Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 10.4.3 - A SPH-ALE method to model multiphase flow with surface tension
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 Material Information
Title: 10.4.3 - A SPH-ALE method to model multiphase flow with surface tension Computational Techniques for Multiphase Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Leduc, J.
Leboeuf, F.
Lance, M.
Parkinson, E.
Marongiu, J.-C.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: SPH-ALE simulations
mesh-less methods
multiphase flows
Riemann solvers
preconditioning
 Notes
Abstract: This work focuses on a new multiphase method for Smooth Particles Hydrodynamics (SPH) simulations which uses many advantages of the arbitrary Lagrange-Euler (ALE) formalism. The use of an acoustic Riemann solver in the SPH-ALE method allows sharp interfaces between fluids and is fully compliant with a purely lagrangian description of the interface. The mass fluxes between volumes of control associated with different fluids are controlled by the interfacial velocity of the Riemann problem which allows to cancel these fluxes. One of the major advantages of the proposed formulation is that it remains stable for very high density ratios (air/water density ratio=1000) without adopting unphysical speeds of sound in both media. In the limit of low Mach number flows, preconditioned Riemann solvers are used to limit the numerical diffusion linked with upwind schemes. Surface tension effects are also included in this model. The proximity of the SPH-ALE formalism with the Finite Volumes method allows an adaptation of the Continuum Surface Force (CSF) formalism. An other model of surface tension is presented. This Local Laplace Pressure Correction model (LLPC) integrates the Laplace law in the Riemann solver between volumes of control associated with different fluids. The acoustic solver and the momentum equations are then modified to take into account the surface tension. The LLPC model recovers sharp variation of pressure at the interface and decreases the intensity of parasitic currents. This approach is successfully validated on well documented academic cases: gravity waves and static/oscillating droplet.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Bibliographic ID: UF00102023
Volume ID: VID00257
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: 1043-Leduc-ICMF2010.pdf

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7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


A SPH-ALE method to model multiphase flows with surface tension


J. Leduc*, F. Leboeuf*, Michel Lance*,

E. Parkinsont and J.-C. Marongiut
Laboratoire de Mecanique des Fluides et d'Acoustique, Ecole Centrale de Lyon,
University of Lyon, Ecully, FRANCE
t ANDRITZ Hydro, Vevey, SWITZERLAND
julien.leduc @ec-lyon.fr and jean-christophe.marongiu @andritz-hydro.com
Keywords: SPH-ALE simulations, mesh-less methods, multiphase flows, Riemann solvers, preconditioning




Abstract

This work focuses on a new multiphase method for Smooth Particles Hydrodynamics (SPH) simulations which uses
many advantages of the arbitrary Lagrange-Euler (ALE) formalism. The use of an acoustic Riemann solver in the
SPH-ALE method allows sharp interfaces between fluids and is fully compliant with a purely lagrangian description
of the interface. The mass fluxes between volumes of control associated with different fluids are controlled by the
interfacial velocity of the Riemann problem which allows to cancel these fluxes. One of the major advantages of
the proposed formulation is that it remains stable for very high density ratios (air/water density ratio=1000) without
adopting unphysical speeds of sound in both media. In the limit of low Mach number flows, preconditioned Riemann
solvers are used to limit the numerical diffusion linked with upwind schemes.
Surface tension effects are also included in this model. The proximity of the SPH-ALE formalism with the Finite
Volumes method allows an adaptation of the Continuum Surface Force (CSF) formalism. An other model of surface
tension is presented. This Local Laplace Pressure Correction model (LLPC) integrates the Laplace law in the Riemann
solver between volumes of control associated with different fluids. The acoustic solver and the momentum equations
are then modified to take into account the surface tension. The LLPC model recovers sharp variation of pressure at
the interface and decreases the intensity of parasitic currents.
This approach is successfully validated on well documented academic cases: gravity waves and static/oscillating
droplet.


