Numerical simulation of rising droplets in liquid–liquid systems: A comparison of continuous and...

International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

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International Journal of Heat and Fluid Flow

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Numerical simulation of rising droplets in liquid–liquid systems:A comparison of continuous and sharp interfacial force models

http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.05.0030142-727X/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author at: Chair of Fluid Process Engineering, University ofPaderborn, Pohlweg 55, 33098 Paderborn, Germany.

E-mail addresses: [email protected] (R.F. Engberg), [email protected](E.Y. Kenig).

Please cite this article in press as: Engberg, R.F., Kenig, E.Y. Numerical simulation of rising droplets in liquid–liquid systems: A comparison of conand sharp interfacial force models. Int. J. Heat Fluid Flow (2014), http://dx.doi.org/10.1016/j.ijheatfluidflow.2014.05.003

Roland F. Engberg a, Eugeny Y. Kenig a,b,⇑a Chair of Fluid Process Engineering, University of Paderborn, Pohlweg 55, 33098 Paderborn, Germanyb Gubkin Russian State University of Oil and Gas, Moscow, Russian Federation

a r t i c l e i n f o a b s t r a c t

Article history:Received 13 December 2013Received in revised form 25 April 2014Accepted 5 May 2014Available online xxxx

Keywords:DropletLiquid–liquid extractionTerminal velocityCFDLevel setInterfacial force model

Simulations of single droplets rising in a quiescent liquid were performed for various initial droplet diam-eters. The continuous phase was water and the dispersed phase was either n-butanol, n-butyl acetate ortoluene, thus resulting in three standard test systems for liquid–liquid extraction. For the simulations, alevel set based code was developed and implemented in the open-source computational fluid dynamics(CFD) package OpenFOAM�. To prevent volume (or mass) loss during the reinitialisation of the level setfunction, two methods published recently were used in the code. The continuum surface force (CSF)model and the ghost fluid method (GFM) were applied to model interfacial forces, and their influenceon grid convergence, droplet shapes and rise velocities was investigated. Grid convergence studies showa reasonable behaviour of the GFM, whereas the CSF model is less reliable, especially for systems withhigh interfacial tension. The results for droplet shape and terminal rise velocity are in excellentagreement with experimental and numerical investigations from literature. The onset of oscillations iscorrectly predicted, and the influence of the smoothing of interfacial forces on velocity oscillations isstudied. Simulations of oscillating droplets remain stable, but the frequencies of the velocity oscillationsdiffer from experimental results.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Transport phenomena at moving boundaries are crucial invarious unit operations of process engineering, e.g. condensation,absorption and liquid–liquid extraction. Therefore, moving bound-aries have been studied extensively for several decades, bothexperimentally and theoretically, and remain subject of ongoingresearch. Mass transfer in liquid–liquid extraction columns occursat moving boundaries of rising droplets. Thus, fluid dynamics ofand mass transfer in droplet swarms are essential for the designof liquid–liquid extraction. To capture these complex phenomenaproperly, fluid dynamics of single moving droplets should be fullyunderstood and mastered first.

Fluid dynamics of a single droplet is governed by physical prop-erties of the system and the size of the droplet. Relatively smalldroplets remain spherical and reach their terminal rise velocityafter a short period of acceleration. Increasing the droplet diameterleads to higher terminal velocities. Consequently, the forces acting

upon the droplet grow and deform the droplet to an ellipsoidal oroblate shape. Droplets of even larger size begin to oscillate. Apartfrom the physical properties of both liquid phases, the terminalvelocity also depends on the purity of the system. In pure systems,the interface remains mobile and the shear forces acting there leadto internal circulations, i.e. toroidal flow patterns in the droplet. Incontrast, impurities accumulated at the interface inhibit thedevelopment of internal circulations. Thus, shear forces increaseand the resulting terminal velocities are lower than in puresystems. For this reason, carefully purified systems were used inrecent experimental investigations.

This work focusses on the numerical simulation of risingdroplets. The most widely-used methods to simulate movingboundaries are moving mesh, front tracking and front capturingmethods. Moving mesh methods (e.g. Bäumler et al., 2011;Tukovic and Jasak, 2012) employ a boundary fitted grid which ismoved to keep the grid nodes exactly at the interface. Thus,boundary conditions at the interface can be implemented directly,resulting in a very exact and sharp representation of the two-phaseflow. Problems may arise when the interface undergoes significanttopological changes, such as strong droplet deformations or thecoalescence of two droplets. Front tracking methods (Unverdiand Tryggvason, 1992) describe the interface with a moving mesh

tinuous

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2 R.F. Engberg, E.Y. Kenig / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx

on a fixed grid. This approach is very accurate but requiresdynamic restructuring of the interface mesh. Front capturingmethods employ markers or indicator functions to locate the inter-face of a two-phase system. The movement of the interface is thendescribed with a simple advection equation for the indicator func-tion, which can be readily discretised with standard numericalmethods, e.g. finite element methods (FEM) or finite volumemethods (FVM). Hence, front capturing methods are robust andespecially suitable for the simulation of strong interface deforma-tions whereupon no time-consuming mesh movement or smooth-ing algorithms are necessary. While various front capturingmethods have been proposed, the prevailing approaches of thiscategory are the volume of fluid (VOF) method introduced by Hirtand Nichols (1981) and the level set method by Osher andSethian (1988).

