7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
Multiscale Simulations of Gas Assisted Liquid Jet Break up
Gaurav Tomar1'2, Daniel Fuster1'2, Stephane Zaleski1'2* and Stephane Popinet3
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F75005, Paris, France.
2 CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F75005 Paris, France.
3 National Institute of Water and Atmosphere research, P.O. Box 14901, Kilbirnie, Wellington, New Zealand.
Affiliation University, Faculty, Department, Institute
Address, City, Postal Code, Country
Email: stephane.zaleski @upmc.fr
Keywords: Multiscale, Atomization, Volume of Fluid
Abstract
The process of atomization involves break up of a liquid jet into small droplets. Atomization of liquid jet, in certain applications,
is assisted by a high speed gas. The efficiency of atomization can be measured based on the droplet distribution in the atomizer
and their size distribution. The size of the smallest droplets is much smaller compared to the liquid jet diameter. Therefore,
several orders of spatial scales need to be resolved to accurately capture the physics in numerical simulations. Resolving the
smallest droplets requires enormous computational resources and time. We propose here to perform numerical simulations of
primary atomization by modelling the smallest droplets as Lagrangian point particles. The liquid jet is simulated using a Volume
of Fluid (VOF) method coupled with a Lagrangian particle tracking for droplet motion. Motion of point particles is governed by
fluid forces such as lift, drag, Faxen, added mass and inertial forces. The interaction of the gas flow with the particles is
accounted for by introducing a momentum source term in the NavierStokes equations. The transformation of a VOF resolved
droplet into a Lagrangian particle is based on a threshold volume criterion. The contiguous lumps of the liquid are identified and
tagged for transformation into Lagrangian particles and the algorithm is fully parallelized to account for correct identification of
droplets in all the processors and also prescribing unique ids to the appearing droplets. We present numerical simulations of a
slow moving liquid jet destabilized by a gas flow. Nozzle is modeled as a thin converging plate over which liquid and gas
interact. The probability density function of the droplet size distribution in different domains suggests different mechanisms
operating at different scales. Droplet size distribution in a zone close to the nozzle has a peak at smaller diameters compared to
distribution taken further downstream of the jet entry. We suggest that the droplet size distribution is altered by an interaction
between the droplets generated during primary atomization and the liquid ligaments downstream.
Introduction
In various applications of industrial and scientific relevance
the physical phenomena occur with interaction between
temporal and spatial scales separated by orders of
magnitude. Widely separated scales are seen in situations
like micro fluids, particleladen flows, sprays, atmospheric
flows and combustion chambers (Kusano (2007) and
Kolobov (2007)). It is practically impossible to implement a
single numerical scheme that resolves the physics spreading
over multitude of temporal and spatial scales. Therefore,
multiscale simulations involve an implementation of a
hybrid scheme stitching together different models in an
algorithm allowing exchange of relevant information for
model closures.
A liquid jet breaks up into small droplets due to an interplay
of different mechanisms active during the atomization
process. The droplets so formed are orders of magnitude
smaller than the diameter of the liquid jet. Numerical
simulations of the atomization process have been performed
(Keller (1994), Tauber (2000) and Kim (2006))
incorporating the complete interfacial dynamics using
sharpinterface tracking algorithms (Brackbill (1992),
Univerdi (1992), Sussman (1994), Zaleski (1995), Popinet
(1999), Pilliod (2004), Renardy (2002)). Three dimensional
temporal simulations of the breakup of a liquid jet by a
coaxial highspeed gas have been reported in Keller (1994),
Zaleski (2003) and Fuster et al.(2009) using the
VolumeofFluid (VOF) method and by Tauber et al. (2000)
using the FrontTracking method. These studies showed the
formation of thin ligaments from the jet periphery which
subsequently breaks up into small droplets. The
computational cost increases substantially on the formation
of droplets (wherever adaptive grid resolution in used for
capturing the smallest features) and simulations can be
continued only by artificially removing the smallest droplets
from regions of the computational domain a little distance
away from the liquid jet. An unaccounted removal of the
droplets effects the measurements of probability density
function (PDF) of droplet sizes. Also, this leads to loss of
physics since the interaction of droplets with the
downstream jet is not being modeled. Therefore, a
multiscale numerical algorithm is required to incorporate
the essential features of the physics due to the smallest
droplets. Recently, Kim et al. (2006) and Hermann (2010)
presented multiscale simulations of break up of a liquid jet
by a coaxial gas jet and a crossstream flow, respectively. A
variant of the LevelSet method was employed to solve the
twophase flow coupled with a Lagrangian spray model to
track the droplets brokenoff from the liquid jet. However,
the splashing of the droplets on the primary jet was not
modeled.
