Modelling od surface tension using ISPH method

Modelling of surface tensionforce for free surface flows in

ISPH methodAbdelraheem M. Aly, Mitsuteru Asai and Yoshimi SondaDepartment of Civil Engineering, Kyushu University, Kyushu, Japan

Abstract

Purpose – The purpose of this paper is to show how a surface tension model and an eddy viscositybased on the Smagorinsky sub-grid scale model, which belongs to the Large-Eddy Simulation (LES)theory for turbulent flow, have been introduced into ISPH (Incompressible smoothed particlehydrodynamics) method. In addition, a small modification in the source term of pressure Poissonequation has been introduced as a stabilizer for robust simulations. This stabilization generates asmoothed pressure distribution and keeps the total volume of fluid, and it is analogous to the recentmodification in MPS.

Design/methodology/approach – The surface tension force in free surface flow is evaluatedwithout a direct modeling of surrounding air for decreasing computational costs. The proposed modelwas validated by calculating the surface tension force in the free surface interface for a cubic-dropletunder null-gravity and the milk crown problem with different resolution models. Finally, effects of theeddy viscosity have been discussed with a fluid-fluid interaction simulation.

Findings – From the numerical tests, the surface tension model can handle free surface tensionproblems including high curvature without special treatments. The eddy viscosity has clear effects inadjusting the splashes and reduces the deformation of free surface in the interaction. Finally, theproposed stabilization appeared in the source term of pressure Poisson equation has an important rolein the simulation to keep the total volume of fluid.

Originality/value – An incompressible smoothed particle hydrodynamics is developed to simulatemilk crown problem using a surface tension model and the eddy viscosity.

Keywords Flow, Viscosity, Simulation, Incompressible smoothed particle hydrodynamics,Surface tension, Free surface flow, Eddy viscosity, Milk crown

Paper type Research paper

Nomenclaturec ¼ colour functiond0 ¼ initial particles spacingFij ¼ interaction particle force�g ¼ gravity accelerationk ¼ turbulent kinetic energyL ¼ length of fluid object�n ¼ unit normal to the interfacerij ¼ distance between particless*ij ¼ strength of proposed force acting

between particlest ¼ time�u ¼ velocity vector of fluidCs ¼ Smagorinsky constant�fi ¼ total force due to interactions

f s ¼ surface tension forceh ¼ smoothing lengthks ¼ curvature of the interfacemi ¼ mass at each particle “i”P ¼ pressure of fluidSab ¼ strain rate tensor componentssij ¼ strength of force acting between

particles�u * ¼ intermediate velocityWij ¼ Kernel function

Greek symbolsa ¼ relaxation coefficientDt ¼ time step

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0961-5539.htm

Received 18 March 2011Revised 22 June 2011

Accepted 19 August 2011

International Journal of NumericalMethods for Heat & Fluid Flow

Vol. 23 No. 3, 2013pp. 479-498

q Emerald Group Publishing Limited0961-5539

DOI 10.1108/09615531311301263

Modellingsurface tension

force

479

h ¼ numerical parametery ¼ kinematic viscosity of fluidr ¼ density of fluids ¼ surface tension coefficient~t ¼ sub particle scale (SPS) stress tensorD ¼ Smagorinsky constantdab ¼ Dirac delta functionf ¼ arbitrary functionyT ¼ eddy viscosityr * ¼ numerical density of fluidf ¼ gradient operator

~tab ¼ sub particle scale (SPS) stress tensorcomponents

Subscriptsa, b ¼ refers to spatial dimensionsj ¼ refers to particle “j”i ¼ refers to particle “i”

Superscript0 ¼ refers to initial value

1. IntroductionSimulating turbulent liquids with splashes is among the desired features in fluidanimations. Very interesting processes, e.g. turbulence, wall fluid interactions, freesurface behaviors, fluid-fluid interactions, take place in different spatio-temporal scales,starting from micro scale, through meso scale to macro scale. The SPH method is a verypopular particle method for simulating processes in the macro scale (Monaghan, 2005),and it has possibility to simulate phenomena from the domain between macro and mesoscales. For the accurate prediction for such phenomena, the SPH method must possessthe ability to calculate the following: surface tension at the interface, stabilized pressure,and turbulence models.