The interfacial forces are of utmost importance for the dynamicbehaviour of moving boundaries. Consequently, the implementa-tion of interfacial forces are crucial for the performance of a CFDcode for two-phase flows, and various methods have been intro-duced. One of the earliest methods is the continuous surface stress(CSS) model of Lafaurie et al. (1994) and Gueyffier et al. (1999).Owing to its simplicity and robustness, the continuum surfaceforce (CSF) model of Brackbill et al. (1992) is probably the mostwidely-used interfacial force model. By spreading the interfacialforces over a volume, the CSF model leads to a smooth transitionin pressure at the interface. In contrast, the ghost fluid method(GFM) of Liu et al. (2000) and Kang et al. (2000) leads to a sharppressure jump.

Dijkhuizen et al. (2005) used a front tracking model to simulatetoluene droplets with initial diameters of 4–12.5 mm rising inwater. Compared with the experimental data of Wegener et al.(2010) for the same system, velocity oscillations occured for largerdiameters in the simulations and showed different amplitudes andfrequencies. Moreover, the rise velocities were lower in the simu-lations than in the experiments of Wegener et al. (2010).

Wang et al. (2008) studied mass transfer from droplets in an-butanol/water system. The moving boundary was described withthe level set method and the interfacial forces were implementedwith the CSF model. For one droplet diameter, numerical resultsfor the rise velocity and a comparison of the simulated and exper-imental droplet rise height were shown.

Bertakis et al. (2010) published terminal velocities of n-butanoldroplets rising in water obtained with the academic code DROPS(cf. Gross et al., 2006). This code employs a level set formulationof the two-phase flow discretised with finite elements; the interfa-cial forces are implemented with the CSF model. The results werein excellent agreement with the authors’ own experimental data.However, the simulated rise time was rather short and coveredonly the onset of velocity oscillations.

Eiswirth et al. (2011) studied toluene droplets rising in waterusing the commercial finite-element code COMSOL Multiphysics3.3a. The two-phase flow was described with a modified level setmethod suggested by Olsson and Kreiss (2005) which is, in fact,very similar to a VOF approach. The CSF model was used for inter-facial forces. Comparing the terminal velocities with their ownexperimental results and those published by Wegener et al.(2007), Eiswirth et al. (2011) found very good agreement for smalldroplets (<1.5 mm) and large droplets (>3 mm) as well as satisfac-tory agreement for medium-size droplets. Moreover, experimentaland simulated droplet shapes were in excellent accordance. How-ever, the simulated rise time was too short to show oscillationswhich were, consequently, not discussed.

Bäumler et al. (2011) investigated the rise of single dropletsexperimentally and numerically. The simulations of three standardtest systems for liquid–liquid extraction were carried out withmoving mesh method implemented in the academic code NAVIER.

Please cite this article in press as: Engberg, R.F., Kenig, E.Y. Numerical simulatiand sharp interfacial force models. Int. J. Heat Fluid Flow (2014), http://dx.doi

The code uses finite element discretisation, and interfacial condi-tions, including the interfacial forces, are implemented with a sub-grid projection method. Experimental results were taken fromWegener et al. (2010) (toluene/water) and Bertakis et al. (2010)(n-butanol/water). Additionally, experiments for n-butyl acetatedroplets rising in water were performed. The agreement of numer-ical and experimental results was excellent for all systems andover the whole range of droplet diameters. However, oscillationsof large droplets led to strong mesh deformation and, eventually,to a collapse of the simulations.

The results presented in this publication were obtained with aCFD-code based on a level set formulation of the two-phase flow.The code was implemented in the open source CFD packageOpenFOAM� (Open Source Field Operation and Manipulation,www.openfoam.com, cf. Weller et al. (1998)). It includes tworecent methods to overcome the volume loss frequently found inlevel set based methods. Simulations of rising droplets in threestandard standard test systems for liquid–liquid extraction(n-butanol/water, n-butyl acetate/water and toluene/water) wereperformed with the continuous interfacial force model ofBrackbill et al. (1992) and with the sharp model of Kang et al.(2000). The influence of the interfacial force model on grid conver-gence, droplet shape, rise velocity and the onset of oscillations wasstudied. Furthermore, the simulation results were compared withrecent numerical and experimental investigations. In this context,a comparison with the precise boundary fitted moving meshmethod of Bäumler et al. (2011) is particularly interesting.

2. Mathematical model and numerical method

The CFD code used for this publication was recently employedto study the influence of Marangoni convection on single risingdroplets (Engberg et al., 2014a). In the following sections, themathematical framework of the level set approach, the governingequations and the numerical details are outlined.

2.1. Level set method

The fundamental idea of the level set approach is to employ afunction to indicate whether a point x is located in the continuousphase Xc or in the dispersed phase Xd. For this purpose, the indica-tor or level set function is defined as follows:

wðx; tÞ < 0 if x 2 Xd ð1Þ

wðx; tÞ > 0 if x 2 Xc ð2Þ

Thus, the interface @X ¼ @Xc \ @Xd is defined implicitly by the iso-contour wðxIÞ ¼ 0, where xI is an arbitrary point on the interface.

In principle, any sufficiently smooth function could be chosen asa level set function. However, it has been shown that a signed dis-tance function is a particularly advantageous choice (cf. Osher andFedkiw, 2003). It has the following properties:

jwðxÞj ¼minðjx� xIjÞ for all xI 2 @X ð3Þ

jrwj ¼ 1 ð4Þ

The movement of an interface in a velocity field v can be describedwith the following advection equation for the level set function w:

@w@tþ v � rw ¼ 0 ð5Þ

Due to the advection, the level set function becomes distorted andloses its properties as a signed distance function (Eqs. (3) and (4)).

The procedure to restore w as a signed distance function iscalled reinitialisation and is realised by solving the following par-tial differential equation:

on of rising droplets in liquid–liquid systems: A comparison of continuous.org/10.1016/j.ijheatfluidflow.2014.05.003

http://www.openfoam.com


Fig. 1. Sketch of the cell and cell face structure (above) and the application of theGFM to discretise pressure (below). A ghost value (�) is inserted in cell i to accountfor the pressure jump at the interface I.

