IOP PUBLISHING FLUID DYNAMICS RESEARCH

Fluid Dyn. Res. 42 (2010) 035503 (18pp) doi:10.1088/0169-5983/42/3/035503

Smooth particle approach for surface tensioncalculation in moving particle semi-implicit method

Hiroki Ichikawa1,2 and Stéphane Labrosse

Laboratoire des sciences de la Terre, École Normale Supérieure de Lyon, Université de Lyon,CNRS UMR 5570, 46 Allée d’Italie, 69364 Lyon Cedex 07, France

E-mail: h-ichikawa@sci.ehime-u.ac.jp

Received 15 January 2009, in final form 25 August 2009Published 28 January 2010Online at stacks.iop.org/FDR/42/035503

Communicated by M Maxey

AbstractWe present here an algorithm to solve three-dimensional multi-phase flowproblems based on the moving particle semi-implicit (MPS) method. Themethod is fully Lagrangian and can treat flows with large deformations ofthe interface such as encountered in the break-up and coalescence of drops. Themean curvature and normal vector of the interface, needed for surface tensioncalculation, are estimated by a blending of smooth particle hydrodynamics(SPH) and MPS differential schemes. The method is applied to two problems:the free oscillation of a droplet and the transition of a falling drop into avortex ring. The results are consistent with theory and experiment. This methodcan be successfully applied to the calculation of a process in planetary coreformation, where centimetre scale liquid metal droplets form an emulsion inliquid silicate.

M This article features online multimedia enhancements.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Computation of multiphase flows is complex, not only because fluid properties arediscontinuous across the interface between the different fluids but also because surface tensionis often an important player in the dynamics. The treatment of surface tension requiresknowledge of the shape of the interface, which of course evolves with the flow.

The Eulerian description of fluid dynamics is at the root of most numerical methods incomputational fluid dynamics because of their relative simplicity in many important problems.

1 Author to whom any correspondence should be addressed.2 Present address: Geodynamics Research Center, Ehime University, 2-5 Bunkyo-cho, Matsuyama 790-8577, Japan.

© 2010 The Japan Society of Fluid Mechanics and IOP Publishing Ltd Printed in the UK0169-5983/10/035503+18$30.00 1

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

Free boundary problems with surface tension can be treated by use of the volume of fluid(VOF) method (Hirt and Nichols 1981), the level set (LS) method (Osher and Sethian 1988,Sussman et al 1994) or the continuum surface force (CSF) method (Brackbill et al 1992).Surface tension is applied to the free surface and depends on the local curvature and normalvector. Both the position of the interface and its shape need therefore to be tracked in the flow,a difficult task using Eulerian approaches. In particular, a successful method must providea good treatment of the advection of the surface and avoid any numerical diffusion. To thatend, high order schemes such as the cubic interpolation pseudo-particle (CIP) method (Yabeand Aoki 1991) and the cubic interpolation volume/area coordinate (CIVA) methods (Tanaka1999) have been proposed. See Scardovelli and Zaleski (1999) for a review of the fluid-dynamical problems in multi-phase situations and some of their numerical solutions.

Free surfaces are, on the other hand, treated naturally by Lagrangian descriptions offluids. Smooth particle hydrodynamics (SPH) (Monaghan 1992) is a Lagrangian method oftenused in the context of astrophysical fluid dynamics. In this method, matter is decomposed ina number of particles, each being represented by a kernel function which can overlap thoseof others. Each quantity at any given position, including density, is obtained by summingthe contributions of each particle, weighted by its kernel. This allows us to treat both verylarge deformations of interfaces and density variations, both being important ingredients ofaccretion dynamics (Benz et al 1986, Speith and Riffert 1999, Canup 2004). On the otherhand, it is not very appropriate to incompressible situations, although some extensions havebeen proposed to that end (Cummins and Rudman 1999, Shao and Lo 2003).

We use another fully Lagrangian method, termed the moving-particle semi-implicit(MPS) method (Koshizuka and Oka 1996), that is well suited to modelling of incompressibleflows. As in SPH, a discrete number of particles is used and each quantity at each point isobtained by summing that of neighbouring particles to which a weight is attributed. However,particles are forbidden to overlap in MPS by using radius-dependent weight functions thatare infinite at the centre. The density increase is so drastic when particles try to overlapthat strong repulsive pressure forces arise. This method has been successfully applied to anumber of problems such as fragmentation (Koshizuka and Oka 1996), two-dimensional (2D)multi-phase flows with surface tension (Nomura et al 2001, Duan et al 2003a, 2003b), andfragmentation with a phase change (Koshizuka et al 1999).

Whereas Lagrangian methods allow a natural tracking of interfaces, computing thegeometrical characteristics of the interface is not straightforward but still necessary tocompute surface tension. This has been done in 2D models (Nomura et al 2001) by calculationof the number density of one phase only. However, in three-dimensional (3D) situations, thetwo principal radii of curvature are required (Landau and Lifshitz 1959) and this cannot beobtained by simple extension of the 2D approach based on curvature calculation in MPS.

The aim of this paper is to present a method allowing computation of the surfacecharacteristics, both radii of curvature and normal vector, in a 3D MPS model. This isachieved by including some features of SPH into the MPS approach. We test this techniqueby computing two different flows: the free oscillation of a droplet (Lamb 1932) and formationof vortex rings during the fall of a droplet in a viscous fluid (Baumann et al 1992).