Traditional treatment of surface tension in SPH method is borrowed from thecontinuum surface force (CSF) model (Brackbill et al., 1992) where interface curvature iscalculated through a colour function (Morris, 2000). This approach gives an accurateestimation of the effects of surface tension, but it is difficult to apply it for the largecurvature cases in the free surface problem because of the error in the calculation of thesecond derivative of a color function. Adami et al. (2010) proposed a new surface tensionmodel for multi-phase problems using a reproducing divergence approximation forgetting accurate surface curvature without full support of kernel function. Anothertreatment of surface tension has been proposed by Nugent and Posch (2000) andTartakovsky and Meakin (2005), in which the free surface tension force is modeled bythe summation of an interaction particle force. In this study, the interaction particleforce is improved by using a smoothing function.

In the incompressible flow analysis by the SPH, there are two trends; a semi-implicittruly incompressible SPH (ISPH) and the classical weakly compressible SPH (WCSPH).Lee et al. (2008) made comparisons between these two formulations. In their report,the ISPH shows much accurate solution especially in the pressure representation. In theISPH based on the projection method, the pressure Poisson equation (PPE) should besolved. The source term in PPE is not unique in the literature. One of them is modeledas a function of density variation, and the other utilizes velocity divergence term. Theimprovement of the ISPH is still continued. Khayyer et al. (2008, 2009) proposed acorrected incompressible SPH (CISPH) method derived based on a variational approachto ensure the angular momentum conservation of ISPH formulations. In their model,improved pressure distribution is coming from improvement of momentumconservation and the second improvement is achieved by deriving and employing ahigher order source term based on a more accurate differentiation. Hu and Adams (2007)pointed that, if only a divergence free velocity field is enforced, density variation, or

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particle clustering could happen due to the spatial truncation error; moreover, thisdensity error could accumulate during the simulation. To overcome such a problem,a new method, where the particle positions are shifted slightly across streamlines, andthe hydrodynamics variables at new positions are interpolated by Taylor series, hasbeen presented by Xu et al. (2009). In this study, a simple modification in the source termof PPE is utilized for generating a smoothed pressure distribution and for keeping thetotal volume of fluid. The modification is analogous to the recent modification in MPS byTanaka and Masunaga (2010).

To resolve all of the vortices during complex flows analysis is not easy and it isbetter to take into account the effects of turbulence model. A brief overview for turbulentincompressible free surface flows using SPH method has been introduced by Violeauand Issa (2007). A 2D large eddy simulation (LES) was applied to the particle methods,where sub particle scale motion was modeled as additional viscosity (Gotoh et al., 2000,2001, 2003; Shao and Gotoh, 2003, 2004). In this study, the LES with the traditionalSmagorinsky turbulence model has been introduced to take into account the effect ofeddy viscosity.

In this paper, an ISPH method with a modified source term is utilized to solve freesurface flows including the surface tension effect. In addition, the eddy viscosity and ourproposed surface tension model have been introduced into ISPH method. The accuracyand the efficiency of the proposed model are investigated in a couple of examples,deformations of cubic droplet and milk crown problem, respectively. Moreover,fluid-fluid interactions have been performed in different two cases. In the first case, thesimulation is introduced including/excluding eddy viscosity to show the effect of eddyviscosity, while in the second case, the simulation is introduced for study the interactionbetween two fluids with different density ratios. Also, the graphical illustrations forthe total volume of fluid during the whole simulation have been discussed.

2. Incompressible smoothed particle hydrodynamics (ISPH) formulationAfter the explanation of the conventional ISPH method, the proposed surface tensionmodel and stabilization in the PPE are presented in this section.