In the present study, we show results of multiscale
simulations of a primary jet breakup using an
EulerianLagrangian approach incorporating the effect of
particles on the coaxial gas jet by performing twoway
coupling, that is, bothway interaction between fluid and
particles. The multiscale algorithm has been implemented in
Gerris (Popinet (2003)), a twophase VOF solver with
balanced force surface tension model and quad/octree
adaptive mesh refinement. The momentum source term in
the NavierStokes equation because of the particles is
smoothed using a Gaussian distribution function. The
algorithm has been validated against various test cases. We
show simulations of a breakup of a liquid jet converting
smallest droplets into particles upon formation and back into
VOF resolved droplets based on their proximity to the VOF
interface. We present a PDF of the droplet sizes during the
atomization process. We show that the interaction of the
droplets formed during primary atomization interact with
the downstream jet and modify the size distribution towards
large droplet sizes.
Nomenclature
c Void fraction field
f subscript for fluid computational cells
F Force on the particle
m Mass of the particle (kg)
n normal at the interface
p pressure (Nm 2)
t time (s)
u Velocity
U Inlet velocity
v Particle velocity
V Volume of the particle
x Particle position
Greek letters
() Twoway coupling particle force
K Curvature
p Density (kg/m3)
U Dynamic viscosity (Pas)
1 Kinematic viscosity (m2 s)
o Surface tension coefficient (N/m)
up Smoothening length scale for particle momentum
Subsripts
g Subscript for gas properties
i Superscript for particle index
1 subscript for liquid properties
Mathematical Formulation
The NavierStokes equations for a twophase
incompressible flow modified to implicitly incorporate the
boundary conditions at the interface, namely shear stress
continuity, normal stress balance including surface tension
force and velocity continuity can be written as,
p[atu + (uV)u] = Vp + V(2pD)+ alKSn + 4p.
(1)
Here = (u,v,w) is the fluid velocity, p(x,t) is the fluid
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
density, u(x,t) is the dynamic viscosity and D is the
deformation tensor defined as D= ((Vu)+(Vu) )/2 .
Dirac delta function, 8., signifies that the surface tension
force is nonzero only at the interface, where a, n and
K representing the surface tension coefficient, the unit
normal and the curvature at the interface, respectively. The
momentum source term p represents the effect of the
dispersed phase simulated using a Lagrangian approach.
The advection equation for density and the
incompressibility condition are given by,
3tp+V(pu)= 0, (2)
V.u = 0. (3)
The density and the viscosity field are obtained as,
p cp, + (1 )pg and (4)
c/u, + (1 c)/g (5)
respectively, where c(x,t) is the volume fraction. Here,
pl, pg and l p g are the densities and viscosities of
the liquid and gas fluids, respectively. The volume fraction
takes values between zero and one. The advection equation
for the density can be written in terms of the volume
fraction as,
tc +V.(cu)= 0. (6)
A twoway coupled EulerianLagrangian approach for the
dispersed phase comprises of computing the external and
fluid forces on the particle and incorporating the effect of
the particles as a source term in the NavierStokes equation
( p). The governing equations of motion of the particles
are given by,
dx.
dx vi (7)
dt
mi = FD +F, + F +F F,+F. (8)
dt
where, mi, xi andvi are the mass, position and the
velocity of the ith particle respectively. The density and
volume of the particle are denoted by pp' and V,
respectively. The different forces FD,FI,FA,FL, Ft,
acting on the particle are the drag, inertial, added mass force,
lift and external forces, respectively (see Refs Magnaudet
and Eames (2000) and Climent (2006) for details).
The momentum source term (p in the Eq.1 is given by,
= lim p g +
S ,i Vi i(g dv'j (Du '2
SVJO i=1V f dt Dt
(9)
where Vf is the control volume containing Np particles
(Climent (2006)).
The methodology adopted for solving the twophase,
sharpinterface, incompressible flow equations is presented
in detail in Popinet (2003), Popinet (2009) and Fuster et al.
(2009). A secondorder accurate discretization staggered in
time has been employed for the volumefraction/density and
pressure fields. The details of the algorithm and its
validation are presented in Fuster et al. (2009). The
Lagrangian particle tracking is performed using the updated
velocity field. The fluid forces which are a function of the
relative velocity between the particle and the fluid are
computed (see Magnaudet & Eames (2000)),. The fluid
velocity is obtained at the particle position using a bilinear
interpolation. Acceleration (Eq. 8) so obtained is integrated
to compute the velocity of the particle and subsequently the
updated particle position (Eq. 7). Equations 7 and 8 are
discretized in time using the first order explicit forward
Euler scheme.