Introduction


Roman symbols
g gravitational constant (m.s 2)
p pressure (N.m 2)
Greek symbols
p density (kg.m 3)
(a surface tension coefficient (N. m 1)
Subscripts
1 left particule
r right particle


High speed water flows as jets from Pelton hydraulic
turbines nozzles, can lead to important interface defor-
mations without phase changes. These kinds of de-
formations are really hard to catch with eulerian nu-
merical methods since the diffusion of the interface
smoothes the sharpness of the physical interface pertur-
bations. On an other hand mesh-less/lagrangian meth-
ods as SPH offers the possibility to naturally simu-
late flows with large deformations. Furthermore pre-
vious work on SPH-ALE formalism brought improve-
ment in term of precision, stability of single phase
flows (Marongiu 2008). Different approaches were de-
veloped to simulate multiphase flows with SPH method
(Grenier 2008), (Colagrossi 2003) or (Hu 2006). Gre-
nier et al. (Grenier 2008) developed a method based on


Nomenclature







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


the SPH-ALE formalism using a volume fraction ap-
proach. The goal of the present study is to develop a
model which can simulate multifluid flows with surface
tension effects using an SPH-ALE formalism without
volume fraction. The advantage of such an approach is
that control volumes always belong to one single phase,
which allows a sharp transition from one fluid to another
and a lagrangian tracking of interfaces.

The SPH-meshless method

The Smoothed Particle Hydrodynamics (SPH) is a
meshless method based on an interpolation technique.
A continuous field f is approximated from discrete data
points through the formula:

(f(x)) = w fj W(xz x, h) (1)
J D,.
W is a regular function (usually called kernel function)
different from 0 on its support D, h is the smoothing
length (linked with the size of Dx) and wj is the weight
of the particle j.


W


Figure 1: SPH approximation


One of the major advantage of this formalism is that
it allows to compute gradient of continuous field using
the gradient of the kernel function with a simple formula
(here written far from the domain boundary):

(V f(x)) -= w fjVW(xj x, h) (2)

The traditional SPH formalism is a purely lagrangian
formalism and suffers usually from instability and lack
of precision. In this paper the numerical method is based
on an Arbitrary Lagrange Euler formalism in the SPH
framework (1) and (2) (Vila 1999)

The SPH-ALE formalism

This formalism considers conservative form of Euler
equations in a moving frame of reference at the veloc-


ity vo.

d j dQ + J j (v vo).ndS=
dt 12 S
0 a s (3)

j Qs.ndS + QvdQd

where 2 is the volume of control, S its boundary, vector of conservative variables and Qs and Qv the sur-
face and volume source terms. If surface terms are re-
stricted to pressure term, the equation (3) can be rewrit-
ten:
LO () + div (FE() vo ) Q (4)
where LOo is the transport operator associated to vo and
FE is the flux vector of the Euler equations.
From this point Vila observed that the discretization of
the conservative formulation leads to the appearance of
one dimensional Riemann problems between each pair
of control volumes (Vila 1999). If we consider two con-
trol volumes i and j, their interaction results in contribu-
tions along the direction joining i and j, with a discon-
tinuous state evolution at mid-point. It can be expressed
as:

at ) x (FE 4 D t), .)

j( (X ), 0) (Ii if x( ) < 0
^j if z("ij) > 0
(5)
where is the unit vector between i and j (from i to j),
xij is the mid-point between i and j, x"3 is the curvi-
linear abscissa along the straight line between i and j,
whose origin is taken at xij and <( and of conservative variables at i and j respectively. Vila
showed that the solution can be expressed as:
z(xci) + Xo(t), 4)4))

S t (6)
Xo (t) .,,-, ,.
I Jo
where steady Riemann problem for Euler equations expressed
in a fixed frame of reference, and vo(xij, t) is the veloc-
ity of the interface between points i and j.
Vila showed that the final set of discrete equations in
the SPH-ALE formalism is:
d
t (Xi) = vo(X, t)
d
-(Wi) = oi Yj (vo (xj) vo(xi))ViWij

d (wiPi) + wi p wj2pEij(VEij vo(Xi, t)).ViWi, =
jdtD

S(ipivi) + i 3 uj2[pE,ijVE,ij (E,i Vo(xj, t))+
dtjEDi
PE,ij]-ViWij = Uwipig
(7)







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


where (PE,ij, VE,ij)t = ij (A) is the upwind solution
of the moving Riemann problem.
This final formalism has different interesting proper-
ties: first it allows an upwind solution for the veloc-
ity which stabilizes the model, mass fluxes exist be-
tween control volumes(which are no more "particles")
and third, one equation is added in comparison with tra-
ditional SPH which takes into account changes of vol-
ume control distributions. Finally this formalism is re-
ally close to the Finite Volume formalism. More de-
tails on this formalism are presents in the work of Vila
(Vila 1999) and Marongiu et al. (Marongiu 2008).