2.1 The governing equations for incompressible flowIn the Lagrange description, the continuity and Navier-Stokes equations can be written by:

Dr

Dtþ r7 · �u ¼ 0; ð1Þ

D �u

Dt¼ 2

1

r7P þ y72 �uþ �gþ

1

r7 · ~tþ

1

r�f s; ð2Þ

where r and y are density and kinematic viscosity of fluid, �u is a velocity vector of fluid,P is pressure of fluid, �g is gravity force, t indicates time, ~t is sub particle scale (SPS) stresstensor and �f s is the surface tension force.

In the incompressible flow, the density is assumed by a constant value with itsinitial value r 0. Then, the aforementioned governing equations lead:

7 · �u ¼ 0; ð3Þ


force

481

D �u

Dt¼ 2

1

r07P þ y72 �uþ �gþ

1

r 07 · ~tþ

1

r 0�f s: ð4Þ

2.2 SPH formulationsA basic concept in SPH method is that any function f attached to particle “i” at aposition ri is written as a summation of contributions from neighbor particles:

f ðriÞ < kfil ¼j

Xmj

rjfjW ðrij; hÞ: ð5Þ

Note that, the triangle bracket kfil means SPH approximation of a function f. Thedivergence of a vector function can be assumed by using the above defined SPHapproximation as follows:

7 · �f ðriÞ < k7 · �fil ¼1

ri j

Xmjð �fj 2 �fÞ ·7W ðrij; hÞ; ð6Þ

and the expression for the gradient can be represented by:

7f ðriÞ < k7fil ¼ rij

Xmj

fj

r2j

þfi

r2i

!7W ðrij; hÞ: ð7Þ

In this paper, quintic spline function (Schoenberg, 1946) is utilized as a kernel function:

W ðrij; hÞ ¼ bd

3 2rijh

� �526 2 2

rijh

� �5þ15 1 2

rijh

� �50 # rij , h

3 2rijh

� �526 2 2

rijh

� �5h # rij , 2h

3 2rijh

� �52h # rij , 3h

0 rij $ 3h

266666664

; ð8Þ

where bd is 7/478 ph 2 and 3/358 ph 3, in two- and three-dimension space, respectively,(Figure 1).

The gradient of pressure and the divergence of velocity are approximated as follow:

7pðriÞ < k7pil ¼ rij

Xmj

pjr 2

j

þpir 2

i

!7W ðrij; hÞ; ð9Þ

7 · �uðriÞ < k7 · �uil ¼1

ri j

Xmjð�uj 2 �uiÞ ·7W ðrij; hÞ: ð10Þ

In addition, the second derivative of the laminar viscous force and the Laplacian ofpressure have been proposed by Morris et al. (1997) by an approximation expression asfollows:

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7 · ðy7 · �uÞðxiÞ < k7 · ðy7 · �uiÞl ¼j

Xmj

riy i þ rjy j

rirj

�rij ·7W ðjri 2 rjj; hÞ

r2ij þ h 2

!�uij; ð11Þ

where h is a parameter to avoid a zero dominator, and its value is usually given byh 2 ¼ 0.0001 h 2. Similarly, the Laplacian of pressure in PPE:

72pðriÞ < k72pil ¼j

Xmj

ðri þ rjÞ

rirj

pij�rij ·7W ðjri 2 rjj; hÞ

r2ij þ h2

!: ð12Þ

The SPS stress tensor is modeled through the traditional Boussinesq eddy viscosityassumption as:

~tab

r¼ 2yT

�Sab 22

3kdab; ð13Þ

where yT is an eddy viscosity is calculated by using standard Smagorinsky model:

yT ¼ ðCsDÞ2j �Sj; ð14Þ

in which, Cs ¼ 0.2 is Smagorinsky constant, D is constant and it taken as smoothingcompact support in this scheme. The local strain rate j �Sj is calculated as Violeau andIssa (2007). The stress tensor of sub-particles scale:

Sab ¼1

2

›�ua

›xbþ

›�ub›xa

� �; ð15Þ

the turbulent kinetic energy k incorporated in the pressure term by definingPE ¼ P þ ð2=3Þrk. As there is no difference in the numerical procedures betweenPE and P, since physically k ! P, from here P will be used instead of PE.