Numerical Scheme
The Lagrangian particle tracking algorithm has been
implemented in twophase, incompressible flow solver
GERRIS. The VOF algorithm used in GERRIS is presented
in detail in Refs[popinet2003,popinet2009,fuster2009]. A
second order staggered time discretization scheme has been
implemented for volumefraction/density and pressure.
Using the classical timesplitting projection method the
discretized equations can be written as:
P A" +u ,Vu r = V (D +D.) +(6 ,n)
C cC
21 = 0
2 +V.(c a)=0O (11)
U1 = U,  Vp 1
V ., = 0
Using equations [] and [], the Poisson
the pressure field can be obtained as,
At Vp =Vu,
P 2
\ "2
(13)
equation governing
In Gerris, Poisson equations is efficiently solved using a
quad/octtree based multigrid solver. The details of the
algorithm are given in Ref[popinet2009]. Lagrangian
particle tracking is performed using the updated velocity
field. Particle acceleration is computed using the fluid forces
computed using bilinear interpolation. Advection algorithm
used for particles is first order explicit forwardEuler
scheme. The twoway coupling force is a source term in the
NavierStokes equations and has been smoothed using a
Gaussian distribution with standard deviation op. To achieve
numerical convergence it is smoothed using a Gaussian
distribution with standard deviation op The standard
deviation for the distribution of the force is taken to be the
maximum of the radius of the particle and the size of the
computational cell containing its center. The particle size is
in general smaller than the cellsize except in special
situations where the grid is extremely refined due to high
vorticity or some other criterion. The smoothed force is
given by,
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
=PeI x 0 ( )D, (15)
where (P is the smoothed force and D is the dimension of
the problem (2 in 2D). The smoothed force is distributed
only in a surrounding stencil of approximately 3,p to
reduce the computational cost. On a quad/octree grid we
achieve the above criterion efficiently by moving to a
coarser level till one of the cellfaces is at least 3,p
distance from the particle thence the closest neighbors in 2
(3 in 3dimensional) directions are identified and the
children of these are traversed up to the finest level where
the source term 4, is defined.
Conversion of small VOF droplets into Lagrangian
Particles and viceversa.
Liquid droplets formed during the atomization process are
orders of magnitude smaller than the jet diameter. In order
to capture and track the droplet motion once formed is
impractically computational expensive more so with the
continued formation of droplets. Therefore, the smallest
droplet captured in VOF simulations need to be transformed
and tracked as Lagrangian particles. First, contiguous lumps
Sof liquid are identified and tagged. The volume of the
tagged droplets is computed and droplets smaller than a
cutoff volume are transformed into Lagrangian particles
and subsequently the grid is coarsed. The average velocity
of the droplet is computed prescribed as the velocity of the
particle. Similarly, particle position is identified by the
centroid of the droplets.
Similarly, a Lagrangian particle is converted back into a
VOF resolved droplet when in proximity of an interface.
This is done in order to capture a physical collision between
the droplets and the liquid interface. The twoway coupling
force is replaced by the velocity impulse introduced in the
computational domain upon the transformation of the
particle into a VOFresolved droplet. Subsequently, the
refined computational grid containing the droplet is
assigned a uniform velocity field corresponding to the
momentum of the particle. The above approximation can be
improved by choosing an approximate analytical solution
for the flow field inside the droplet. In addition, we perform
2D computations. The restriction to 2D flow is not realistic
everywhere but constitutes only a heuristic exploration of
what can be done. The drag, lift and other force laws used
for studying the motion of the droplets are, however,
threedimensional.
Simulations were parallelized and in order to prescribe
unique tags (particleids) to the transformation of droplets in
various parts of the full computational domain we employ
the following algorithm:
1. Transform VOF droplets into Lagrangian particles
2. Broadcast the number of newly formed particles
to all other processors in an arrary idadd[i]
where 'i' is the processor id (PID).
3. idadd[i] = idadd[i] + idadd[i1]
)
4. Loop over number of local particles in processor
'i' newly added:
particleid = GMAXID + idadd[i] 
Here GMAXID is the previous time step
global maximum particleid.
5. Broadcast GMAXID = mpi_reduce(MAXID,
...,MPI_MAX,...)
Transportation of particles preserving their unique tags as
they move from one processor to another is achieved in
three steps:
o First, particles moving out of the portion of the
computational domain prescribed to a particular
processor to a neighboring processor are
identified.
o Processors exchange information of the number
of particles to be exchanged among them.
o Subsequently, the information packets are sent
across containing data of the particles to be
exchanged and the neighboring processor
expecting the packet receives it.
