Study of Dynamic Break-Up of

Capillary Jet Using Level-Set Method

Absar Lakdawala1, Vinesh H Gada1 and Atul Sharma1∗

1Department of Mechanical Engineering,Indian Institute of Technology Bombay, Mumbai, India - 400076 ∗Corresponding

author. Phone: 022 2576 7505, Fax: +91 22 2572 6875 Email: atulsharma@iitb.ac.in

Abstract

Capillary jets, created due of interfacial instabil-ity in two-�uid system under the e�ect of surfacetension, are widely encountered in nature. In thepresent work, dynamic breakup of a axisymmet-ric liquid jet injected vertically into another im-miscible liquid is investigated numerically at var-ious injection velocity, for four di�erent disperse�uids with water as continuous �uid. For thispurpose, an in house Level Set (LS) methodologybased code is used. The two �uid �ow simulationmethodology uses �nite volume method for solu-tion of Navier-Stokes and �nite di�erence methodfor solution of LS equations. An excellent agree-ment is found between the present simulation anda published theoretical and experimental work fordetached bubble diameter. Furthermore, di�erentbreakup modes reported in the published experi-mental results are captured in the present sim-ulations. The results show a greater sensitivityof jet length and drop diameter to the Reynoldsnumber. However, viscosity ratio has negligiblee�ects on breakup dynamics. The results showsthat the transition from dripping mode to jettingmode takes place at Weber number 2.2.Keywords: Capillary Jet, Drop Dynamics,

Level Set Method.

1 Introduction

Injection of a liquid, from an ori�ce, in anotherimmiscible liquid results in drop formation. Withincreasing injection velocity, Vi, there are three

types of drop formation modes. First, drippingmode at lower Vi, the drop forms close to the ori-�ce and no jet is formed; second, jetting modeat intermediate range of Vi, a jet is formed upto a certain streamwise direction and then breaksinto drops; and third, atomization mode at higherVi, where many non-uniform droplets are formednear the ori�ce. The �rst (second) transitionfrom �rst (second) to second (third) mode oc-curs at a certain critical value of the injectionvelocity called as jetting (atomization) velocity,Vi,jet (Vi,a). After the second transition, anothertransition from 2-D axisymmetric to 3D �ow oc-curs. With increasing Vi, it is found that thereis a monotonic increase (decrease) in the drop-formation/jet-breakup length, L, in the 2-D (3-D)�ow regime. The �ow transitions depend on thedelicate balance among buoyancy, inertia, viscousand interfacial tension force which further dependupon various �uid and �ow properties. The for-mation of liquid jet injected into another immisci-ble liquid and the breakup of the jet into dropletis of fundamental importance in many industrialliquid-liquid contact processes. The breakup ofthe jet increases the interfacial area, hence en-hancing heat and/or mass transfer, and some-times chemical reactions.

There are numerous analytical [1-6], experi-mental [5-11] and numerical [12-17] studies onthe drop formation in the liquid-liquid system;mainly to predict the break-up length and thesize of the droplet. Mansour and Lundgren [12]used Boundary Element Method (BEM) to solvebreakup of liquid jet into air; limited to inviscid

1

and irrotational �uid conditions which are inher-ent characteristics of BEM. Homma et al. Us-ing front tracking method, [13] simulated jet pro-duced in 6 di�erent liquid-liquid systems. Themost popular interface capturing techniques arethe Volume of Fluid (VOF) [18] and the LevelSet (LS) Method [19], or combinations of thesetwo (CLSVOF). Using VOF method, Richards etal. [14, 15] and Xiaoguang [16]carried out directnumerical simulations for the formation of an ax-isymmetric jet and its breakup into droplets inliquid�liquid systems. Using CLSVOF method,Chakraborty et al. [17] simulated bubble detach-ment from submerged ori�ce in quiescent liquidunder normal and reduced gravity. Using LSmethod, no such work is found for the presentproblem.

The objective of the present work is to testthe capability of LS method for the simulationof drop formation process. Furthermore, the ob-jective is to study the e�ect of Reynolds num-ber and viscosity ratio, on the drop diameter andthe jet break-up length, for four di�erent liquid-liquid systems. The rest of the paper is orga-nized as follows: the physical description of theproblem is done in Section 2. The mathemati-cal formulation and numerical methodology arediscussed in Section 3 and 4, respectively. Sec-tion 5 discusses the results obtained from grid-independence and code-validation study. Finally,the results obtained and conclusions drawn fromthe present work are discussed in Section 6 and7, respectively.