Figure 1.Particles approximations

in support domain

W

i

j


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483

Then, by introducing equation (13) into momentum equation (4), yields a momentumequation as:

D �u

Dt¼ 2

1

r07P þ y e7

2 �uþ �gþ1

r 0�f s; ð16Þ

with:

k7 · ðy e7 · �uiÞl ¼j

Xmj

riy e;i þ rjy e;j

rirj

�rij ·7W ðjri 2 rjj; hÞ

r2ij þ h2

!�uij;

where y e;i ¼ y i þ yT;i .

2.3 Projection-based ISPHThe ISPH method based on the projection method is summarized here. First, particlesposition �rni in the previous time step is given. In the predictor step, an intermediatevelocity �u*i is calculated based on the momentum equation without the pressuregradient term as follows:

�u*i ¼ �uni þ y e72 �uni þ �gþ

1

r 0f�si

� �D t ð17Þ

The pressure field is obtained by solving the PPE with divergence of velocity in thesource term, as follows:

7 ·1

r07pnþ1

i

� �¼

1

Dt7 · �u*i ; ð18Þ

Next, in the corrector step, the velocity at time n þ 1, �u nþ1i is calculated as follows:

�unþ1i ¼ �u*i 2

Dt

r 07pnþ1

i ; ð19Þ

and the particle positions are updated at time n þ 1, �rnþ1i as follows:

�rnþ1i ¼ �rni þ Dt �unþ1

i ð20Þ

The modification in the PPE will be presented in Section 2.5.

2.4 Modelling of surface tension force for free surface flowsSurface tension plays a significant role in bubble dynamics and droplet spreading. Inthe current point of view for transport in the microscopic scale, molecules in the fluidare attracted by neighboring molecules. Inside the fluid, these attractive forces arecancelled out and molecules are in the balanced state, while on the free surface, a netforce in the direction of the surface normal to the fluid is exist.

In the CSF model (Brackbill et al., 1992), the interface curvature is calculatedthrough a colour function (Morris, 2000) as follows.

A colour function “c” is used to determine the surface location, and distinguishbetween fluids at immiscible fluids. The surface tension force is given by:

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�f s ¼ sks �n; ð21Þ

where s is the surface tension coefficient, ks is the curvature of the interface and �n is theunit normal to the interface.

The unit normal �n is obtained by the gradient of colour function as follows:

�n ¼7c

½c�; ð22Þ

where [c ] is the jump in c across the interface. The curvature of the surface can becalculated:

ks ¼ 27 · �n: ð23Þ

This approach gives an accurate estimation of the effects of surface tension but it isdifficult to handle large curvature cases correctly. This is due to the fact, that particleshave few neighbors in the free surface and moreover the second derivative of a colorfunction is sensitive to particle disorder.

In this paper, the other surface tension model with an interaction particle force,which was originally proposed by Nugent and Posch (2000) and improved byTartakovsky and Meakin (2005), is adapted. The surface tension force is given by:

�f s ¼r0

mi

�fi ð24Þ

where �fi is the total force due to interparticle interactions acting on any particle “i”,and this total force is calculated by the summation of an interaction particle forces as:

�fi ¼j

X�Fij; ð25Þ

and the interaction particle forces are given by:

�Fij ¼sijcos 1:5p

3h jrj 2 rij� � �ri2�rj

jrj2rij; jri 2 rjj # 3h;

0; jri 2 rjj . 3h;

8<: ð26Þ

where sij is the strength of the force acting between particles “i” and “j”. This force isrepulsive for distances less than h (jrj 2 rij # h), attractive for distances between hand 3h (h , jrj 2 rij # 3h) and zero for distances larger than 3h (jri 2 rjj . 3h).