The multifluid model

General solution to the Riemann problem

The resolution of the system (7) needs the solution of
the nonlinear Riemann problem of the form (5). In this
work the behaviour of the fluid is governed by the Tait
equation of state:


and the local speed of sound c is given by:
aPO
and the local speed of sound c is given by:


S (p + B)


Pressure and density derivatives are then linked by
speed of sound as:


Op

O p
at


adp
c20
C:c


SPI*

)v(1)
(2)
(2) *\


P,
vr(*
(2)
V,


Figure 2: Riemann problem scheme



if pi < pr, the Riemann problem will be structured by
a shock wave between the state I and I*, a rarefaction
wave between the state r and r*, and a contact wave
between 1* and r*. Considering low Mach number
flows and considering that the transport field is chosen
so that 0 <|| vo II problem has to be found in the star region. It means
that the vector (,., 'E,j)t of the equation (7) is
equal to (p*,v() '-' )t (p* is computed with the
Tait equation from p*, and v(2)* = v () if x/t < v1)*,
otherwise v(2)* = v2 ).

Linearized approximate Riemann solver for multi-
phase simulations

Consider here that the contact discontinuity is the in-
terface between two fluids, to determine the solution of
the Riemann problem, the method described by Murrone
is applied (Murrone 2008). It consist in a linearization
of the problem in each side of the contact discontinuity:


1 3.(q* q,) =0
1(q ) 0
It,.(q* o,


It results the following Riemann problem written fo
variable U (p, v(1), v(2))

BU BU
au + A(u) =S
at A(U)a O


The left eigenvectors of the matrix
(-1/pc, 1, )t,12 (0,0, 1)t and 1 -
(11) From (12), we gets two equations (13):


A are 11
S(1/pc1, .)t


where v(1) is the velocity normal to the interface
between i and j, v(2) is the tangential velocity and
V () pc2 0
A = 1/p v(1) 0 is the jacobian Matrix of
0 0 v(1)
the system. The eigenvalues of the jacobian matrix A
are: A1 v(1) c, A2 v() and A3 v(1) + c.
Since we are considering low Mach numbers in our
applications, A, represents a non linear wave moving
to the left and A3 a non linear wave moving to the
right. Fig. 2 shows the structure of the solution of the
Riemann problem using the Tait equation. It means that


1(v )) V (p+ --)
prCr(V)* V(1)) ( Pr)


Since v(1) and the pressure are continuous at the in-
terface between two fluids (if surface tension effects are
neglected), one gets:


{ (i)* V 1)* -v*
p**r
P1 Pr P =












Then the state star (*) is computed as:


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010

1mM


P1C1 + PrCr
CPyCl P(v(1)-
pl cl + Pr cr
Plcl + prce


V(1))
v^ )


(15)
The tangential velocities v(2) are conserved through
the left and right wave and discontinuous through the
contact wave.

Solution of the multiphase Riemann problem

The ALE property is used to impose the velocity of
the interface vo equal to the resulting velocity from the
Riemann solver v* if the left and the right states are as-
sociated with two different fluids. In this case the term
VE,ij v(xij, t) in the equation (7) is identically null
and the mass exchanges between the volumes of control
are blocked. From this point since the mass fluxes are
blocked at the interface by imposing the interface veloc-
ity between two volumes of control, we then assume im-
plicitly that the interface is described with a lagrangian
form of the ALE formalism. This solver has the advan-
tage to be based on velocity and pressure which are con-
tinuous at the interface without surface tension. The re-
sulting density is computed from the pressure using the
Tait equation. The first terms of the equations (15) show
that the resulting velocity is balanced upwind on the side
of the heaviest fluid while the pressure is weighted on the
side of the lightest fluid, as expected across an interface
between a liquid and a gas for instance. In more prac-
tical terms the heavy fluid (water) imposes the velocity
and the light fluid (air) imposes the pressure.
A special treatment is also applied to the equation for
the weight wc (7) when the two states correspond to dif-
ferent fluids. This equation is in this case:


d (W'i)
dt


Ui wj2(vE,ij o(xi))ViW, (16)
j D,


The weight variations are no more supported sym-
metrically by the two fluids. For an heavy fluid
.,, .) vo (xi) is nearly equal to 0 which implies that
the light fluid will support all the weight variations.