2. Dynamics of multi-phase flows

Surface tension effects result from intermolecular forces close to an interface between twofluids, on a length scale so small that, at the scale of hydrodynamics, it is best viewed asacting on the surface separating two immiscible fluids, to which we will attribute labels0 and 1. In this study, we will assume a constant surface tension coefficient, so that the surface

2

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

tension force per unit surface can be written as (Brackbill et al 1992)

F6(x) = −n̂σ(∇6 · n̂), (1)

where n is the normal vector that points from fluid 0 to fluid 1, σ is the surface tensioncoefficient and ∇6 = ∇ − n̂(n̂ · ∇). (∇6 · n̂) in the surface tension force is linked to the mainradii of curvature of the interface, R1 and R2, by (e.g. Brackbill et al (1992), Scardovelli andZaleski (1999))

− (∇6 · n̂) = −(∇ · n̂) =1

R1+

1

R2≡ χ. (2)

With the surface delta function δ6 , we can introduce this term into the local form of themomentum balance equation, assuming Newtonian rheology,

DuDt

= −1

ρ∇ p +

1

ρ∇ · {η∇u + η(∇u)T

} + g −σ

ρn̂(∇6 · n̂

)δ6, (3)

where u, η, p and g are fluid velocity, dynamic viscosity, pressure and gravitationalacceleration, respectively, and incompressibility has been assumed,

∇ · u = 0. (4)

Note that even if each phase is incompressible and satisfies (4), it does not mean that thisequation must be satisfied everywhere. Indeed, in the case where mass exchange betweenphases is allowed, e.g. when fusion or crystallization occurs, the velocity need not becontinuous across the interface. This situation is excluded from the present study and thevolumes of both phases are constant, as well as that of their sum. Therefore, equation (4) isassumed valid everywhere.

3. Method

3.1. Generalities about the MPS method

In this section, we briefly introduce the MPS method and explain the algorithm used tocompute the surface tension term in 3D models. A more comprehensive presentation of theMPS method can be found in Koshizuka and Oka (1996).

The system is approximated as being composed of a number of discrete particles thatmove under the action of the forces applied. We therefore need to compute each term of(3) as felt by each particle. Each quantity is computed by summing the contributions of theneighbouring particles to which a distance-dependent weight w(r) is attributed. As a firstexample, the number density at the position of particle α (any given quantity q at the positionof particle α will be denoted qα), which is imposed to be a constant equal to n0 in order toachieve incompressibility, is therefore

〈n〉α

=

∑β 6=α

w(‖rβ− rα

‖) = n0, (5)

where rγ is the position vector of particle γ .Differential operators are expressed by computing weighted differences. For example, the

gradient of φ at the position of particle α is

〈∇φ〉α

=d

n0

∑β 6=α

[φβ

− φα

‖rβ − rα‖2(rβ

− rα)w(‖rβ− rα

‖)

], (6)

3

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

where d is the number of space dimensions of the problem. Similarly, the divergence is

〈∇ · v〉α

=d

n0

∑β 6=α

[(vβ

− vα) · (rβ− rα)

‖rβ − rα‖2w(‖rβ

− rα‖)

]. (7)

The Laplacian is

〈∇2φ〉

α=

2d

λn0

∑β 6=α

[(φβ

− φα)w(‖rβ− rα

‖)], (8)

with

λn0=

∑β 6=α

‖rβ− rα

‖2w(‖rβ

− rα‖), (9)

which assumes a constant number density and gives a conservative scheme (Koshizuka andOka 1996).

Several possible choices of weight function can be made (e.g. Ataie-Ashtiani and Farhadi(2006)) and we choose

w(r) =

{ re

r− 1, if 06 r 6 re,

0, if re 6 r.(10)

This means that the interaction between particles is limited to interparticle distances r 6 re.The value of re has to be defined for each operator and has been discussed for 2D cases byKoshizuka and Oka (1996) who proposed to use re = 4l0 for the Laplacian and re = 2.1l0

for all other operators, l0 being the distance between first neighbours in the cubic latticearrangement. For 3D calculations, we adopt re = 2.5l0 for the Laplacian and re = 2.1l0 forthe other operators (following a personal communication of S Koshizuka).

3.2. Treatment of the viscous jump

Even if, as assumed here, the viscosity is uniform in each phase, it can be discontinuous acrossthe interface between phases and this deserves special attention. In particular, the momentumequation (3) includes the divergence of the stress tensor, which, if decomposed in its differentcomponents, features the derivative of the viscosity:

∇ · τ =∂

∂x j

(η

∂ui

∂x j

)+

∂

∂x j

(η∂u j

∂xi

), (11)

where we have adopted the summation notation, as is the standard in tensor calculus. The firstterm can be computed similarly to the Laplacian, as given in equation (8):

〈∇ · (η∇ui )〉α

=2d

λn0

∑β 6=α

[(ηα + ηβ

2

)(uβ

i − uαi )w(‖rβ

− rα‖)

]. (12)

Taking into account the continuity equation (4), the second term of (11) can be written as

∂

∂x j

(η∂u j

∂xi

)=

(∂η

∂x j

)(∂u j

∂xi

). (13)

If locally we adopt a system of coordinates such that one of the axes is the normal to theinterface, n̂, then the viscosity variation is restricted to this coordinate and one can write(

∂η

∂xi

)(∂u j

∂xi

)=

(∂η

∂xn

)(∂un

∂xi

)= δ6[[η]]

(∂un

∂xi

), (14)

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

where [[η]] = ηn+ − ηn− is the viscosity jump across the interface, in the direction of thenormal vector n̂. The gradient can be computed according to (6) but un = u · n̂ requirescomputation of the normal vector. This computation and that of the surface delta functionare explained in the next section.