2 Physical and Mathematical

Description of the Problem

The present simulations are done for the prob-lem shown in Figure 1. It shows an axisymmetricview of a stationary cylindrical tank of length Land diameter D2 = 10 containing �uid �1� (wa-ter); with a �uid �2� injected vertically upwardsfrom a circular hole (ori�ce) at the bottom of thetank. The diameter of the ori�ce, D1 = 1, istaken as the non-dimensional length scale. The�uid is injected with a fully-developed velocitypro�le: Vi = 2Vi (1−4R2

1) where Vi, considered as

Figure 1: Liquid-Liquid drop-formation con�gu-ration.

the non-dimensional velocity scale, is the averageinjection velocity. The �gure also shows the for-mation of an an axisymmetric jet and its breakupinto droplets; with the jet length represented asLj. Both the �uids are considered incompressibleand immiscible. The continuous �uid 1 is takenas the reference �uid.

In the present study, �ow is assumed to be ax-isymmetric. Figure 1 shows the computationaldomain consisting of tank-wall, tank-axis, inletand outlet; where no-slip (U = 0, V = 0), sym-metric (U = 0,∂V/∂R = 0), in�ow (U = 0, V =Vi)and out�ow (∂2U/∂Z2 = 0, ∂2V/∂Z2 = 0)boundary conditions are used, respectively.

3 Mathematical Formulation

In case of single-phase �ow, Navier-Stokes equa-tions alone are solved to obtain velocity, pressureand temperature. However, in case of two-phase�ow, in addition to Navier-Stokes equations, thegoverning equation of the interface is solved. Forthe present work, Level Set (LS) method [19] isused to model the interface. Although origins ofthe LS method lie in mathematical sources, phys-ical interpretation of various functions used in LSmethod and conservation law based derivation ofLS governing equation and continuity equationfor two-phase �ow simulation is presented in our

2

recent work Gada and Sharma [20].

3.1 Level Set Method

The Level Set interface representation is basedupon concept of implicit surfaces, wherein a LevelSet function (φ) is de�ned in a domain having a�xed value at the interface. LS function is takenas a signed normal distance function measuredfrom the interface such that it has negative val-ues in �uid 2 and positive values in �uid 1 andthe interface is de�ned at LS function value ofzero. The LS �eld is smooth leading to accuratecalculation of interface normal and curvature andat any instant of time, the exact interface posi-tion is determined by locating the zero Level Set.Additionally, LS method avoids logical di�cultiesof interface tracking as no special treatment isneeded to capture breakup/merging of �uid bod-ies.

3.2 Single Field Formulation

In the present work, the Navier-Stokes equationsare coupled with LS method by invoking the sin-gle �eld formulation; wherein a single velocity andpressure �eld is de�ned for both the �uids. Con-stant material properties are taken, but not equalfor each phase i.e. the bulk �uids are incompress-ible. Moreover, the surface tension force at inter-face is modeled as volumetric source term in themomentum equation; non zero only at the inter-face. The surface tension coe�cient is assumed tobe constant and its tangential variation along theinterface is neglected. It is assumed that interfaceis thin and mass-less with no-slip in tangential ve-locity.The non-dimensional single �eld governing

equations for simulation of two-phase �ow are:

Volume Conservation (Continuity)Equation:

∇ · ~U = 0 (1)

Mass Conservation (Level-Set Advec-tion) Equation:

∂φ

∂τ+ ~U · ∇φ = 0 (2)

Momentum Conservation Equation:

∂(ρm ~U)∂τ

+∇ ·(ρm~U ~U

)=

−∇P + 1Re∇ · (2µmD)− ρm

Fr2j + 1

Weκnδε (φ)

(3)where rate of deformation tensor, D =

0.5(∇~U +

(∇~U

)T). Furthermore, ρm and µm

are the mean non-dimensional density and vis-cosity, respectively; calculated as

ρm = Hε (φ) + χ (1−Hε (φ))µm = Hε (φ) + η (1−Hε (φ))

(4)

where the Heaviside function is calculated asH (φ) = 0 if φ < 0 or H (φ) = 0.5 if φ = 0 orH (φ) = 1 if φ > 0. However, while solving theNavier-Stokes equations, numerical instabilitiesarise if the step change in Heaviside function isused to calculate the �uid properties at cell/facecenters. Thus, the interface of �nite thickness,2ε is considered, de�ned as −ε < φ < ε, whereε is the half thickness of interface and is takenas a factor of grid spacing. For the present workthe interface thickness is taken as 2ε = 3∆x. Toavoid the step change in properties, a smoothenedHeaviside function [19] is used