Gravity wave test case

The numerical developments are evaluated on the
gravity wave test case. The geometry is described on
figure 3 and the reference data are given in table 1. Wall
on the top and bottom of the domain and periodicity on
the side of the domain are used as boundary conditions.
Three different discretization are used to perform the
simulation: 60*60, 80*80, and 120*120.


Figure 3: Geometry for gravity waves test case


0.45
0.4
0.35
E 0.3
~0.25
0.2
a 0.15
0.1
0.05


0 0.5 1 1.5 2 2.5
time [s]

Figure 4: Evolution of the kinetic energy depending on
discretization


The error on frequencies are 2.66%, 0.791% and
0.50% for the different discretizations (Fig 5, Ax[m]
1/60, 1/80 and 1/120 respectively) These differ-
ences are of the same order of magnitude in com-
parison with the results obtained with eulerian codes
(Bonometti 2005). An important numerical diffusion
is responsible for the decay of the kinetic energy (it is
recalled that no physical viscosity is modelled in the
present study). This phenomenon is sensitive to the nu-
merical discretization (see figure 4).


Preconditioning of Riemann Problems

The previous results show a satisfactory prediction of
the wave frequency but also an important numerical dif-
fusion. Preconditioning techniques are widely used in
Finite Volumes methods to decrease this excessive nu-


(1) (1)
pc* i + PrcrV
PiCl + PrCr
* PiClPr + Pr CrPtI
SPiCl + PTrCr












g,
.....










S 0.01

Figure 5: Frequency error depending on discretization


Table 1: Configuration gravity waves


domain length L [m] 1
initial deformation a/L [-] 0.01
density fluid 1 [kg.m 3] 1000
density fluid 2 [kg.m 3] 1
sound speed 1 [m.s 1] 5
sound speed 2 [m.s ] 15
surface tension [N.m 1] 0
gravity [m.s 2] 9.81


medical diffusion obtained with Godunov type schemes
for low Mach number flows, and were introduced by
Murrone et al. (Murrone 2008) for multiphase flows.
The linear Riemann solver in (11) can thus be modified
by multiplying the Jacobian matrix A associated to the
Euler equations by the preconditionner of Turkel:

32 0 0
P 0 1 0 (17)
0 0 1

where 3 is of the order of magnitude of the Mach
number. For the following results, preconditioning
is applied for both single and multi-phase interac-
tions: interactions between two particles of the same
fluid are treated by a VFRoe-Turkel Riemann solver
(Marongiu 2009) and those involving two particles of
different fluids by a preconditioned acoustic Riemann
solver.
The results on figure 6 show a positive effect of pre-
conditioning on the numerical diffusion. This effect is
smaller for fine discretization. The preconditioning has
also a positive effect on frequency determination since


UV.4

d 0.3

0.2

0.1

0


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Non preconditioned 80*80 -
Preconditioned 80*80 ------
Non preconditioned 120*120 ..........
Preconditioned 120*120
A




; l\ \*



0 0.5 1 1.5 2 2.5
time [s]


Figure 6: Comparison between nonpreconditioned and
preconditioned schemes


the errors are 0.61% for the 80*80 discretization (instead
of 0.79% without preconditioning) and 0.47% for the
120*120 discretization (instead of 0.50% without pre-
conditioning).


Surface tension models

Surface tension effects can lead to different multiphase
phenomena such as coalescence or capillary waves.
This topic was considered in various studies with SPH
methods (Hu 2006). Two models are studied here,
one based on the Continuum Surface Force (CSF,
(Brackbill 1992)) method and one based on a novel
approach related to SPH-ALE formalism and the use of
Riemann solvers, the Local Laplace Pressure Correction
(LLPC).