3.3. Treatment of the surface tension

Nomura et al (2001) proposed a method to treat surface tension in 2D models within the MPSframework. The curvature and normal vectors are obtained by computing the distribution ofnumber density of one phase, which is equivalent to fitting the local interface by an arc of acircle. Adapting this approach to 3D models is not straightforward since it requires us to findthe directions of the principal curvatures. However, this is not necessary to compute surfacetension, which only depends on the mean curvature, which is simply the divergence of thenormal vector according to (2).

We adopt a two-step approach, the normal vector being computed with a SPH scheme,which is reviewed by Monaghan (1992), in a way resembling that proposed by Morris (2000),whereas the mean curvature and surface delta function are obtained by MPS. SPH providesa normal vector determination that is much less noisy than that obtained from MPS. Indeed,this allows smoothing of the interface which is then thickened compared to its description inMPS.

Only the fundamentals of the SPH are recalled here. A physical quantity A(r) at anyposition is obtained by summing the contributions Aα of the neighbouring particles weightedby their kernel functions, W :

A(r) =

∑β

mβ

ρ(rβ)Aβ W (‖r − rβ

‖, h), (15)

with

ρ(r) =

∑β

mβ W (‖r − rβ‖, h), (16)

where mβ is the mass of particle β and ρ(r) is the mean density as a function of position. Weadopt a spline kernel, whose 3D form is

W (r, h) =1

πh3

1 −3

2

( r

h

)2+

3

4

( r

h

)3, if 06

r

h< 1,

1

4

[2 −

( r

h

)]3, if 16

r

h< 2,

0, if 26r

h.

(17)

h determines the width of the kernel function, and therefore the effective thickness of theinterface. We use h = 2.5l0 in this study. In order to compute the normal vector usingn̂ = ∇θ/‖∇θ‖, where θ is a colour function that takes the value 0 in fluid 0 and 1 in fluid 1,the colour function is smoothed as (Morris 2000)

〈θ〉(r) =

∑β

mβ

ρβθβ W (‖r − rβ

‖, h), (18)

and its gradient is obtained as (Monaghan 1992)

〈∇θ〉(r) =1

ρ(r)

∑β

mβ(θβ− θ(r))∇W (‖r − rβ‖, h). (19)

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

The normal vector is then obtained by dividing this gradient by its norm. The use ofSPH to compute the normal vector is justified by the need to smooth the interface. Adirect computation using (6) gives a very jagged structure essentially reflecting the localarrangement of particles. Using SPH, not only is the colour function smoothed but alsothe gradient operator is applied to kernel functions which have the proper regularitycharacteristics. In addition, it has the advantage of being based on particles and is thereforestraightforward to couple with the MPS method. On the other hand, instead of computing thecurvature with SPH (Morris 2000), which could result in over-smoothing and underestimate,we prefer to derive it from (2), using the MPS scheme for the divergence:

χα=

d

n0

∑β 6=α

(n̂β− n̂α) · (rβ

− rα)

‖rβ − rα‖2w(‖rβ

− rα‖). (20)

The surface delta function, with the proper normalization, could be obtained from(Brackbill et al 1992)

δ6 =

∥∥∥∥∇θ

[[θ ]]

∥∥∥∥ . (21)

However, this leads to a peak as broad as that defining the kernel function W (cfequation (17)). We adopt a narrower peak delta function based on the surface tracking methodproposed by Nomura et al (2001). Particle α of fluid 1 is on the interface if

〈θ〉α

=

∑β 6=α

θβw(‖rβ − rα‖) < ξn0, (22)

where ξ (<1) is a constant. The case for a particle in fluid 0 is treated by replacing θ by1 − θ . Now, this criterion is used to test whether a given particle is part of the interface or notand if the force terms that include δ6 are to be included. However, this interface has a finitethickness and a normalization factor must be applied. In a 3D problem, we find empiricallythat ξ = 0.97 gives an interface thickness tδ = 1.15l0. This thickness has to be partitionedbetween the two phases, in a way that should depend on their physical parameters, like theviscosity for example (Bercovici and Ricard 2003). We chose to apply all of the surfacetension to one phase, which is the relevant limit if the two fluids have very different viscositiesor densities, because the thickness of the interface is only about one layer of particles. Wethink that this choice is not critical in the case studied here where surface tension only actson the normal to the interface. In situations where surface tension is variable, this questionshould be investigated more thoroughly.