R.F. Engberg, E.Y. Kenig / International Journal of Heat and Fluid Flow xxx (2014) xxx–xxx 3

@w@s¼ w0ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

w20 þ Dx2

q ð1� jrwjÞ þ eF ð6Þ

where s is pseudo-time, Dx is grid spacing and w0 is the initialcondition

w0 ¼ wðx; s ¼ 0Þ ð7Þ

It has been shown that the zero level set contour (wðxÞ ¼ 0) is oftendisplaced during the reinitialisation (cf. Sussman and Fatemi, 1999).To prevent these artificial volume changes, a forcing term eFrecently suggested by Hartmann et al. (2010) is included in thereinitialisation equation (6). The formulation of the term eF can befound in Hartmann et al. (2010).

Eq. (6) is discretised in pseudo-time with a simple explicit Eulerscheme with a predictor–corrector procedure for the forcing termeF . For the spatial derivatives, the third-order WENO scheme ofJiang and Peng (2000) is used. To reinitialise w, 10 pseudo-timesteps with a step size of Ds ¼ Dx

2 were performed for each (physical)time step.

Despite the forcing term eF , preliminary simulations showedthat the volume was not conserved. As the droplet size has a signif-icant influence on the terminal velocity, it is very important toconserve the droplet volume exactly. Hence, an interface shiftwas implemented as proposed by Gross (2008). After the reinitiali-sation, a new level set function wnew is calculated by substracting acorrection ~w from the reinitialised level set function w:

wnew ¼ wðx; tÞ � ~w ð8Þ

The correction ~w is determined iteratively in such a way that thevolume described by wnew is equal to the initial droplet volume.The shifted level set field wnew is then defined as the new level setfunction and used for the subsequent calculations. It was found thata rather small correction of ~w � 10�3 Dx applied every second timestep was required to conserve the volume. Thus, one may safelyassume that the solution is not disturbed by the interface shift.

Peng et al. (1999) introduced the concept of a localised level setfunction: They pointed out that it is sufficient when the level setfunction is a distance function in the vicinity of the interface. Con-sequently, only cells in a confined region close to the interface haveto be reinitialised, reducing the computational effort substantially.Outside the reinitialised region, the level set function is set to aconstant value. In this work, the level set function was reinitialisedin four consecutive cells on each side of the interface.

2.2. Navier–Stokes equations and moving reference frame

In this work, three systems, each consisting of two immiscibleisothermal liquids, are studied. The liquids can be considered asNewtonian and incompressible. Hence, the fluid dynamics isdescribed with the mass continuity and the Navier–Stokes equa-tions. The equations describing the two-phase system are appliedin single-field formulation, i.e. one set of equations for the wholedomain including both the continuous and dispersed phase:

r � v ¼ 0 ð9Þ

@ðqvÞ@tþqaþðqv �rÞv ¼r� g rvþrvT

� �� rp�qgþFr ð10Þ

where q denotes density, g viscosity, p pressure and g gravitationalacceleration. A modified pressure without hydrostatic contributionp� ¼ p� qg � x is used in the simulations to simplify the specifica-tion of boundary conditions (cf. Rusche, 2002). To simulate long risetimes, the centre of the computational domain is kept approxi-mately at the centre of mass of the droplet. The acceleration terma in the momentum Eq. (10) accounts for this moving reference


frame and is determined with a procedure based on a method givenby Rusche (2002) (cf. Engberg et al., 2014b).

In the present work, the discontinuities of physical properties atthe interface are ‘‘smeared’’ over a small area to ensure numericalstability. Therefore, a smoothed Heaviside function is employed:

H�ðwÞ ¼0 if w < ��12þ

w2�þ 1

2p sin pw�

� �if jwj 6 �

1 if w > �

8><>: ð11Þ

In Eq. (11), parameter � determines the thickness of the transitionregion from one phase to another. In terms of the grid spacing Dx,a typical value of the smoothing parameter of � ¼ 1:5Dx was chosenfor the simulations.

The density q is calculated by

q ¼ qd þ ðqc � qdÞH�ðwÞ ð12Þ

The viscosity g is interpolated with a method suggested by Kothe(1998). For this purpose, the arithmetic and the harmonic meanof viscosities are calculated with the Heaviside function (11)interpolated to the cell faces (cf. Fig. 1). These mean values are thenaveraged according to the relative orientation of the interfacenormal and the cell face normal.

Two different models are used for the surface force at theinterface due to the interfacial tension Fr: the continuum surfaceforce (CSF) model of Brackbill et al. (1992) and the ghost fluidmethod (GFM) of Kang et al. (2000).

For the CSF model, the interfacial force is multiplied with asmoothed delta function d� and, consequently, transformed into avolume force acting in a small region at the interface:

Fr ¼ ðr�jrwÞd�ðwÞ ð13Þ

where r denotes the interfacial tension and �j the curvature of theinterface. The smoothed delta function d� is calculated with

d�ðwÞ ¼0 if jwj > �1

2�þ 12� cos pw

�

� �if jwj 6 �

(ð14Þ

The curvature j is defined as the divergence of the interface unitnormal nI calculated from the level set function:



Fig. 2. The wedge-shaped computational domain is 12d0 high and 5 d0 wide. Toresolve the fluid dynamics close to the droplet, the grid is refined in the centre andbecomes gradually coarser towards the boundaries.


nI ¼rwjrwj ð15Þ

j ¼ �r � nI ð16Þ

To obtain a cell averaged value, the curvature is integrated overeach finite volume Vi:

�ji ¼ �1Vi

ZVi

r � nI dV ð17Þ

Applying the Gaussian theorem yields:

�ji ¼ �1Vi

ZS

nI � dS ð18Þ

Eq. (18) can be readily approximated within the finite volumeframework as a sum over all faces of a cell.

