0_wˆ 2U

…o_U=o_{y† jmax}: …19†

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

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

eigen-functions determined by linear instability theory. They are normalized so that their real parts are equal to one at y ˆ0. There is no phase di€erence between the funda-mental mode and the subharmonic mode, which as-sumes that perfect pairing takes place with both modes being introduced simultaneously. In our simulations, a_ˆ0:8620 is used. This corresponds to having the most unstable wave at the wavelength, k_ˆ2p₌a_ˆ7:289. To match the initial ®elds of Moser and Rogers [17], we set A1;0ˆ …0:019043166;ÿ0:012901† and A1=2;0ˆ

…0:019043166;ÿ0:0052838_†. The Reynolds number,Re_ˆ u₁d0_w=mˆ250, is used throughout in this study.

The initial interface is evenly distributed around the midpoint in vertical direction. Periodic boundary con-ditions for velocity and density are used in the stream-wise direction. Free slip wall boundary conditions are applied in the vertical direction

o_u

[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 vor-ticity-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 do-main size in theydirection, the domain truncation e€ect is less in¯uential for the vorticity-based scheme than for the velocity-based scheme. Therefore, results from vor-ticity-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 demon-strated quantitatively by comparison with spectral re-sults of the evolution of the momentum thicknessdm.dm

is a measure of the momentum loss due to the presence of the sheared mixing layer. It is de®ned as

dmˆ

The time evolution of the momentum thickness is shown in Fig. 2. All results have been properly

nor-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=kˆ1:5. The dash-dotted line is the single-phase LBM simulation, and the short-dashed line is the two-phase LBM simu-lation with no surface tension e€ects. In these two LBM simulations, the vertical domain sizes are set to 2L=k_ˆ1:0. 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_ˆ2:0. 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 re-sults. This is possibly due to the ®nite vertical domain size e€ect since the bigger domain simulation (long da-shed 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

In Fig. 3, we present the contour plots of the vorticity ®elds for ®ve values of the surface tension at tˆ21:5, when all the momentum thicknesses reach their ®rst local maximum. The ®rst plot is the result of a single-phase 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 com-pared with the single-phase plot. This delay might be due to the fact that the nearly incompressible assump-tion in the LBM multi-phase model is not exactly sat-is®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,tˆ21:5. The contour plot with j_ˆ0:01, which corresponds to capillary number Caˆu0l=rˆ28:8, is quite similar to the zero

Fig. 2. Comparison of momentum thicknessdmof single-phase mixing layers from four di€erent numerical methods. The solid line is the re-sult 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_ˆ2:0. All simulations have the same initial perturbations andReˆu1d0w=mˆ250. All results

have been nondimensionalized, using the length scale d0

w and the

surface tension plot, except for the occurrence of several small disturbances. However, when the j increases to 0.1, i.e.Caˆ2:88, the two pairing vortices start to dis-tort and more small vortices appear. Whenjreaches 0.5 (Ca_ˆ0:58), the two pairing vortex cores almost disap-pear. Instead, small elongated vortices appear in the core region. When the surface tension is large enough to

reach jˆ1:0 and Caˆ0:29, 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 tˆ4:56 to

tˆ36:45, are plotted in Fig. 4 for the cases Caˆ 1 …jˆ0†, Caˆ2:88 …jˆ0:1†, and Caˆ0:29 …jˆ1:0†, 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

is continuous and smooth. In later stages, many layers of interfaces are trapped in a relatively circular and compact core region. ForCaˆ2:88…jˆ0:1†, 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.

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., Caˆ0:29 …j_ˆ1:0_†, even in the early stages, quantitative di€er-ences 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 inter-facial 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 Caˆ2:88j_ˆ0:1_†case fall o€ from the elongated in-terface. 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 sepa-rates from the stretched drop. The pinch-o€ that occurred for Caˆ0:29 <sub>…</sub>jˆ1:0<sub>†</sub> 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 self-intersecting is also valid. For the zero surface tension case, since the interface always follows streamlines, self-intersecting is impossible. However, for the nonzero surface tension case, because of the resistance of surface tension, the interface cannot keep up with the stream-lines. Therefore, self-intersecting becomes quite possible. From plots 5 and 6 of Caˆ0:29 j_ˆ1:0_†, 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 asym-metry 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 sur-face tension or due to our numerical approximation.