In this study, the interaction particle forces are slightly modified by:

�F*ij ¼s*ij cos 1:5p

3h jrj 2 rij� � �ri2�rj

jrj2rijWij; jri 2 rjj # 3h;

0; jri 2 rjj . 3h;

8<: ð27Þ

Note here that, the nonzero range with 3h is dependent on the range of nonzero value inthe weight function defined in equation (8). In addition, strengths sij and s*ij may dependon the surrounding temperature. Both of original interaction force and the proposedforce define an appropriate direction of the surface tension force on the free surfacewithout special treatments. The proposed interaction function with normalization


force

485

F*ij=jF*maxij j and original function are plotted in Figure 2. It is clear that, the proposed

interaction particle force function is much smoother than its original function.

2.5 Stabilizations of pressure evaluation in PPEIn the general ISPH approach, it is difficult for numerical density ri to keep its initialvalue r 0:

ri ¼j

XmjW ðrij; hÞ ð28Þ

Hu and Adams (2007) pointed that, if only a divergence free velocity field is enforced,the numerical density variation, or particle clustering could happen due to thespatial truncation error; moreover, this density error could accumulates during thesimulation. Then, they improved ISPH by combining both a divergence free velocityfield and a density-invariant field (ISPH_DFDI) which includes internal iterations ateach time increment. In this context, Xu et al. (2009) proposed a particle shiftingtechnique (ISPH_DFS), in which the particle positions are shifted after solving pressurePoisson. Both techniques need some additional calculations to overcome the numericaldensity variation problem.

In this study, a modification is performed without additional calculations, only thesource term in PPE is slightly changed from the original ISPH as follows:

7 ·1

r07pnþ1

i

� �¼

1

Dt7 · �u*i þ a

r 0 2 r*ir 0Dt 2

; ð29Þ

Note that, the proposed scheme couples the divergence-free condition and a relaxeddensity-invariance condition. A special case, using a ¼ 0, leads to the originaldivergence-free scheme. The additional term with relaxation coefficient a has animportant role to keep uniform particle distribution and to conserve the total volume offluid during numerical analysis. The efficiency will be discussed in Section 3.3.3.

Figure 2.Particle-particleinteraction

0.0 0.5 1.0 1.5 2.0 2.5 3.0

–1.0

–0.5

0.0

0.5

1.0

Attractive

Repulsive

Nor

mal

ized

Inte

ract

ion

forc

es

|ri – rj|/h

Fij*

Fij

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2.6 Boundary conditionThe boundary condition on the rigid bodies has an important role to prevent penetrationand to reduce error related to truncation of the kernel function. In this work, two types ofboundary conditions, i.e. a rigid wall and a free surface condition are taken into account.

2.6.1 Wall boundaries. In the literature, there are several techniques for treatmentwall boundaries, for example, by using:

. mirror particles (Cummins and Rudman, 1999).

Takeda et al. (1994) and Morris et al. (1997) have introduced a special wall particlewhich can satisfy imposed boundary conditions. Recently, Bierbrauer et al. (2009)described a consistent treatment of boundary conditions, utilizing the momentumequation to obtain approximations to velocity of image particles:

. Repulsive force (Monaghan, 1994) is exerted in solid particles to prevent inner fluidparticles from penetrating the wall by employing an artificial repulsive force.Moreover, with repulsive forces, only one layer of particles are placed on the wall.

. Dummy particles (Shao and Lo, 2003) are regularly distributed at the initial stateas shown in Figure 3.

The dummy particles have zero velocity through the whole simulation. In this study,dummy particles technique has been performed for wall boundary, in which dummyparticles are regularly distributed at the initial state and have zero velocity through thewhole simulation process. Moreover, the PPE is solved for all particles including thesedummy particles to get an enough repulsive force preventing penetration.

2.6.2 Tracking free surface boundary. Detection of free surface has an important rolein the ISPH method for free surface flow, because the pressure values on free surfaceparticles should be equal to zero as the Dirichlet boundary condition. In this study,surface particles are simply judged by the total number of neighboring particles.