Before treating the dynamical problem, the first test of the method is purely geometricand aims at checking the accuracy of the curvature computation. The particles are placed in avolume having the shape of a sphere of radius r , approximated by a cubic lattice allocation,and the mean curvature is computed from the model as

χ =1

S

∫ (∇ · n̂

)δ6 dV '

l30

4πr2

∑β

〈∇ · n̂〉βδ

β

6 (23)

and compared to the theoretical one, 2/r . Figure 1 shows the result of this comparison fordifferent values of r and the agreement is quite good. For r = 4.058l0, the result starts todeviate slightly because of the decreasing quality of fitting a sphere with a cubic lattice. Thelength parameters entering the surface tension calculation are h = 2.5l0 and re = 2.5l0 andare both about half of 4.058l0. This gives an idea of the limit in the curvature that can beaccurately computed with this method.

6

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

0.1

0.2

0.3

0.4

0.5

4 8 12 16 20

t δ= 1.15 l0t δ= 1.2 l0

Cur

vatu

re m

ultip

lied

by δ

Σ n

orm

aliz

ed b

y l0

Radius normalized by l0

Figure 1. Mean curvature χ̄ =∑

β 〈∇ · n̂〉βδ

β6l3

0/4πr2 calculated from the model with two valuesof tδ , 1.15 and 1.2 and the equivalent value (2/r ) with respect to equivalent radius of a sphere.The equivalent radius (r ) is calculated from the total volume of the particles in the sphere.The equivalent radius and the mean curvature are normalized by the mean spacing l0 and 1/ l2

0 ,respectively. We use tδ = 1.15 in the following calculation because it has better agreement withthe theoretical line in the case of small spheres, though tδ = 1.2 is more accurate in the case oflarge spheres and converges to the theoretical line.

3.4. Algorithm for time stepping

Having described the computation of the different terms in the balance equation, let us nowdiscuss the flow evolution. From the velocity at one time step, a first guess for the solutionat the next time step is computed using the momentum equation (3) without any pressurecontribution. The new positions of particles are then obtained using a simple first-orderexplicit integration and the deviation from uniform number density (5) is used to solve thepressure, satisfying the following Poisson equation:

∇2 P = −

ρ0

1t2

n?− n0

n0, (24)

n? being the number density calculated with the first-guess particle positions, 1t the timestep and ρ0 the constant density. The solution is obtained by a conjugate gradient method(Hestenes and Stiefel 1952). The velocity is then corrected to take into account the pressureforce and the positions of particles are corrected. The time step 1t is chosen to satisfy a CFLcondition for both the advection and diffusion terms, with l0 as characteristic distance.

It is worth noting that only one pressure iteration is computed and used to correctthe velocity of particles. More iterations could be used to improve the accuracy of theincompressibility constraint but one should realize that local fluctuations due to imperfectorganization of the particles are necessary to avoid locking of their position in a lattice.However, the errors in the incompressibility do not accumulate with time because the particlepositions are corrected at every time step to reach the required precision.

7

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

(f)(e)

(a)

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

(c)

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

–1.0 –0.5 0 0.5 1.0x

–1.0

–0.5

0

0.5

1.0

y

(d)

(b)

Figure 2. Free oscillation of an initially square drop at four different times, without any viscosity.Length scale of the figures are l. (a)–(d) is the result with 20 × 20 particles in the drop.(e) and (f) are for 15 × 15 and 10 × 10 particles, respectively. (a) t = 0. (b) t = π1/4/4

√15. (c)

t = π1/4/2√

15. (d) t = π1/4/√

15. (e) t = π1/4/√

15. (f) t = π1/4/√

15. Only fluid 0 (inside) isvisualized.

4. Benchmark problems

4.1. Free oscillation of a droplet

As a first test of the method, we computed the free oscillation of a 2D droplet initiallyhaving a square shape (figure 2) in the absence of gravity and viscosity. This problem has

8

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

been theoretically solved (Lamb 1932, section 273) and we compare the results of our modelcalculation to this solution.

The square initial state is such that the curvature is null everywhere except at cornerswhere it is infinite. But the integral of surface tension including the corner is finite. The surfacetension force at the corners leads to the deformation of the square toward an equilibriumcircular shape but inertia makes it overshoot that position and reach a square shape that is justthe initial shape rotated by 45◦. The same process takes the droplet back to its starting stateand it oscillates with a constant period. The scale T of the oscillation period is easily obtainedby dimensional analysis, by equating inertia and surface tension, U 2/L ∼ σ/(ρL2), L beingthe side length of the square, U = L/T , the velocity scale of the problem, ρ = ρ0 + ρ1, ρ0 andρ1 the density of the inner and the outer fluids, respectively. Therefore, the oscillation periodscales as T =

√L3(ρ0 + ρ1)/σ . The exact period for an infinitely small perturbation is (Lamb

1932, section 273),

P =π1/4

√15

T . (25)

We use T as a unit of time and the only non-dimensional control parameter is ρ0/(ρ0 + ρ1) =

2/3. The calculation is performed in a 5L × 5L square domain with 8 particles in the thirddimension, in which no dynamics is supposed to take place. All boundaries are periodic. Wesolved the problem for different resolutions, as measured by the number of particles describingthe length of the square L , 10, 15 or 20.

Figure 2 shows the positions of the particles at different times: initial, a quarter of thepredicted period, half the period and a full period. This shows that for more that 15 × 15particles in the drop, the period is correctly recovered. However, one can see that the anglesare rounded off even after half a period. Clearly, this type of singularity can only be smoothedby the way surface tension is computed. Of course, this effect is stronger for lower resolutionand, for example, the calculation with 10 × 10 particles in the square shows that not even thefirst oscillation is completed (figure 2(f)).