The pressure gradient at a cell face iþ 3=2; j between cellsiþ 1; j and iþ 2; j (cf. Fig. 1) can be discretised with

@p@x

� �iþ3=2;j

�piþ2;j � piþ1;j

Dxð19Þ

However, problems arise when the interfacial tension leads to adiscontinuity in pressure between the cell centres. Liu et al.(2000) and Kang et al. (2000) showed that the approximation ofthe pressure gradient (19) can be modified by inserting a ghostvalue pi þ rjI;iþ1=2 in cell i (cf. Fig. 1) to account for the discontinu-ity. Thus, the following expression is obtained:

@p@x

� �iþ1=2

� piþ1 � pi

Dx� rjI;iþ1=2

Dxð20Þ

where jI;iþ1=2 is the curvature interpolated to the interface (cf. Liuet al., 2000; Francois et al., 2006):

jI;iþ1=2 ¼�jijwiþ1j þ �jiþ1jwijjwij þ jwiþ1j

ð21Þ

Brackbill et al. (1992) derived the CSF model under the assumptionof a gradual change in pressure at the interface. In contrast, the GFMis a discretisation technique which takes a sharp pressure jump atthe interface into account. Therefore, the CSF model leads to asmooth transition in pressure between the phases, whereas theGFM produces a pressure jump from one cell to another, which iscloser to reality. Both models are used in conjunction with single-field formulations of the mass continuity and the Navier–Stokesequations.

The implementation of the GFM is more demanding andincreases the computational effort. While the CSF model can beeasily applied to arbitrary grids, the GFM, as currently imple-mented, is limited to structured orthogonal grids. With appropriateinterpolation techniques, however, the GFM may be extended toarbitrary grids.

2.3. Discretisation, grid and boundary conditions

Eqs. (5), (9) and (10) were solved within a finite volume frame-work. As is customary in OpenFOAM�, a collocated variablearrangement was used, i.e. level set function, pressure and veloci-ties were stored at the cell centres. Convective terms v � rð Þ in Eqs.(5) and (10) were discretised with the NVD Gamma schemeintroduced by Jasak et al. (1999) and time was discretised with afirst-order accurate implicit Euler scheme. The time steps wereconstant during the simulations and in the range of 10–100 msleading to maximum Courant numbers of 0.25.

The pressure–velocity coupling is achieved with a PressureImplicit with Splitting of Operators (PISO)-algorithm (cf. Issa,1986) adopted from the VOF-code interFoam provided with


OpenFOAM� 1.7.1. For this algorithm, the continuity Eq. (9) isreformulated as a Poisson equation for pressure, providing acorrection for the velocity field. The discretised Navier–Stokes-equations and the pressure equation are solved sequentially in apredictor–corrector procedure. A detailed description of the algo-rithm was presented by Rusche (2002). In all simulations, threecorrector steps were performed with the PISO-loop.

The droplets considered in this work are small enough to rise ina vertical path (cf. Bäumler et al., 2011). Thus, axial symmetry ofthe droplet was assumed, effectively reducing the computationaleffort. The wedge-shaped computational domain can be seen inFig. 2. The reference frame is translated in the z-direction, as inthe study of Bäumler et al. (2011).

In terms of the initial droplet diameter d0, the computationaldomain is 5d0 wide and 12d0 high. As can be seen in Fig. 2, the gridis structured and refined in the vicinity of the droplet. The size ofthe refined region is d0 � 2d0 except for the simulation of thelargest n-butanol droplet. This droplet (d0 ¼ 4 mm) deforms signif-icantly and requires a refined region of 1:5d0 � 2d0. The grid spac-ing in the refined region is discussed in Section 3.2.

The boundary conditions were chosen as follows: due to themoving reference frame, the velocity at the top and the outerboundary of the computational domain is equal to the frame veloc-ity (cf. Fig. 3). At the bottom of the computational domain, a zerogradient condition is employed for velocity. Zero gradient condi-tions are also appropriate for the localised level set function atthe top, bottom and outer boundary. The boundary conditions forthe modified pressure are a constant reference pressure at the bot-tom and zero gradient conditions at the top and sides.

As can be seen in Fig. 2, the droplet is initialised as a sphere atrest in the middle of the domain.

The physical properties are taken from Misek et al. (1985) andare listed in Table 1.

3. Results

3.1. Static droplet

Before the more difficult simulation of deformable droplets isapproached, it appears reasonable to analyse a problem havingan analytical solution to assess the two interfacial force models.The static droplet is a popular benchmark for interfacial force mod-els. A 2D ‘‘droplet’’ with a diameter of d0 ¼ 4 mm was initialised in



Fig. 3. Schematic representation of the computational domain with the boundaryconditions for the simulations.


a quiescent continuous phase. The size of the two-dimensionaldomain was 8 mm � 8 mm. Both phases had similar densities(1000 kg/m3) and viscosities (10�3 Pa s) and the interfacial tensionwas 0.073 N/m. The gravitational acceleration was not taken intoaccount. Therefore, no flow should develop and the pressure jumpat the interface is given by the Young–Laplace-equation:

Dpexact ¼ rj ¼ 2rd0¼ 36:5 Pa ð22Þ

A time step of Dt ¼ 10�6 s was calculated. All systems of equationswere solved with a tolerance of 10�12.