Continuum surface force

This classical model arises from the work of
(Brackbill 1992) who rewrites this surface effect as a
volume force. A mathematical development allows to
rewrite this volume force as


(c Vc
F8 V.1


VcK1)


where c is the smoothed color function (c=0 or 1 far from
the interface and varies continuously near the interface).
The previous equation (18) considers that the force is
symmetrically applied to the two fluids. As it was done
by Brackbill (Brackbill 1992) the previous force will be
balanced between the two fluids when there is an impor-
tant density ratio between the fluids. For air-water inter-
face, it will model the fact that surface tension is mainly
arising from interactions between water particles.












To keep the conservation property of the previous
model, no fluxes of surface tension are exchanged be-
tween volumes of control from different fluids and a
constant coefficient (2 P1 ) is added based on the
PD1 +PD2
reference density of each fluid. The final expression for
the surface tension effect on particle i in this formalism
and for a high density ratio is given by:

Fti = 2 PD1 : Wj (Fj Fi) VWi (19)
PD, +PD2 cD1

where D1 is the group of particles from the same fluid
as the particle i, pD, is the reference density of the par-
ticle i and j (which are from the same fluid) and

Vci < Vci
F, v |Vc|I (20)


Since the coefficient PD1 is constant, this
PD1 PD2
formalism is conservative. For low density ratios the
computation of surface tension effects is based on the
first formalism of the volumic force (18). The limit of
the density ratio to use the equation (18) or the equation
(19) is around ten but nor clearly defined since the use
of the formalism (18) allows to use the fluxes on each
side of the interface to compute the force on a particular
control volume. On the contrary the equation (19) uses
the fluxes between control volume from the same fluids.

Local Laplace pressure correction LLPC

The previous surface tension model applied to the
acoustic multiphase model has the major drawback that
a volume force representing a surface phenomenon is
applied to a sharp density interface. Indeed Brackbill
(Brackbill 1992) developed the original model on a
smoothed density transition through the interface. In
order to take benefit from the sharp interface obtained
with the lagrangian feature of SPH, a novel surface
tension model is developed which can reproduce the
sharp pressure jump predicted by Laplace law. In the
presence of surface tension, the difference of attraction
between particles is creating a resulting pressure jump
at the interface. Between two control volumes from
different fluids, the pressure continuity written in (14)
is then no more valid to describe this effect. As a con-
sequence the pressure jump is directly included in the
acoustic Riemann solver between two control volumes
corresponding to 2 different fluids (Perigaud 2005):

S l)* (1)* *(21)
r (21)

where K, is the local surface curvature from the point of
view of I (Kr = -Kl) and a the surface tension coeffi-


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


cient. The second equation of (21) can be written as:


S Pi Pr
Pi - cTl = p + --rKl
Pi + Pr Pi + Pr


We can introduce a pressure p*:

Pl *
P = pi p- acl = Pr
P1 + Pr


Pr
P1 + Pr


c(Kr (23)


Introducing this last equation for the resolution of the
linearized Riemann problem, one gets:


(1) (1) -
P* 1 ;c1 + Crcr ) (Pr -Prst) (Pi
PlCl + PrCr PlCl + Prcr
* PlCl(pr -Prst) + PrCr(Pl Pst)
PiCl + PrCr
Plce PrCr
PlClprCr(V +) -V 1))
plCl + PrCr


with:


Pist

Prst


P1
P1 + Pr
Pr
P- + P Kr
P1 + Pr


Pist)







(24)



(25)


The equations (25) are balancing the effect of surface
tension from one side to another depending on the den-
sity ratio. Indeed if we are considering a static liquid
droplet at rest in a stagnant gaz, the Laplace law predicts
a pressure jump /' ) through the interface. Figure
7 shows that the pressure terms in the Riemann solver
(24) are canceling: There are no flux momentum created
through the Riemann solver by the pressure jump for a
static droplet.


pressure


Kt < 0
P, = 1

Plst = 1g Kl




Pt Pist -
Pl--
Pg


r > 0
i- 9


aK


Figure 7: Example of pressure treatment for LLPC
model (see eq. 24)

In this model the fluxes linked with surface tension
are blocked at the interface (as for the CSF model for
large density ratios). As figure 7 shows, effects of sur-
face tension are removed from pi and Pr before comput-
ing the pressure through the Riemann solver (24). The