Figure 3.Wall boundary condition

Pressure

Wall particle

z3h

Inner particle Dummy particle


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Liu and Liu (2003) have investigated the number of neighboring particles to estimatean efficient variable smoothing length for the adaptive analysis. In the case of a simplycubic patterned lattice, h is usually chosen as larger than 1.2 times of the initial particledistance d0. They showed that the number of neighboring particle within the supportdomain khwith k ¼ 2 for cubic spline kernel function should be about 21 in 2D simulations.We checked a threshold for judging free surface particles for quintic spline kernel functionwith k ¼ 3, and the threshold should be about 27 in two dimension (Figure 4).

3. Results and discussionsIn the following section, the numerical examples have been introduced to validate thecurrent scheme. We demonstrate the capabilities of the current ISPH method forsimulating milk crown problem.

3.1 Deformation of cubic-droplet in free surfaceIn this test, the surface tension driven deformation of an initially cubic droplet hasbeen investigated. The cubic droplet in the free surface under null-gravity has beenchanged into the spherical shape as time goes. Note that, only the pressure gradientterm and surface tension force generate the driving force in this example. The effect ofsurface tension has been studied for one phase only without taking into account theeffect of surrounding air as a multiphase flows. Figure 5 shows the effect of surfacetension in the free surface cubic-droplet. The initial particle size is taken as d0 ¼ 0.01and the number of particles 97,336. It is clear that, the cubic shape changes intosmallest shape as spherical shape under the effect of surface tension force. Theperiodicity of the oscillation may depend on droplet radius, strength of surface tension,and other parameters used in the simulation.

3.2 Milk crownA milk crown is generated under certain parameters when a droplet falls on a thinliquid film as shown in the schematic diagram in Figure 6. The forming condition of

Figure 4.Free surface boundarycondition

3h

Inner fluid particle

Free surface particle

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the milk crown has been clarified experimentally (Gunji et al., 2003). The origin of themilk crown is immediately formed as soon as the droplet collides with the liquid film.

The stabilized ISPH with the surface tension force and an eddy viscosity canpresent the “crown shape” as shown in Figures 7, 9 and 10. Several resolutions modelhave been studied for this problem, d0 ¼ 0.01, 0.005 and 0.004, and Table I show thenumber of particles for each particle size. The initial schematic diagram for the currentmilk crown problem has been shown in Figure 6. The formation of the crown shape hasbeen introduced for the particle size d0 ¼ 0.004 cm at several time steps in Figure 7. It isobserved that, as the milk droplet collides the liquid film, leads to formation in milkcrown. The milk crown formation depends on the time step, surface tension, viscosityand speed of collision. The time step is very short and the surface tension has cleareffect and the strength of particle forces is about 1 £ 1026. It is seen that, as liquid dropcollides thin layer, the milk crown is formed immediately, and it appears clearly as timegoes. Figure 8 shows the shape of milk crown, which is taken by high speed camera inthe laboratory experiment for Gunji et al. (2003). Figure 8 shows the comparison

Figure 5.Deformation of free

surface cubic-dropletunder the effect of

surface tension

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 6.Initial schematic diagram

for milk crown problem

280 cm/s

0.5 cm

0.1 cm

2.4 cm

4 cm


force

489

Particle size (cm) Number of particles

0.01 336,3150.005 2,706,1210.004 5,071,910

Table I.The resolution of the milkcrown problem

Figure 7.Crown shape at particlesize d0 ¼ 0.004 cm

T = 0.0015 s T = 0.00175 s

T = 0.00225 s T = 0.0025 s

T = 0.00275 s T = 0.003 s

T = 0.00325 s T = 0.0035 s

T = 0.00375 s T = 0.00425 s

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between the milk crown shape for different particle sizes as d0 ¼ 0.004, 0.005 and 0.01.It is observed that, the milk crown shape is formed even if the resolution of model is nothigh in ISPH method. For the case of medium resolution model, particle size d0 ¼ 0.005,the effect of surface tension has been studied separately. In Figure 9, the milk crownshape has been drawn for two cases, in the first case, the effect of surface tension isincluded, while in the second case, the effect of surface tension has been cancelled. Thedifferent in milk crown shape for the two cases shows clearly contribution of surfacetension in milk crown formation (Figure 10).