We checked the accuracy of incompressibility from the definition of number densityof MPS (5) for several resolutions and obtained RMS fluctuation of the order of 1%,independently of resolution. This can be explained by the averaging process with re thatscales with the average interparticle distance, re = 2.5l0. The incompressibility constraint istherefore enforced on a scale that is independent of the total number of particles. Increasingthe ratio re/ l0 could improve this constraint but is more time consuming and may lead tosmoothing.

We computed the total kinetic energy at each time step, and it is represented as a functionof time for different viscosities (figure 3(a)). The results are smoothed by the same weightfunction as that used for the flow calculation in order to eliminate small scale fluctuations.Starting from a resting drop having a null kinetic energy, it reaches a maximum when the dropis circular, the surface energy being therefore minimal. Inertia makes the surface rebound andthe kinetic energy is transformed back to surface energy. The same process should proceed incycles indefinitely, but in the absence of viscosity, the kinetic energy increases. This is causedby a numerical oscillation that can be largely suppressed by adding some viscosity. In thiscase, the decrease of the kinetic energy comes from a combination of the actual viscous forceand a numerical viscous force that results from the smoothing of the interface by the SPH andthe treatment of pressure.

Figure 3(b) shows the total kinetic energy for different resolutions in the case of aviscosity η = 0.003 16

√σ L(ρ0 + ρ1). The results are mostly independent of resolution for

9

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

0 0.5 1.0Dimensionless time

0

0.2

0.4

Dim

ensi

onle

ss k

inet

ic e

nerg

y

η = 0η = 0.001 η = 0.00316η = 0.01η = 0.1

0 0.5 1.0 1.5Dimensionless time

0

0.05

0.10

0.15

0.20

0.25

0.30

Dim

ensi

onle

ss k

inet

ic e

nerg

y

20 particles30 particles40 particles

(a) (b)

Figure 3. Time evolution of the total kinetic energy in the oscillating drop computation.Horizontal axis denotes dimensionless time and vertical axis denotes dimensionless kinetic energy.(a) Calculations of 20 particles per L . Dimensionless viscosity, η, of each case is labeled, where η

is normalized by√

σ L(ρ0 + ρ1). Both fluids have same value of η. (b) Calculations of 20, 30, 40particles per L at η = 0.003 16.

the cases investigated here and agree well. The periods of oscillations computed by Fourieranalysis are within 10% of the predicted value.

4.2. Formation of a vortex ring by a falling droplet

The second benchmark we consider is the formation of a vortex ring during the fall of adroplet in another fluid. This situation has been studied experimentally and the conditions forthe formation of a vortex ring have been identified by Baumann et al (1992):

M =η0

η1> 5–8, (26)

J

R< O(1), (27)

where subscripts 0 and 1 denote the fluid properties inside and outside the droplet,respectively, J = σaρ1/η

21 is the Chandrasekhar capillary number (Chandrasekhar 1961,

section 111), R = Uaρ1/η1 is the Reynolds number, U and a are representative velocity andlength scales, respectively. J/R = σ/η1U is the ratio between surface tension, which is actingto keep the droplet as spherical as possible, and the viscous drag from the ambient fluid, whichis deforming the droplet and is the driving force for the transition to a vortex ring. It is worthnoting that the droplet will break up in the case of very small J/R with any M . If J/R = 0and M ∼ 1, a vortex ring will be formed (Baumann et al 1992). We use, as length scale, theinitial radius a of the droplet and for velocity scale U the Stokes velocity of a non-deformingfluid sphere (Batchelor 1970),

U =1

3

a2g

η1(ρ0 − ρ1)

(η1 + η0

η0 + 32η1

). (28)

The actual velocity of the droplet is affected by its deformation and internal circulation. Inparticular, if M is large enough, the internal circulation drives the rupture of the membrane atthe base of the droplet, which then becomes a vortex ring.

10

Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

Table 1. Parameter set for the calculations of a falling drop. (J/R)obs is computed using for thevelocity scale U the stationary velocity of the drop. In cases of vortex ring formation, U is takenas the stationary velocity before the transition. The shape is either spherical cap (sc) or vortex ring(vr), in which case Tb is the dimensionless time at which the transition occurs. Tb is expected tohave errors as large as ±4. L i is the length of the numerical domain in direction i . N is number ofnumerical particles. ‘∗’ in the (J/R)obs column means that the drop does not reach a steady statevelocity, in which case, the velocity is estimated from the velocity just before the breakup time Tb.In all cases, the outside fluid has η1 = 0.049 Pa s and ρ1 = 922 kg m−3.