Hε (φ) =

1 if φ > ε

φ+ε2ε

+ 12π

sin(πφε

)if |φ| ≤ ε

0 if φ < −ε

In the momentum equation (Eq. (3)), ~n =∇φ/ |∇φ| and κ = −∇ · n are the interfaceunit normal vector and curvature, respectively.Furthermore, in the momentum equations, δε (φ)is smoothened Dirac delta function determinedfrom the smoothened Heaviside function as

δε (φ) = dHε(φ)dφ

=

{12ε

+ 12ε

cos(πφε

)if |φ| ≤ ε

0 otherwise

Note that, the interface thickness will be neg-ligible on a fairly �ne grid. For the above equa-tions, the non dimensional variables are expressedas

~U =~u

vi, R =

r

D1

, Z =z

D1

, τ =tviD1

, P =p

ρ1v2i

3

χ =ρ2

ρ1

, η =µ2

µ1

where D1 and vi are the oriface-diameter andmean velocity of the jet at inlet, respectively.The non-dimensional governing parameters arethe Reynolds number (Re), Froude number (Fr), Weber number (We) are de�ned as

Re =ρ1viD1

µ1

, F r =vi√gD1

,

We =ρ1v

2iD1

σ

3.3 Subsidiary Equation : Reini-

tialization

The LS �eld obtained after solving the advectionequation (Eq. (2)), in general, will not remaina normal distance function �eld. For accuratecalculation of the smoothened Heaviside functionand delta function, it is necessary to maintainthe constant width of the di�used interface alongthe interface at all times, this is ensured by reini-tializing the advected Level Set function �eld tosigned normal distance function �eld without al-tering the location of interface obtained after ad-vection step. In the present work, a constraintbased PDE reinitialization procedure Sussman etal. [22] is used.

4 Numerical Methodology

The present work uses an inhouse code, whosenumerical methodology is discussed in detail byGada and Sharma[21]. The governing equationsare discretized on a Cartesian MAC-type stag-gered grid arrangement to avoid pressure veloc-ity decoupling. Pressure is located at the cen-troid of CVs (known as cell/CV center) and thecell centers for velocity components are staggeredwith u and v velocity located on vertical and hor-izontal face centers of pressure CVs, respectively.The Level Set function grid point is also stag-gered with respect to other grid points and islocated at the vertices of pressure CVs enabling

the use of uniform stencil everywhere in thedomain while implementing higher order advec-tion schemes such as Essentially Non-Oscillatory(ENO) and Weighted Essentially Non-Oscillatory(WENO).

A Finite Volume Method (FVM) semi-explicitpressure projection method is used to solve theNavier-Stokes equations. The continuity equa-tion is treated implicitly whereas the advection,di�usion and all body forces in momentum equa-tion are treated explicitly. The advection anddi�usion terms in momentum equation are dis-cretized using 2nd order TVD Lin-Lin and cen-tral di�erence scheme, respectively. The den-sity, viscosity and mass �ux are calculated at LSnodes �rst, and their values are interpolated torequired locations. Furthermore, since a semi-explicit method is used, the time step size calcu-lation is based on CFL, grid Fourier number andcapillary time step restriction.

LS advection equation is solved explicitly andit is discretized using Finite Di�erence Method(FDM) with 3rd order Runge-Kutta and 5th orderupwind WENO scheme for temporal and spatialterms, respectively. The reinitialization equationis solved using 3rd order ENO scheme.

5 Results and Discussion

The present study is done for the continuous�uid 1 as water, which is taken as the refer-ence �uid; whereas four di�erent dispersed �uids,shown in Table 1, are considered for the presentnon-dimensional study. Note that all the dis-persed �uids are less-dense and more-viscous (ex-cept heptane) than the continuous �uid.

By varying the injection velocity from 10− 50,10 − 40 and 10 − 30 and 5.2 − 17 cm/s for thesystem number 1, 2, 3 and 4, respectively; thenumerical study is done for various Re and Weshown in 2. Thus, eighteen combinations of thenon-dimensional numbers are simulated for thenon-dimensional ori�ce diameter of D1 = 1. Notethat the Re and We in the table are obtainedby considering the dimensional ori�ce diameterd1 = 2.54mm, to benchmark the present resultsfor heptane-water system.