Fig. 5 shows a comparison of vorticity evolution from tˆ4:56 totˆ36:45, 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 concen-trations appear on the interfaces. Plots 2, 3, and 4 for both the Caˆ2:88 …jˆ0:1† and the Caˆ0:29 …jˆ1:0† 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 rela-tively 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 in-creasing values of surface tension. In Fig. 6, the mo-mentum thickness is plotted as a function of time for Caˆ 1(solid line), Caˆ2:88 (dotted line), Caˆ0:58 (dash-dotted line), andCaˆ0:29 (dashed line). For zero surface tension,dmincreases 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,dmincreases faster. Therefore, momentum loss

two reasons for this. First, because of surface ten-sion, 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 Caˆ0:29 at tˆ36:45, 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 dissipa-tions occur on the interface due to the small vortices induced by the surface tension. Therefore, ¯ow ki-netic energy decreases faster and causes a loss of ¯ow

momentum. We will discuss this again in the fol-lowing section.

Fig. 7 presents the time evolution of the mixing layer density thickness for the above four cases. The density thicknessddis de®ned as the vertical width of the region

in which two di€erent ¯uids interact with each other. The results show a behavior ofdd that is like that of the

momentum thickness dm, since both evolutions are

di-rectly 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

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, tˆ36:45 andtˆ145:8, 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

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 de-rive the following equation for the ¯uid kinetic energy:

1

Then, the total kinetic energy equation can be written by integrating Eq. (23)

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 inte-grations on boundaries are small. Therefore, for the zero surface tension case, the total kinetic energy should al-ways 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 Caˆ2:88 (dotted line), Caˆ0:58 (dot-ted±dashed line), and Caˆ0:29 (dashed line), respec-tively. 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 dissipa-tion no longer monotonically decays, and the contri-bution from the surface tension becomes quite interesting. When the surface tension is strong enough

Fig. 7. Time evolution of density thicknessdd for di€erent surface tensions. The solid line is the zero surface tension case. The dotted line isCaˆ2:88j_ˆ0:1_†. The dot-dashed line isCaˆ0:59j_ˆ0:5_†, and the dashed line isCaˆ0:29j_ˆ1:0_†:

(Ca_ˆ0:29, 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 contribu-tion to the kinetic energy becomes positive. The fol-lowing 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 in-terface. 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-di-mensional vorticity equation (x_{ˆ r} u) can be derived by taking the curl of Eq. (17)

dx dt ˆ ÿ

rq rp

q2 ‡mr

2_x

‡j_r /

q

rr2_/:

…25_†

Fig. 8. Mean velocity pro®les for four di€erent cases withjˆ0,jˆ0:1,jˆ0:5, andjˆ1:0, respectively. The solid lines are the initial mean velocity pro®le. The dotted lines are forCaˆ 1 …jˆ0†. The dashed-dotted lines are forCaˆ2:88…jˆ0:1†. Short dashed line are forCaˆ0:58

The equation forx2_{can be shown to be} Since our study is nearly incompressible, barotrop-ic, and both phases have a same density, the pressure terms can be neglected. Thus, the equation for total enstrophy, X_{ˆ h}x2_{i, satis®es the following equation}

(which is similar to the total kinetic energy equation, Eq. (24)):

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 forCaˆ2:88, the dot-dashed line is forCaˆ0:58, and the dashed line is for Caˆ0:29. For the zero surface tension case, like total kinetic energy E, the total enstrophy X decays mono-tonically 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 continu-ously 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 forCaˆ0:29 (dashed line) in the ®gure appear at exactly the same time as the peaks in Fig. 10, (tˆ45:6 andtˆ59:2). 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. Sincerr2_{/in Eq. (27) is related to the local curvature,}

the surface tension term usually becomes large. There-fore, 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 con-tributions from the right-hand side of the enstrophy

Fig. 10. Time evolution of kinetic energy dissipation from viscosity and surface tension e€ect.evis the viscous dissipation, andesis the dissipation due to surface tension. The solid line is zero surface tension case. The dotted line isCaˆ2:88j_ˆ0:1_†. The dot-dashed line isCaˆ0:59j_ˆ0:5_†. The dashed line isCaˆ0:29j_ˆ1:0_†.

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 isCaˆ2:88…jˆ0:1†. The dot-dashed line isCaˆ0:59 (jˆ0:5†. The dashed line isCaˆ0:29…jˆ1:0†.