No. η0 σ a (m) ρ0 M J/R R (J/R)obs Shape Tb L x (m) L y (m) L z (m) N

(Pa s) (N m−1) (kg m−3)

1 0.388 0.002 67 0.0085 970 7.9 0.391 25.6 1.37 vr 71.6 0.044 45 0.044 62 0.089 91 244 5962 0.291 0.002 71 0.0085 970 5.9 0.292 25.9 1.39 vr 68.6 0.044 45 0.044 62 0.089 91 244 5962l 0.291 0.002 71 0.0085 970 5.9 0.292 25.9 1.40 vr 38.1 0.044 45 0.044 62 0.1344 365 7722w 0.291 0.002 71 0.0085 970 5.9 0.292 25.9 1.16 vr 38.1 0.062 63 0.062 12 0.089 91 479 8182h 0.291 0.002 71 0.0085 970 5.9 0.292 25.9 1.41 vr 141 0.044 02 0.044 37 0.089 55 658 3123 0.096 0.002 75 0.0085 960 2 0.387 22.0 1.46 sc – 0.044 45 0.044 62 0.089 91 244 5964 0.486 0.001 68 0.0085 971 9.9 0.195 26.0 0.872 vr 53.5 0.044 45 0.044 62 0.089 91 244 5965 0.971 0.002 41 0.0085 971 19.8 0.294 25.6 1.33 vr 107 0.044 45 0.044 62 0.089 91 244 5965h 0.971 0.002 41 0.0085 971 19.8 0.294 25.6 1.38 vr 177 0.044 02 0.044 37 0.089 55 658 3126 0.194 0.002 16 0.0085 970 4 0.297 26.4 1.20 sc – 0.044 45 0.044 62 0.089 91 244 5967 0.971 0.004 82 0.0085 971 19.8 0.615 25.6 0.236 sc – 0.044 45 0.044 62 0.089 91 244 59631 0.388 0.002 67 0.011 970 7.9 0.233 55.5 1.30∗ vr 48.8 0.056 80 0.057 01 0.1149 244 59632 0.291 0.002 71 0.011 970 5.9 0.174 56.1 1.04∗ vr 27.1 0.056 80 0.057 01 0.1149 244 59633 0.096 0.002 75 0.011 960 2 0.236 47.7 1.01 sc – 0.05680 0.057 01 0.1149 244 59634 0.486 0.001 68 0.011 971 9.9 0.117 56.3 0.810∗ vr 56.3 0.056 80 0.05 701 0.1149 244 59635 0.971 0.002 41 0.011 971 19.8 0.176 55.4 1.01 vr 87.6 0.056 80 0.05 701 0.1149 244 59636 0.194 0.001 94 0.011 970 4 0.177 57.3 0.825 vr 42.8 0.056 80 0.05 701 0.1149 244 596

We simulated this process using the same parameters as those used in the experimentalstudy of Baumann et al (1992), except the radius. Indeed, we computed solutions for slightlysmaller droplets (a = 0.850 and 1.10 cm instead of a = 1.37 cm in the experiment) becauseof our limitation in domain size. Our computational domain is about 9 cm × 4.5 cm × 4.5 cmin the case of a = 0.850 cm and 11.5 cm×5.75 cm×5.75 cm in the case of a = 1.10 cm using244 596 particles, initially at rest in hexagonal close-packed arrangement. Boundaries in alldirections are periodic. We chose the standard of pressure as p0 = −ρagz, where ρa is theaverage density of the system. This system is characterized by four dimensionless numbers,J/R, M , R and ρ0/ρ1, complemented by those relevant to the initial geometry. Table 1summarizes the cases that were computed.

Let us first discuss the difference between a case where the transition to a vortex ringoccurs (case 5, figure 4, M = 19.8, J/R = 0.294 and associated movie) or does not (case 3,figure 5, M = 2.0, J/R = 0.396 and associated movie). In both cases, we start with a sphericaldrop of dimensional radius a = 0.850 cm, which is then deformed by viscous drag and takesthe shape of a spherical cap (Clift et al 1978) as an internal circulation develops. In case 5,this circulation is strong enough to overcome the surface tension at the base of the dropletand leads to the transition to a vortex ring (figure 6). The formation of a vortex ring stronglyaffects the velocity of the drop, as can be seen in figure 7: the velocity first evolves rapidlyduring its initial deformation from its initial spherical shape, then reaches an almost constantvelocity until the instability to the vortex ring situation develops and the velocity decreasesagain. The decrease of the velocity follows due to the expansion of the vortex ring (figure 4)which leads to an increase of surface to volume ratio.

We ran the simulation for many different sets of parameters M and J/R in order toobtain a phase diagram (figure 8) and compared our results to that of Baumann et al (1992).

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

(a) (b)

(c) (d)

(e) (f)

Figure 4. Evolution of the drop shape with time in the case (No. 5) of a transition to a vortex ring.(a–d) Top view. (e, f) Side view. (a) t = 0 s or t? = 0. (b) t = 5.0 s or t? = 94.1. (c) t = 7.5 s ort? = 141.1. (d) t = 10.0 s or t? = 188.1. (e) t = 0 s. (f) t = 10.0 s, where the dimensionless time,t? = U/a for U given in (28). Only fluid 0 (inside) is visualized.M An MPEG movie of this figure is available from stacks.iop.org/FDR/42/035503/mmedia.

We observed the same qualitative behaviour in our simulation as in the experiments,with small quantitative deviations in the phase boundary between situations leading to theformation of a vortex ring and those not leading to a vortex ring. More precisely, the transitionto a vortex ring seems favoured in our simulations compared to the experiments.

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

(a) (b)

(c) (d)

(e) (f )

Figure 5. Evolution of the drop shape in the case (No. 3) of a spherical cap. (a–d) Top view. (e, f)Side view. (a) t = 0 s or t? = 0. (b) t = 5.0 s or t? = 80.9. (c) t = 7.5 s or t? = 121.3. (d) t = 10.0 sor t? = 161.8. (e) t = 0 s. (f) t = 10.0, where the dimensionless time, t? = U/a for U given in (28).Only fluid 0 (inside) is visualized.M An MPEG movie of this figure is available from stacks.iop.org/FDR/42/035503/mmedia.