For a quantitative comparison of the results, four quantitieswere defined (cf. Francois et al., 2006):

� jvmaxj, i.e. the magnitude of the maximum velocity.� Dptot ¼ �pd � �pc , where �pd and �pc are the arithmetic averages of

the pressures in the dispersed (w 6 0; Nd cells) and the contin-

uous (w > 0; Nc cells) phase, respectively: �pd ¼ 1Nd

PNdi¼1pi and

�pc ¼ 1Nc

PNci¼1pi.

� Dppart ¼ �ppart;d � �ppart;c. For the arithmetic averages, only cellswith H�;i ¼ 0 (for �ppart;d) or H�;i ¼ 1 (for �ppart;c) are taken intoaccount. Thereby, the transition region of the CSF model is notincluded.� Dpmax ¼ pmax � pmin, where pmax is the maximum and pmin the

minimum pressure in the computational domain.

Table 1Physical properties of the investigated systems: Density q, dynamic viscosity g and interfaciphase.

System qd (kg/m3) gd (mPa s)

Toluene/water 862.3 0.552n-Butyl acetate/water 877.2 0.820n-Butanol/water 845.4 3.280


The relative error E of the pressure differences was calculatedaccording to

E ¼ jDp� Dpexact jDpexact

ð23Þ

The results for various grid spacings are compiled in Table 2.Regarding the maximum velocities, the results indicate that spuri-ous currents are reduced with grid refinement when the GFM isused. In contrast, with the CSF model, grid refinement increasesthe magnitude of the maximum velocity. Moreover, jvmaxj is in allcases larger with the CSF model than with the GFM.

Due to the transition region, Etot is large for the CSF model oncoarse grids and decreases with grid refinement. Consequently,the error is significantly smaller when the transition region is notincluded (Epart), and a reasonable agreement with the exact pres-sure jump is achieved. Nevertheless, the error increases with thelast level of refinement. The same conclusions apply to Emax.

For the GFM, Etot is relatively small in all cases and decreaseconsistently with grid refinement. A comparison of Etot and Epart

shows that no significant deviations from the exact pressure differ-ence occur at the interface. While Emax is larger for the GFM than forthe CSF model on coarse grids, it shows a better grid convergencewith the GFM. On the finest mesh, Emax obtained with the GFM isone order of magnitude smaller than the corresponding error ofthe CSF model.

For this simple test case, results obtained with the GFM arequantitatively better in most cases. In contrast to the CSF model,the GFM shows a better and more consistent convergence withgrid refinement. These conclusions disagree with those ofFrancois et al. (2006) who observed a similar behaviour of bothmodels when a good curvature approximation was used. At thesame time, our results are in accordance with the conclusions ofKang et al. (2000), as well as some other works. In comparison withthe numerical results of Francois et al. (2006), two factors mightcontribute to the different findings: Francois et al. (2006) used adiscretisation procedure which led to an exact balance of interfa-cial and pressure forces in the static case. As noted by Rusche(2002), however, this is not guaranteed for the current implemen-tation of the PISO-algorithm. Moreover, the smoothed deltafunction with � ¼ 1:5Dx distributes interfacial forces over abroader region than the VOF-code of Francois et al. (2006). Theresults show that under these circumstances, a sharp approachleads to a better balancing of interfacial and pressure forces and,consequently, to a reliable grid convergence.

3.2. Grid convergence of the terminal rise velocity

To assess the grid convergence of the terminal rise velocity,droplets with an initial diameter of d0 ¼ 2 mm were simulatedfor all systems with various grid spacings. The resulting terminalrise velocities are compiled in Table 3.

For the n-butanol/water system, grid-independent results areobtained with a grid spacing of d0=Dx ¼ 60 and d0=Dx ¼ 80 forthe GFM and the CSF model, respectively.

al tension r at temperature #, where d denotes the droplet phase and c the continuous

qc (kg/m3) gc (mPa s) r (mN/m) # ð�C)

997.0 0.890 35.0 25996.6 0.904 14.0 25986.5 1.390 1.63 20



Table 2Comparison of the CSF model and the GFM for the static droplet test case.

Dx (lm) CSF: jvmax j (lm/s) Etot (%) Epart (%) Emax (%) GFM: jvmaxj (lm/s) Etot (%) Epart (%) Emax (%)

400 2.17 12.5 1.085 1.929 1.140 0.971 0.961 3.546200 4.15 6.55 0.183 0.822 0.294 0.230 0.232 0.988100 9.83 3.48 0.017 0.664 0.119 0.055 0.055 0.270

50 19.7 1.56 0.050 0.782 0.044 0.025 0.026 0.071

Table 3Comparison of the terminal rise velocity of a droplet with d0 ¼ 2 mm in all three testsystems obtained with the CSF model and the GFM and various grid spacings.

d0=Dx n-Butanol/water n-Butyl acetate/water Toluene/water

CSF GFM CSF GFM CSF GFM

20 55.92 55.98 80.19 84.49 83.26 102.7540 56.44 56.57 87.53 88.01 98.74 110.8760 56.59 56.63 88.90 88.61 106.24 112.2880 56.68 56.63 88.11 88.72 112.09 112.90

100 56.68 56.63 88.82 88.80 112.47 112.97120 – – 88.96 88.85 112.18 112.96

Fig. 4. Simulated droplet shapes of the n-butanol/water system at times 0.1 s, 0.2 s,0.3 s, 0.4 s, 0.5 s and 0.6 s (left to right); the initial droplet diameters are 2 mm,3 mm, 3.48 mm and 4 mm (top to bottom). The GFM was used for the simulations.