3.3 Fluid-fluid interactionsIn this section, interactions between free falling fluid body and liquid in tank havebeen performed. The falling fluid object with density r2 moves down under the effect ofgravity, it collides with the fluid in tank which has density r1 ¼ 1:0 gm=cm3. After thecollision, splash is appearing and the two fluids are mixing together. The initialschematic diagram of the current simulation as Lee et al. (2010) is shown in Figure 11.In this figure, the width and height of falling fluid object is equal to L ¼ 20 cm, thewidth and height of the liquid in tank is 5 L and the distance between the fallingfluid object and liquid is equal to L. This interaction has been performed in several casesas follows.

3.3.1 Including/excluding eddy viscosity. To show the effect of eddy viscosity insplashes and deformations in free surface, which is induced by fluid-fluid interactions,the simulation is performed at the same densities for the two fluids as r1 ¼ r2 ¼1 gm=cm3 and the effect of surface tension force has been cancelled. In the first case,the eddy viscosity is included and the results have been introduced in Figure 11(a). In thesecond case, the eddy viscosity is cancelled and the viscous term contains kinematicviscosity only and the results have been introduced in Figure 11(b). It is clear that,there are differences between the two cases; the deformation in free surface issmaller in the case of included eddy viscosity compare to the second case and splashformation is much better in the case of included eddy viscosity. Finally, as eddy viscosityincluded, the deformation in free surface interface is reduced and splash is formedclearly.

3.3.2 Different densities ratios. In this simulation, the density of falling fluid body ischanging, while the density of fluid in tank is keeping as water density and the eddyviscosity effect is taken. In the first case, the density of falling fluid object r2 ¼0:5 gm=cm3 is smaller than the density of water in tank and density ratio becomer2=r1 ¼ 0:5: In this case, the falling fluid object collides the water in tank and it keepsabove the water until steady state as shown in Figure 12. It is clear that, the resultantsplash is small compare to the normal splash “density ratio r2=r1 ¼ 1” and the lowerfluid keeps above the heavier fluid without mixing.

Figure 8.Crown shape at time

T ¼ 0.003 s for particlesizes d0 ¼ 0.004, 0.005 and

0.01 cm, respectively

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In the second case, the density ratio between falling fluid object and liquid in tank isequals to r2=r1 ¼ 1:5. Figure 13 shows the interaction between the two fluids until itreaches to steady state, in this figure, the splash is higher than the normal case and alsothe mixing between the two fluids is rapidly occurs and the heavier fluid goes downuntil it reaches the bottom.

Figure 9.Milk crown shape fortwo cases of surfacetension effects

T = 0.00175 s

T = 0.00225 s

T = 0.00275 s

T = 0.00325 s

T = 0.0035 s

(b) Excluding surface tension(a) Including surface tension

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3.3.3 Keeping the total volume of fluid. In this section, the effect of relaxation coefficientfor keeping the total volume of fluid during the interactions between two fluids isdiscussed. Figure 14 shows the effect of relaxation coefficient on the height of free surfaceat different time instants T ¼ 2, 4, 8 and 10 s, respectively. The theory value of height isabout 104 cm after collisions. It is observed that, as relaxation coefficient decreases, theheight of free surface decreases, which leads to error in height compare to theory valueand the simulation cannot keeps the total volume of fluid. Note, the suitable relaxationcoefficient can keeps the total volume of fluid during the whole simulation with smallerror. From Figure 14, the range of relaxation coefficient is about 0.01 # a # 0.25, thisrange is judged from the error in height in the current simulation. The comparisonbetween the smallest and largest chosen values of the relaxation coefficient for differenttime instants is introduced in Table II. At relaxation coefficient equals to 0.0001, the errorof height is about 11.5 percent at time 2 s and the error increases as time increases,it reaches to 20.61 percent at time 10 s, while at relaxation coefficient equals to 0.25,the error of height is about 1.25 percent at time 2 s, and at time 4 s, the error of height isabout 6.67 percent, in this case, the error comes from increase the height than normal.Also, at time 8 s, the error of increased height is about 4.98 percent and at time 10 s, theerror of decreased height is about 1.75 percent. Then, the relaxation coefficient should bechosen so that the fluid volume does not increase nor decrease.