4

Table 1: Fluid properties for the two-�uid sys-tem used in the present work. Continuous �uidis taken as water with ρ1 = 996 kg/m3and µ1 =0.958mPa.s

Sys.No.

DispersedFluid

σ(mN/m)

χ=

ρ2/ρ1

η=

µ2/µ1

1. 70%para�noil 30%Heptane

44.8 0.825 7

2. 80%para�noil 20%Heptane

45.4 0.846 16.383

3. 90%para�noil 10%Heptane

44.4 0.868 36.848

4. Heptane 36.2 0.686 0.410

Table 2: Various non-dimensional governing pa-rameters considered in the present study.CaseNo.

Sys.No.

Velocitycm/s

Re We

1. 1 10 254.72 0.542. 1 20 509.44 2.183. 1 30 764.15 4.94. 1 40 1018.87 8.725. 1 50 1273.59 13.626. 2 10 254.72 0.547. 2 20 509.44 2.158. 2 30 764.15 4.849. 2 40 1018.87 8.610. 3 10 254.72 0.5511. 3 20 509.44 2.212. 3 30 764.15 4.9513. 4 5.2 132.45 0.1814. 4 8 203.77 0.4315. 4 10 254.72 0.6716. 4 13 331.13 1.1417. 4 15.48 394.30 1.6218. 4 17 433.02 1.95

5.1 Domain-Size-Independence

and Grid-Independence Study

Initial simulation were carried out for domain-sizeindependence study. A domain size of R2 = 5and L = 30,50 and 75 (for Re =≤ 500, 500 <Re ≤ 700 and Re > 700, respectively) are foundsu�cient for the domain-size independent results

Grid-independence study is done for heptane-water system at Re = 254.72, We = 0.67, Fr =0.6335, η = 0.4102 and χ = 0.6857 (Case no.4 in Table2). On a domain size of 5 × 30, uni-form grid size of 32 × 192 , 64 × 384, 128 × 768and 256 × 1536 are considered. For the last two�nest grids, negligible di�erence in the temporalvariation of jet length and instantaneous interfaceshape is seen in Figure 2(a) and (b), respectively.Thus, a uniform grid size of 128 × 768 with cellwidth∆R = ∆Z = 5/128 is considered su�cient�ne for the grid-independent results and is usedin further simulation.

5.2 Code-Validation Study

Figure 3 shows a comparison of drop diameter,obtained by present simulation with theoretical-model and experimental results of Miester andScheele [5]; for the Heptane - Water system (Sys-tem no 4 in Table 1). The �gure also shows anerror-bar of ±5% for the experimental results;with excellent agreement between the present andthe published results.. For the drop deformedfrom the spherical shape, the drop diameter iscalculated as the mean of major and minor di-ameter of the drop, which is expected to yield anerror of less than 5% of drop volume if the diam-eter ratio is less than 1.7 [15].

5.3 Qualitative Results

As an injection velocity of 5, 20 , 30 and 40 cm/svarious breakup modes of the drop is seen in Fig-ure 5(a)-(d) for �uid-system 4, 3, 2 and 1, respec-tively. With increasing velocity, transition fromdripping (4 (a)) to jetting (4 (b)-(d)) modes isseen in the time dependent motion of the inter-face. In the jetting mode, the drop which aredetached periodically (from the jet) are of uni-

5

τ = 0.51

128

x76

8

256

x15

36

τ = 0.16(b)

τ = 0.32

τ

L j

0 5 10 15 200

1

2

3

4

5

32 x 19264 x 384128 x 768256 x 1536

(a)

Figure 2: Grid re�nement test for (a) jet lengthand (b) shape of the jet and droplet; at Re=254,We=0.67, η = 0.41023, χ = 0.68574.

V (cm/s)

Dd

2.5 5 7.5 10 12.5 15 17.52.1

2.3

2.5

2.7

Present Results

MS Model [5]

MS Experimental [5]

Figure 3: Comparison of drop diameter obtainedby present simulation with MS theoretical andexperimental results [5]. Note that the error barcorresponds to ±5% of the experimental results.

form and non-uniform size, seen in Figure4(b)and (c) respectively. The detached drop in theframe number 5 as compared to frame 4 of theFigure 4(d) is of smaller size, indicating that thedrop losses a satellite-drop which merges with thejet. Multiple-node breakup identi�ed experimen-tally by Meister and Scheele [5] is seen in the�gures for jetting mode.