Fig. 5. The streamlines of a n-butanol droplet with d0 ¼ 4 mm rising in water reveala vortex in the wake of the droplet and two internal circulations.

Table 4Aspect ratios of the system n-butanol/water from this study obtained with the CSFmodel (ECSF ) and the GFM (EGFM) and simulation results from Bäumler et al. (2011)(ENav ier)

d0 (mm) ECSF EGFM ENavier

1.00 0.95 0.95 0.951.50 0.84 0.84 0.842.00 0.66 0.66 0.662.50 0.48 0.47 0.473.00 0.35 0.36 0.36

Table 5Aspect ratios of the system n-butyl acetate/water from this study obtained with theCSF model (ECSF ) and the GFM (EGFM) and simulation results from Bäumler et al. (2011)(ENav ier)


1.00 0.99 0.99 0.991.50 0.97 0.97 0.972.00 0.92 0.91 0.922.50 0.79 0.79 0.803.00 0.65 0.64 0.653.50 0.55 0.56 0.55

Fig. 6. Simulated droplet shapes of the toluene/water system at times 0.1 s, 0.2 s,0.3 s, 0.4 s, 0.5 s and 0.6 s (left to right); the initial droplet diameters are 2 mm,3 mm and 4 mm (top to bottom). The GFM was used for the simulations.

Table 6Aspect ratios of the system toluene/water from this study obtained with the CSFmodel (ECSF ) and the GFM (EGFM) and simulation results from Bäumler et al. (2011)(ENav ier).


1.00 1.00 1.00 1.001.50 0.99 0.99 0.982.00 0.96 0.95 0.952.50 0.89 0.88 0.883.00 0.77 0.76 0.753.50 0.65 0.65 0.644.00 0.58 – 0.57


Slightly finer grids are required for the n-butyl acetate/watersystem. The interfacial tension of the n-butyl acetate/water systemis almost one order of magnitude higher than that of the n-butanol/


water system. Consequently, the interfacial force model is far moreimportant in this case. While the GFM shows reasonable grid con-vergence, the convergence behaviour of the CSF model is less clear.

The toluene/water system possesses the highest interfacialtension among the three systems investigated. Therefore, it is themost challenging test system for the interfacial force models in this



Fig. 7. Qualitative representation of the transient droplet rise velocity as a functionof time. The lower curve (a) applies for circulating droplets and the upper curve (b)for oscillating droplets. The maximum velocity vmax and the characteristic meanvelocity vcha are marked.


study. The results show a very good convergence for the GFM, but aless reliable behaviour of the CSF model.

The grid convergence study affirms the conclusion of the staticdroplet test case: while grid convergence was achieved with theGFM in all cases, the CSF model was less reliable. This findingreflects the consistent reduction of spurious currents with gridrefinement achieved with the GFM – a quality which is particularlyimportant for systems with high interfacial tension (e.g. toluene/water).

Furthermore, the results show that a grid spacing of d0=Dx ¼ 80leads to good results which deviate only marginally from those

Fig. 8. Terminal rise velocity of n-butanol droplets in water: (a) The numerical results frexperimental and numerical results from Bertakis et al. (2010). (b) Relative deviation froGFM (�). (c) Relative deviation from the experimental results of Bertakis et al. (2010) forresults of Bertakis et al. (2010) for the CSF model (þ) and for the GFM (�).


obtained with the finer grids. This conclusion was also confirmedfor other droplet diameters. Hence, the simulation results pre-sented in following sections were obtained with this grid spacing.

3.3. Droplet shape

The droplet shape is defined by the isocontour wðx; tÞ ¼ 0 whichcan be obtained in the post processing procedure. In Fig. 4, somedroplet shapes for the n-butanol/water system are shown. Theshapes agree well with those presented by Bertakis et al. (2010).In accordance with Bäumler et al. (2011), we observed a vortexin the wake of droplets with d0 > 2 mm. Furthermore, we founda secondary internal circulation for d0 ¼ 3:48 mm and d0 ¼ 4 mmwhich is presented in Fig. 5. This phenomenon can be explainedby the very flat shape in conjunction with the vortex in the wakeof these droplets.

The aspect ratio E, i.e. the ratio of the two principal axes, wasdetermined graphically from the droplet shape for steady-stateconditions. Both for the CSF model and the GFM, the agreementwith the simulation results of Bäumler et al. (2011) is excellent,as can be seen in Table 4. Oscillations were observed for dropletswith d0 P 3 mm. This is in line with Bertakis et al. (2010) andBäumler et al. (2011) and will be discussed further in Section 3.5.

The aspect ratios obtained for the n-butyl acetate/water systemare given in Table 5. A comparison with the aspect ratios deter-mined by Bäumler et al. (2011) shows very good agreement forboth interfacial force models. In accordance with Bäumler et al.(2011), shape oscillations occur for droplets with d0 ¼ 4 mm insimulations with both interfacial force models.

om this work in comparison with simulations from Bäumler et al. (2011) as well asm the simulation results of Bäumler et al. (2011) for the CSF model (þ) and for thethe CSF model (þ) and for the GFM (�). (d) Relative deviation from the simulation



Fig. 9. Terminal rise velocity of n-butyl acetate droplets in water: (a) The simulation results from this work in comparison with experimental and numerical results fromBäumler et al. (2011). (b) Relative deviation from the simulation results of Bäumler et al. (2011) for the CSF model (þ) and for the GFM (�). (c) Relative deviation from vmax ofBäumler et al. (2011) for the CSF model (þ) and for the GFM (�). (d) Relative deviation from vchar of Bäumler et al. (2011) for the CSF model (þ) and for the GFM (�).