Several reasons, physical and numerical, can be invoked to explain these smalldifferences. Among the physical reasons, one can consider the slightly different setups: in oursimulations, the boundaries are all periodic, which means that the falling droplet is influencingitself through the flow produced in the surrounding fluid. We could run the calculation with

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

(a) (c)(b)

Figure 6. Cross section of the side view along the centre of a droplet transitioning to a vortexring. This is the same case (No. 5) as that of figure 4. The particles within 0.1 cm of the centreplane are visualized. The vectors denote the velocity relative to the average velocity of thedroplet. (a) t = 2.5 s or t? = 47.0. (b) t = 5.0 s or t? = 94.1. (c) t = 7.9 s or t? = 148.6, wherethe dimensionless time t? = U/a for U given in (28). Only fluid 0 (inside) is visualized.

0 50 100 150 200Dimensionless time

0

0.1

0.2

0.3

0.4

Dim

ensi

onle

ss v

eloc

ity

0 50 100 150 200Dimensionless time

0

0.05

0.1

0.15

0.2

0.25D

imen

sion

less

vel

ocity

(a) (b) Vortex ring is formed

Vortex ring is formed

Figure 7. Time series of the average dimensionless velocity of the drop. The velocity scale U isgiven in (28). The timescale is a/U . (a) Case 3 (solid line) and case 33 (broken line). (b) Case 5(solid line) and case 35 (broken line).

rigid vertical boundaries instead; however, we would need to use a much larger computationaldomain than that allowed by our present facilities. Clearly, the smaller horizontal length scaleused here leads to a larger vertical shear which should favour the transition to vortex rings.However, the opposite is reported in experiments of Baumann et al (1992) and we ran case 2in a wider domain (case 2w) to test this idea. Figure 9 shows the evolution of the drop velocitywith time in different calculations with the same parameters as case 2 and shows that in thecase of a wider domain, the transition to a vortex ring, which is marked by a slowing downof the drop’s fall, occurs earlier in this case. Therefore, if anything, a wider domain seems tofavour the transition to a vortex ring.

Another physical difference might be in the length of the experiment. In our numericalsimulations, the vertical periodicity allows the computation to run for as long as one wants itto. Of course, if the droplet reaches a steady state shape and velocity, the computation neednot be prolonged. The difficulty is clear: the growth rate of the vortex ring instability can besmall, particularly close to the phase boundary between the two regimes, and in some cases,a very long computation may be necessary to allow the vortex ring formation. The problem ispartly resolved by the last issue to be addressed: the finite numerical resolution.

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

Case 2

Case 5

Case 3

Figure 8. Phase diagram of vortex ring formation on the (M, J/R) space. Black open circles andtriangles are the result of the experiment of Baumann et al (1992). Filled circles denote cases withvortex ring formation in the simulation whereas filled triangles denote cases where the drop keepsa spherical cap shape.

0 50 100 150 200Dimensionless time

0

0.1

0.2

0.3

0.4

Dim

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onle

ss v

eloc

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NormalLong domainWide domainHigh resolution

0 50 100 150 200Dimensionless time

0

0.05

0.10

0.15

0.20

0.25

0.30

Dim

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NormalHigh resolution

(a) (b)

Figure 9. (a) Evolution with time of the average dimensionless velocity of the drop in case 2with four different settings. The line named normal denotes the calculation done with the standardsetting. The line named long domain is the calculation in a domain that has a 1.5 times largervertical scale (case 2l). The line named wide domain is the calculation in a domain that is 1.4times wider in each horizontal direction (case 2w). The line named high resolution is the casecalculated with higher resolution, 658 312 particles instead of 244 596, with the same domain size(case 2h). (b) Evolution with time of the average dimensionless velocity of the drop in case 5(named normal) or with a higher resolution version (named high resolution).

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

Figure 10. Time snapshot of the calculation of the metal separation process in the magma ocean(e.g. Ichikawa et al (2009)). The domain size is 7.5 cm × 7.5 cm × 12 cm with 562 331 numericalparticles. All boundaries are periodic. The surfaces represent the metal phase and the ambientsilicate phase is not visualized. The density of metal and silicate are 7800 and 3750 kg/m−3,respectively. The surface tension coefficient, viscosity of both phases and gravitational accelerationare 1 N m−1, 1 Pa s, and 10 m s−2, respectively. The resulting average velocity of metal phase isabout −0.3 m s−1 and the average drop size is about 1 cm.