Figure 10.Schematic diagram offluid-fluid interaction

problem

L

L

5L

33

70

106

143

180

–4–4 18 39 61 82 104

L


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Figure 11.Fluid-fluid interaction

(a)T = 0.25 s T = 0.3 s T = 0.35 s T = 1 s T = 2 s

(b)

T = 0.25 s T = 0.3 s T = 0.35 s T = 1 s T = 2 s

Notes: (a) Including eddy viscosity (b) excluding eddy viscosity

Figure 12.The fluid-fluidinteraction with densityratio r2=r1 ¼ 0:5

T = 0.25 s T = 0.3 s T = 0.35 s T = 0.5 s T = 0.8 s

T = 1s T = 2s T = 4 s T = 6 s T = 8 s

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Finally, the different cases of previous interactions and keeping the total volume offluid in the interactions until reach to steady state show clearly the effective ofproposed model.

4. ConclusionAn incompressible smoothed particle hydrodynamics is developed to simulate milkcrown problem using a surface tension model and the eddy viscosity. In addition, thesource term in the PPE is slightly modified to generate a smoothed pressuredistribution and to keep the total volume of fluid. For the modeling of surface tensionforce on the free surface, the interaction particle force has been reformulated with asmoothed function. This surface tension model was validated by simulating the cubicdroplet under null-gravity and the milk crown problem. From our numerical tests, thesurface tension model can handle free surface tension problems including highcurvature without special treatments. Finally, a fluid-fluid interaction problem wassimulated to discuss the effect of the eddy viscosity with Smagorinsky model. Theeddy viscosity has clear effects in adjusting the splashes and reduces the deformationof free surface in the interaction. Finally, the proposed stabilization appeared inthe source term of PPE has an important role in our simulation to keep the total volumeof fluid.

In the futures work, a calibration procedure of the strength in the interaction particleforce should be developed, and it may need to discuss a dependency on thesurrounding temperature.

Figure 13.The fluid-fluid interaction

with density ratior2/r1 ¼ 1.5

T = 0.25 s T = 0.3 s T = 0.35 s T = 0.5 s T = 0.8 s

T = 1 s T = 2 s T = 4 s T = 6 s T = 8 s


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Time (s) Relaxation coefficient Average height Error in height (%)

2 a ¼ 0.0001 92.04 11.5a ¼ 0.25 102.69 1.25

4 a ¼ 0.0001 87.64 15.73a ¼ 0.25 110.92 6.67

8 a ¼ 0.001 83.03 20.16a ¼ 0.25 109.14 4.98

10 a ¼ 0.0001 82.55 20.61a ¼ 0.25 102.17 1.75

Table II.Comparison between twovalues of relaxationcoefficient at differenttime instants

Figure 14.The effect of relaxationcoefficient on the height offree surface at differenttime instants T ¼ 2, 4, 8and 10 s, respectively

0 20 40 60 80 100707580859095

100105110115

Hei

ght o

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Further reading

Katsuya, N., Koshizuka, S., Oka, Y. and Obata, H. (2001), “Numerical analysis of droplet breakupbehavior using particle method”, Journal of Nuclear Science and Technology, Vol. 38,p. 1057.

Corresponding authorMitsuteru Asai can be contacted at: [email protected]

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