Some droplet shapes for the toluene/water system are shown inFig. 6. The qualitative agreement of the simulated shapes of thiswork with the experimental and numerical results of Eiswirthet al. (2011) is apparent. In comparison with the CSF model, theGFM leads to aspect ratios slightly closer to the results of Bäumleret al. (2011) (cf. Table 6). Similar to Bäumler et al. (2011), we foundshape oscillations for droplets with d0 ¼ 4:4 mm in the simulationwith the CSF model. In simulations with the GFM, oscillations occu-red already for droplets with d0 ¼ 4 mm. This finding will be dis-cussed further in Section 3.5.

3.4. Terminal rise velocity

In this section, the results for the drop rise velocity are pre-sented and compared with experimental and numerical resultsfrom the literature. A qualitative representation of the transientdroplet rise velocity is shown in Fig. 7. The droplet rise velocityusually increases to a maximum vmax before slowing down andreaching the terminal, or characteristic mean velocity vcha (cf.Wegener et al., 2010). The upper curve (b) in Fig. 7 depicts the risevelocity of a larger droplet with oscillations around the meanvelocity vcha.

In Fig. 8, terminal rise velocities for n-butanol droplets in waterare shown together with the results of Bertakis et al. (2010) andBäumler et al. (2011) as well as the corresponding relative devia-tions. With both interfacial force models, an excellent agreementwith the numerical results of Bäumler et al. (2011) is achieved.Both numerical and experimental values of the terminal rise veloc-ity published by Bertakis et al. (2010) show a distinct maximum


obtained for a droplet diameter of approximately 2.5 mm and arapid decrease for larger droplets. In contrast, our simulationsand those of Bäumler et al. (2011) predict a less pronounced max-imum velocity for larger droplets with initial diameters in therange of 3.1–3.5 mm. Recently, Komrakova et al. (2013) publishedterminal rise velocities of n-butanol droplets in water obtainedwith a Lattice Boltzmann method. In their study, the maximum risevelocity was found for the droplet with d0 ¼ 3:8 mm. As noted byBäumler et al. (2011), the simulated times of Bertakis et al.(2010) were slightly too short to reach the terminal rise velocityfor large droplets. Consequently, deviations are found for dropletswith d0 P 3 mm.

The terminal rise velocities for n-butyl acetate droplets in waterfrom this work, the results of Bäumler et al. (2011) and their relativedeviations are shown in Fig. 9. The results for the n-butyl acetate/water system show the difference between the interfacial forcemodels more clearly: A very good agreement with the simulationresults of Bäumler et al. (2011) is obtained with the GFM, whilethe CSF model leads to larger deviations, cf. Fig. 9(b). The maximumdeviation is found for the smallest droplet (d0 ¼ 1 mm) which pos-sesses the highest jump in pressure at the interface. Furthermore, areasonable agreement with the experimental results of Bäumleret al. (2011) is achieved with both models, cf. Fig. 9(c), (d).

Our results for the terminal rise velocities for toluene droplets inwater and those of Wegener et al. (2010) and Bäumler et al. (2011)as well as their relative deviations are shown in Fig. 10. The resultsobtained with the CSF model deviate distinctly from those with theGFM and the data of Bäumler et al. (2011). Similar to the previoussystem, the result for the smallest droplet (d0 ¼ 1 mm) obtained



Fig. 10. Terminal rise velocity of toluene droplets in water: (a) The numerical results from this work in comparison with simulations from Bäumler et al. (2011) andexperimental results from Wegener et al. (2010). (b) Relative deviation from the simulation results of Bäumler et al. (2011) for the CSF model (þ) and for the GFM (�). (c)Relative deviation from vmax of Wegener et al. (2010) for the CSF model (þ) and for the GFM (�). (d) Relative deviation from vchar of Wegener et al. (2010) for the CSF model(þ) and for the GFM (�).


with the CSF model shows the largest deviations, cf. Fig. 10(b). Withthe GFM, a very good agreement with the simulation results ofBäumler et al. (2011) is achieved. For both interfacial force models,the simulated terminal rise velocities agree well with the experi-mental maximum velocities of Wegener et al. (2010).

Transient rise velocities of toluene droplets with d0 ¼ 2 mm andd0 ¼ 3:5 mm simulated with the GFM are presented in Fig. 11 withthe corresponding experimental data of Wegener et al. (2010). Inboth cases, a good agreement is achieved. Deviations during the

Fig. 11. Transient rise velocity of toluene droplets in water: Simulation results withthe GFM (lines) and experimental data of Wegener et al. (2010) (symbols).


acceleration phase can be explained with different initial condi-tions: While the simulated droplets are initialised as spheres, theexperimental droplets are produced with a capillary and deformduring the detachment. In all cases, the acceleration phase is ter-minated after 1 s or less.

The results presented in this section are in line with the findingsof Section 3.1: In comparison with the CSF model, the GFM reducesspurious currents. This is especially important in cases with highinterfacial tension and large curvatures (i.e. for small droplets).

Table 7Numerical results for the terminal rise velocity in mm/s for all cases considered in thiswork.

d0 n-Butanol/water n-Butyl acetate/water Toluene/water

CSF GFM CSF GFM CSF GFM

1.00 27.9 28.0 33.4 34.9 37.8 41.11.50 44.2 44.4 57.3 59.8 72.8 73.61.56 46.2 46.2 – – – –1.79 52.4 52.5 – – – –2.00 56.7 56.6 88.1 88.7 112.1 112.92.12 58.6 58.7 – – – –2.26 60.0 60.1 – – – –2.48 61.4 61.5 – – – –2.50 61.5 61.6 116.6 116.1 153.3 155.53.00 63.2 63.3 124.8 125.2 179.4 180.83.06 63.4 63.4 – – – –3.48 64.3 64.3 – – – –3.50 – – 123.5 123.7 182.0 182.04.00 63.4 63.3 121.0 121.0 176.0 176.34.40 – – – – 171.8 171.8




The numerical values for the terminal rise velocity of all casesconsidered in this work are compiled in Table 7.