Because of the discontinuous nature of our model fluid, the rupture of the membraneat the base of the droplet is easier than in natural fluids. Indeed, once only one or twolayers of particles form the membrane, the smoothing process of the numerical simulationtends to make its breaking easier than in real experiments. Once broken at one point, surfacetension leads to a rapid collapse of the membrane. In order to investigate this effect, we haveperformed some higher resolution calculations close to the boundary in the phase diagram.Case 2 is found to lead to a vortex ring but is close to the domain boundary in the phasediagram (figure 8). Figure 9(a) shows the time evolution of the fall velocity in that casefor four different settings: the standard case with same resolution and domain size as othercalculations, a vertically longer domain with standard resolution, a laterally wider domainwith standard resolution, and a standard domain size with a higher resolution. The results showthat the size of the computation domain does not affect greatly the time until the break-up. Onthe other hand, the higher resolution calculation significantly delays the transition. This effectof using particles with a finite size could therefore explain in part the quantitative difference

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Fluid Dyn. Res. 42 (2010) 035503 H Ichikawa and S Labrosse

between the experiments and calculations. Figure 9(b) shows the result of two calculationswith different resolutions for case 5, which also leads to the formation of a vortex ring but isfurther away from the phase boundary than case 2. Here again, the higher resolution delays thetransition to a vortex ring but the effect is smaller than for case 2. As a result, higher resolutionis needed for more accurate calculations especially with parameters close to the boundary inthe phase diagram.

5. Conclusion

We have presented a new algorithm to treat 3D multi-phase flows, based on a particleapproach, the MPS method, blended with SPH for the treatment of the surface tension. Themain advantage of this method is its versatility and simplicity in following a sharp interface.We applied this method to two different test problems for which a comparison can be madewith either theoretical or experimental analysis. In the case of the free oscillation of a droplet,our model predicts correctly the oscillation frequency. This problem can in addition help toquantify the limitation of the method.

The method was also applied to a fundamental fluid dynamics problem, the formation ofvortex rings with surface tension. The results concerning the condition for this formation weresuccessfully compared to experimental results of Baumann et al (1992).

This method can successfully calculate the system including interaction of many dropletssuch as fragmentation and coalescence (figure 10) (Ichikawa et al 2009). This calculationsimulates one of the important processes in the formation of planetary cores taking place inthe magma ocean, where metal and silicate phases are totally molten and immiscible. Themetal phase forms droplets that fall to the bottom of the magma ocean. In the calculation, thesize and velocity distribution of the metal droplets, which are needed to estimate the resultingcomposition of the mantle and core, are calculated. This method can therefore be used tomake some significant advances in problems of considerable importance for understandingthe differentiation of planets.

Acknowledgments

HI has been supported as a JSPS research fellow during this work. SL benefited from grants of‘Predictability of the Evolution and Variation of the Multi-scale Earth System: An integratedCOE for Observational and Computational Earth Science’ to visit the University of Tokyo andthe Earthquake Research Institute. This research was also supported by ANR project BEGDyand Région Rhones-Alpes (programme CIBLE) through a post-doc grant provided to HI.The authors wish to express their gratitude to Seiichi Koshizuka, Kei Kurita and anonymousreferees for many useful discussions.

References

Ataie-Ashtiani B and Farhadi L 2006 Fluid Dyn. Res. 38 241–56Batchelor G K 1970 An Introduction to Fluid Dynamics (Cambridge: Cambridge University Press)Baumann N, Joseph D D, Mohr P and Renardy Y 1992 Phys. Fluids A 4 567–80Benz W, Slattery W L and Cameron A G W 1986 Icarus 66 515–35Bercovici D and Ricard Y 2003 Geophys. J. Int. 152 581–96Brackbill J U, Kothe D B and Zemach C 1992 J. Comput. Phys. 100 335–54Canup R M 2004 Icarus 168 433–56Chandrasekhar S 1961 Hydrodynamic and Hydromagnetic Stability (Oxford: Oxford University Press)Clift R, Grace J R and Weber M E 1978 Bubbles, Drops and Particles (New York: Dover)

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Cummins S J and Rudman M 1999 J. Comput. Phys. 152 584–607Duan R Q, Koshizuka S and Oka Y 2003a Nucl. Eng. Des. 225 37–48Duan R Q, Koshizuka S and Oka Y 2003b J. Nucl. Sci. Technol. 40 501–8Hestenes M R and Stiefel E 1952 J. Res. Nat. Bur. Stand. 49 409–36Hirt C and Nichols B 1981 J. Comput. Phys. 39 201–25Ichikawa H, Labrosse S and Kurita K 2009 J. Geophys. Res. submittedKoshizuka S, Ikeda H and Oka Y 1999 Nucl. Eng. Des. 189 423–33Koshizuka S and Oka Y 1996 Nucl. Sci. Eng. 123 421–34Lamb H 1932 Hydrodynamics 6th edn (New York: Dover)Landau L D and Lifshitz E M 1959 Fluid Mechanics (Oxford: Pergamon)Monaghan J J 1992 Annu. Rev. Astron. Astrophys. 30 543–74Morris J P 2000 Int. J. Numer. Methods Fluids 33 333–53Nomura K, Koshizuka S, Oka Y and Obata H 2001 J. Nucl. Sci. Technol. 38 1057–64Osher S and Sethian J A 1988 J. Comput. Phys. 79 12–49Scardovelli R and Zaleski S 1999 Annu. Rev. Fluid Mech. 31 567–603Shao S and Lo E Y M 2003 Adv. Water Resources 26 787–800Speith R and Riffert H 1999 J. Comput. Appl. Math. 109 231–42Sussman M, Smereka P and Osher S 1994 J. Comput. Phys. 114 146–59Tanaka N 1999 J. Comput. Fluid Dyn. 8 121–7Yabe T and Aoki T 1991 Comput. Phys. Commun. 66 219–42

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