Fig. 11. Time evolution of the total enstrophyX. The solid line is zero surface tension. The dotted line isCaˆ2:88…jˆ0:1†. The dot-dashed line is Caˆ0:59jˆ0:5_†, and the dashed line isCaˆ0:29jˆ1:0_†.

equation are positive for suciently large surface ten-sion during interface break-up.

Fig. 13 shows the density distribution for Caˆ0:29 at two times, tˆ54:72 and tˆ59:28. 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 tˆ54:72 and some formed droplets appear at tˆ59:28. There are more small droplets at tˆ59:28 than at tˆ54:72. 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 vis-cous vorticity dissipation. The bottom one is the distri-bution of the viscous dissipation for x2_{. All these}

Fig. 14. Distributions of vorticity, dissipation ofE, dissipation ofx, and dissipation ofx2_{(from top to bottom, respectively) at two times,t}

distributions are strongly related to the interface distri-butions. 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 ex-planations.

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_ˆ2:88. In this ®gure, we plotted density distributions at three times, t_ˆ27:33, tˆ31:89, and tˆ36:45, on two di€erent meshes,

512513 and 10241025. At tˆ27:33, all the structures look same except for the long thin ®lament between the pair. On the ®ne grid (10241025), 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, tˆ31:89 and tˆ36:45, are well con-verged, although the small structures show some dif-ferences. 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 num-ber requires more study.

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 vor-ticity distributions on surface tension in late stages, we examined the correlation functions of density and vor-ticity. The correlation functions of density and vorticity are de®ned as follows:

Cq…r† ˆ

R

/r‡x†/x†dx

h/2i ; …28†

Cx…r† ˆ

R

xr‡x†xx†dx

hx2_i ; …29†

where/is the index function of the two phases andxis 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 Caˆ2:88 (dot-dashed line) decay slower than the zero surface tension case. The correlations of Caˆ0:29 (dashed-line) decay the slowest. This implies that the characteristic size of the density distribution increases with increasing Ca number in late stages. This obser-vation 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

larger the size of droplets. For large separations, r, the correlations of the zero surface tension oscillate rapidly. The correlations ofCaˆ2:88 (dot±dashed line) oscillate slower. The correlations of Caˆ0:29 (dashed line) os-cillate 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 sur-face tension, the organized ¯ow pattern is broken, the interface distribution becomes irregular and the corre-lations oscillate much slower.

Fig. 16. Density correlation for three surface tensions,Ca_{ˆ 1}j_ˆ0_†,Ca_ˆ2:88j_ˆ0:1_†, andCa_ˆ0:29j_ˆ1:0_†, att_ˆ145:8. The solid line is forCa_{ˆ 1}, the dot±dashed line is forCa_ˆ2:88, and the dashed line is forCa_ˆ0:29.

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 correla-tions of Caˆ2:88 decay rapidly. The correlation of Caˆ0:29 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.

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 in-troduced 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 instabil-ities 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 re-sults match previous spectral rere-sults quantitatively. When the surface tension becomes more important (Ca becomes smaller), the surface tension and ¯ow ®eld interact strongly. Pinch-o€ of the interface occurs and droplets are formed. When the energy feed-back from the interface becomes signi®cant, organized ¯ow patterns disappear and broken interfaces dominate the late stage. Surface tension can cause the ¯ow kinetic energy to decrease rapidly. Part of this energy is transformed into surface potential energy. The ma-jority of the contribution of surface tension to kinetic energy evolution is negative. However, during inter-face pinch-o€, the surinter-face potential energy can be converted back into kinetic energy of the ¯ow ®eld. Thus, for small enough Ca, sometimes the contribu-tion of surface tension to kinetic energy could be positive during the interface pinch-o€. The contribu-tion of surface tension to the enstrophy is always positive. This positive contribution generates signi®-cant vorticity near the interface and breaks the stable vortex core that appeared in the zero surface tension case. Since the enstrophy does not decay obviously under the in¯uence of strong surface tension, large dissipations of both kinetic energy and enstrophy can

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€ mech-anism would also be interesting. Certainly, the most important and interesting study is of the three-dimen-sional immiscible multi-phase Kelvin±Helmholtz insta-bility. Finally, a direct comparison with experiments is very much needed.

Acknowledgements

We are grateful to Dr. Lian-Ping Wang, from the University of Delaware, Dr. Xiao-Ling Tong from Mississippi State University, and Dr. Que-Hong Qian from Columbia University for providing spectral results and helpful discussions.

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