3.5. Oscillations and stability

In simulations of the n-butanol/water system, oscillations occurfor droplets with d0 P 3 mm. However, the amplitude of theoscillations decreases and eventually, a constant rise velocity anddroplet shape is reached. This behaviour is shown in Fig. 12 for adroplet with d0 ¼ 4 mm, simulated with both the CSF model andthe GFM.

In contrast, the n-butyl acetate droplet with d0 ¼ 4 mm and thetoluene droplet with d0 ¼ 4:4 mm show oscillations with constantamplitude and frequency. As mentioned in Section 3.3, toluenedroplets with an initial diameter of 4.4 mm show oscillations whenthe CSF model is used, whereas with the GFM, oscillations occur forsmaller droplets (d0 ¼ 4 mm). This finding can be explained withFig. 13. In the simulations discussed previously, the smoothingparameter of the transition region was � ¼ 1:5Dx. When thesmoothing parameter is increased to � ¼ 3Dx, toluene dropletswith d0 ¼ 4:4 mm simulated with the CSF model do not oscillate,which contradicts experimental findings. In contrast, oscillationsare found in the rise velocity simulated with the GFM. Apparently,simulation results are very sensitive to the smoothing of interfacialforces and excessive smoothing may even lead to an erroneousprediction of droplet dynamics.

Fig. 12. Transient rise velocities of a n-butanol droplet with d0 ¼ 4 mm rising inwater, simulated with the CSF model and the GFM.

Fig. 13. Transient rise velocities of a toluene droplet with d0 ¼ 4:4 mm rising inwater, simulated with the CSF model and the GFM. In both cases, the smoothingparameter of the transition region was set to � ¼ 3Dx.


Comparing the simulated rise velocities with the experimentalresults of Wegener et al. (2010), we find a similar amplitude ofapproximately 10–15 mm/s in the velocity oscillations. Neverthe-less, the frequencies (or periods) of the velocity oscillationsobtained from the simulations are very different from experimen-tal results: The order of magnitude of the periods in our simula-tions is 0.1 s, whereas Wegener et al. (2010) determined periodsof approximately 1 s experimentally. For oscillating droplets, axialsymmetry cannot be assumed and droplets should be simulatedfully three-dimensional. Consequently, it is likely that 3D effectswhich cannot be reproduced in our simulations are responsiblefor the different frequencies.

4. Conclusions

In this paper, the results of numerical simulations of singledroplets rising in a quiescent liquid are presented. Three standardtest systems for liquid–liquid extraction were investigated:n-butanol/water, n-butyl acetate/water and toluene/water inwhich water represents the continuous phase. Assuming axialsymmetry of the droplet, we simulated droplets with variousdiameters up to the onset of oscillations. For the simulations, aCFD code based on a level set formulation of the two-phase flowwas developed and implemented in the open-source CFD packageOpenFOAM�. A well-known drawback of the level set method isthe volume change during reinitialisation. Applying two recentlypublished methods, we were able to overcome this problem. Fur-thermore, two interfacial force models were applied in this work:the widely-used CSF model, which leads to a smooth transitionin pressure at the interface, and the GFM, which results in a sharpjump. The influence of the interfacial force model on the shape andthe dynamic behaviour of the droplets was studied.

For the theoretical test case of a static droplet as well as for ris-ing droplets in all three test systems, grid convergence wasachieved with the GFM. In contrast, the CSF model showed a lessreliable behaviour with respect to grid refinement.

Regarding droplet shapes and terminal rise velocities, goodagreement with recent numerical investigations by Bäumler et al.(2011) and Bertakis et al. (2010) was found with both interfacialforce models. The results obtained with the GFM agree particularlywell with the numerical results of Bäumler et al. (2011). The com-parison of terminal rise velocities, droplet shapes and onset ofoscillations obtained from our simulations with experimentalresults published by Bertakis et al. (2010), Wegener et al. (2010)and Bäumler et al. (2011) is equally satisfactory for both interfacialforce models. The only distinct discrepancy was found in then-butanol/water system for droplets with d0 P 3 mm. Bertakiset al. (2010) found a pronounced maximum in the terminal risevelocity for droplets with d0 � 2:5 mm, both experimentally andnumerically. Similar to Bäumler et al. (2011), our simulations showa less pronounced maximum velocity for larger droplets withinitial diameters in the range of 3.1–3.5 mm. Komrakova et al.(2013) found the maximum rise velocity for droplets withd0 ¼ 3:8 mm. Future experimental and numerical investigationsshould clarify the tendency for larger droplets in this system.

The onset of oscillations predicted with our simulationscompares favourably with numerical and experimental resultsfrom literature (Bäumler et al., 2011; Bertakis et al., 2010;Wegener et al., 2010). However, it was shown that excessivesmoothing of interfacial forces with the CSF model can lead to anunphysical stabilisation of the droplet. In contrast, this problemdoes not occur with the GFM.

The results presented in this paper show that a satisfactory sim-ulation of droplets can be achieved with both the CSF model andthe GFM. Nevertheless, the GFM results in lower spurious currents




which, furthermore, are consistently reduced with grid refinement.These properties are reflected in the simulated rise velocities andin a reliable performance with respect to grid refinement.

Acknowledgement

We would like to thank Dr. Mirco Wegener for the experimentaldata.

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