It is found that the dripping mode appearswhen the Weber number is low (here at We =0.18); interfacial tension is dominant over inertialforce which causes the jet end to form a big spher-ical bulb. At higherWe, a jet is formed as the in-ertial force dominates over surface tension force.At intermediate We (= 2.2), the jet breaks upinto uniform droplets whose size is almost equiv-alent to the volume between the nodes of thefastest growing capillary wave. For We ≥ 4.2,the e�ect of the inertial force begins to contributefurther to the instability of the jet, resulting innon-uniform droplets and merging of drop withthe jet.

5.4 Quantitative Results

Figure 5 shows the e�ects of Reynolds numberand viscosity ratio on drop diameter (Dd) andbreakup length (l) for di�erent two-�uid systems.The results in the dripping (jetting) mode areshown by un�lled (�lled) symbols in the �gure.With increasing Re, Figure 5(a) shows that thedrop diameter increases in the dripping mode (ex-cept for the 70% Para�n oil 30% Heptane - Watersystem) and decreases in the jetting mode; witha negligible change for the heptane-water system.This is due to the dominance of viscous force overinertia force for the dripping mode and vice-versafor the jetting mode. Figure 5(c) shows increasein the jet break-up length with increasing Re.This is because higher Reynolds number meanshigher inertia force, which helps the formation ofjet and the mode shifts from dripping to jetting.

With increasing viscosity ratio, Figure 5(b)shows an increase in the drop diameter with aslight drop at higher η, for the Re correspond-ing to the dripping mode. Whereas for the jet-ting mode, shown by �lled symbol in the �gure,there is negligible change in drop diameter with

6

(b)

(c)

(d)

(b)

(a)

Figure 4: Temporal variation of the interface: (a)dripping (Heptane-Water system at Re=132.45,We=0.18, η = 0.41, and χ = 0.68), (b) jet-ting with uniform droplets (90% Para�n oil 10%Heptane - Water system at Re=509.44, We=2.2,η = 36.84, and χ = 0.8684), (c) jetting withnon-uniform droplets (70% Para�n oil 30% Hep-tane - Water system at Re=764.15, We=4.84,η = 7, and χ = 0.8253); and (d) jetting with asatellite drop from the parent drop merging withthe jet (80% Para�n oil 20% Heptane -Watersystem atRe=1018.87,We=8.72, η = 16.38, andχ = 0.8463).

Re

Dd

100 400 700 1000 13002

2.5

3

3.5

4

4.5

(a)

Re100 400 700 1000 13002

6

10

14

18

22

Re

Lj,

brea

k

100 400 700 1000 13002

6

10

14

18

22

(c) η

L j,b

reak

0 10 20 30 402

4

6

8

10

12

14

(d)

η

Dd

0 10 20 30 402

2.5

3

3.5

4

4.5Re = 254Re = 509Re = 764

(b)Re

Dd

100 400 700 1000 13002

2.5

3

3.5

4

4.570% Paraffin oil 30% Heptane80% Paraffin oil 20% Heptane90% Paraffin oil 10% HeptaneHeptane

(a)

Figure 5: E�ect of (a,c) Re and (b,d) viscosity ra-tio on (a,b) drop diameter and (c,d) jet break-uplength. Note that the un�lled (�lled) symbol cor-responds to dripping (jetting) mode. The motionof interface in dripping mode Re=132, We=0.18,η = 0.41, and χ = 0.68

increasing η. Figure 5(d) shows negligible changein jet breakup length with increasing η, exceptfor Re = 509 when η increases from 0.4 to 7.

6 Conclusion

Level set method based simulation of the dropbreakup problem is done for the �rst time; usingan inhouse code on a staggered grid. It shows anexcellent agreement with the published theoreti-cal and experimental results, for the drop diame-ter. The study is done for four di�erent dispersed�uid and water as the continuous �uid. Di�er-ent breakup modes: dripping, jetting with uni-form drop, jetting with non uniform drop, jettingwith nonuniform drop and merging; reported inthe published experimental results are capturedin the present simulations. The e�ect of Reynoldsnumber and viscosity ratio on the drop diameterand jet break-up length is studied. It is shownthat a full numerical simulation such as the oneemployed in the present work is necessary to cap-ture the physics involved under conditions corre-

7

sponding to inlet velocities above jet formation.

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