7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


Table 2: Configurations static droplet-bubble simula-
tions


droplet radius [mm] 10
surface tension coeff. [Nm 1] 0.072
density Fluid 1 [kg.m 3] 1000
density Fluid 2 [kg.m 3] 1


pressure components arising from surface tension terms
have to be added in the momentum equation when the
two volumes of control are from different fluids. The
momentum equation is then:

d
dt (wpivi)
wi wjuj2[PE,ijvE,ij ) (vE,ij -o0) +PE,ij + Pijst] ViWj
3
= wipi9
(26)
where/ A(i) '- TKi and A(ij) 1 if i and j
are from different fluids, otherwise and A(ij) 0. Since
we are computing low Mach number test cases, we get
that the solution of the Riemann problem (i, i, VE,ij)
is equal to i / v*) computed for the interaction between
i and j. It has to be noted that this model is close from
the one developed by Francois et al. (Francois 2006)

Validation

Two test cases were chosen in order to validate the
different models: a static droplet surrounded by air and
a droplet with an initial deformation. The first test case
is useful to test the ability of the model to reproduce the
Laplace Law without developing high spurious currents.
The second test case allows to compare oscillation
frequency with the value predicted by the theory in the
case of small deformations.

Static droplet
As it could be expected, the pressure jump is more accu-
rately reproduced with the LLPC method than with the
CSF method. On figure 9 the pressure jump occurs di-
rectly from a particle of air to a particle of water. The
pressure jump is much smoother with the CSF method
since three layers of volumes of control are needed to
perform the interface transition (Figure 8).
Parasitic currents are also reduced by the use of
LLPC model in comparison with the CSF model. Figure
10 show the evolution of parasitic currents over a period
(0.5s) that can be considered as long with respect to a
typical time scale for flows in Pelton turbines (3.10 3s).


Pressure(x,y)[Pa]


Figure 8: Pressure field for CSF model

Pressure(x,y)[Pa]

4*"^fti'tm" *


Figure 9: Pressure field for LLPC model



Oscillating droplet
The test case of a free oscillating droplet is performed
with the LLPC model. At t=0s, an initially deformed
droplet begins to oscillate under the effect of surface ten-
sion. The initial deformation is around 1% (27) (differ-
ence rate between longer and smaller diameters).


r =o(l+ 1+ (3cos(20) +1)
4


The evolution of kinetic energy is shown on figure 11,
for preconditioned Riemann solvers. The theoretical pe-
riod is 2.86810-1s. This model gives smaller periods
(higher frequencies) than expected by the linear theory.
The errors are 2.82% for the discretization (correspond-
ing to 10 volumes of control in the diameter) and 0.674%
with 20 control volume along the diameter.

Conclusions

The previous study demonstrates the interest of using the
SPH-ALE formalism with an acoustic Riemann solver
to simulate multiphase flows. It gives a model with
a sharp density interface, stable for high density ratios
and which respects a physical ratio of sound speed be-
tween the fluids. The Local Laplace Pressure Correction











le-08


le-09


Sle-10

S le-ll


le-12

le-13


0 0.1 0.2 0.3 0.4 0.5
time [s]


Figure 10: Evolution of parasitic currents


3.5e-08

3e-08

2.5e-08

2e-08

1.5e-08
le-08


5e-09 KV I '

0 0.1 0.2 0.3 0.4 0.5
time [s]


0.6 0.7


Figure 11: Oscillating Droplet: kinetic Energy evolu-
tion



(LLPC) model for surface tension gives a sharp pres-
sure variation at the interface when CSF formalism re-
quires 3 volumes of control to establish the good level of
pressure inside a droplet. The parasitic currents are also
strongly reduced with the LLPC model. The use of pre-
conditioned Riemann solvers enable a reduction of the
numerical diffusion. This methods will now be used on
more complicated academic test cases and also on three
dimensional industrial cases.


Acknowledgements


The authors wish to thank the ANRT (association na-
tionale recherche technologies these 407/2007 for help-
ing to fund this work.


L_ .---" CSF 10
CSF 20
LLPC 10
LLPC 20


7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 -June 4, 2010


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