Results and Discussion
The implementation of the Lagrangian particle tracking
algorithm is validated by comparing the particle trajectories
obtained from the implementation in Gerris with those
obtained from a RungeKutta integration. The particles
move in a constant velocity field given by,
u = C, cos(rx) sin(zy) and v = C, sin(Qx) cos(zy), where,
C, is chosen to be 0.1. Particles move under the following
fluid forces:
(i) Drag force: FD
3 CD (Re')pV' v'u (v' u)
4d,(
where di is the diameter of the particle 'i'. The drag
coefficient is a function of the local particle Reynolds
number based on the diameter of the particle (see
Magnaudet & Eames (2000)
1+0.15, Re
16 p if Re <50,
SRe
D Re 2.21
48 Re otherwise.
Re
(ii) Lift force: F =pCV' (v' u)xw where
w = Vxu is the vorticity and CL is the lift coefficient.
The lift coefficient CL is a weakly increasing function
of the Reynolds number and tends quickly to a value
of half. We use CL =1/2.
(iii) Inertial force: F, = pV' + (u V)u
(iv) Added mass
forces:
F = pV'C u+(U.V)u dv', where CM is the
I t dt
added mass force constant which depends on the
particle shape. For spherical particles the value of CM
is 1/2.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
(v) External forces: Gravitational force on particles is
given by, F = (p p)g .
X
Figure 1. Particle tracking in a vortex flow
Figure 1 shows the particle trajectory for different grid
resolutions. The cross in the figure marks the center of the
vortex. The solution for a grid resolution 28 x 28 converges
to the trajectory of the particles obtained from RungeKutta.
Forces on the fluid due to particles are modeled as a source
of momentum in the NavierStokes equation. Particles have
been modeled as point particles and therefore the
momentum source term is a point force. In our numerical
algorithm we diffuse the force using a Gaussian function
(Eq.(15)) We discuss here a test case to validate the
numerical modeling of diffused point force. An
axisymmetric jet of liquid is generated by a pointsource of
momentum in a fluid (Batchelor (1970) and Landau (1944)).
The axisofsymmetry is in line with the force. Since the A
nondimensional analysis yields a Reynoldsnumber like
parameter, F (2r7pv2), governing the flow. Here, F has
the units of force and therefore is a measure of strength of
the point source of momentum acting on the fluid at the
origin (Batchelor (1970)). Due to the point source of
momentum, a jet is formed by the entrainment of the
slowmoving fluid pushed rapidly away from the origin.
Here, v is the dynamic viscosity. The edge of the so formed
jet can be defined conveniently by the position where the
streamlines are at the minimum distance from the axis. The
azimuthal coordinate of the minimum point of the
streamlines can be obtained from the relation below,
F/(2p2 = 32 cos 0 4 cos0 8
Spv log / \+
3 sin2 9 cos 0, l+cos90) cos90
(16)
Since there is no inherent length scale in the problem, the
smoothening distance of the point force is important as it
becomes the length scale. Figures 2(a) and (b) show the
streamlines forF/(2frpv2) =50 and the comparison in the
computed axialvelocity with the analytical solution for
different grid resolutions. The quad tree grid is refined in a
small radius (<0.05) with levels 8, 9 and 10. The force F is
diffused in a small region with a standard deviation 2hi,
where hi, corresponds to the minimum grid width (1/28 for
104 103 102 10
r
(b)
Figure 2. (a) Streamlines for F/(27;pv2) =50 and grid
resolution around the point force of 1/210 (b) Comparison of
computed axialvelocity (at 0 = 900 ) for different
resolutions with the analytical solution.
grid level 8). The flow develops in time to show variation of
rm in space where m =1.468,1.144and0.974 for
levels 8, 9 and 10, respectively. The theoretical result is
0 =24'37' and we obtain 0 =30'12' with a force
diffused using a standard deviation of 210 (quadtree level
10). The variation in the vicinity of the smoothed force varia
is r .
We perform a simulation of particle plumes to demonstrate
the implementation of algorithm in parallel. Particle to fluid
density ratio is 0.01, volume of the particle is 105m3,
dynamic viscosity of the underlying fluid is 105 Pas and
the ratio of the particle plume width to domain width is 0.05
(width of the plume being 10cm). A serried of particles is
injected with zero velocity with a frequency of 2100 per
second (an array of 21particles). Adaptive mesh refinement
has been used with vorticity as the cost function. The
smallest grid size is 210 and the largest correspond to 2 .
Figure 3 shows a plume of light particles moving upward
under buoyancy. The simulation is performed on four
processor nodes with lines in the figure depicting the
divided computational domain. Particle data are transported
from one processor to the neighboring processor when
particles buoy out from one processor to another. The color
in the graph shows the strength of the vorticity. Momentum
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
exchange with the underlying fluid leads to flow leading to
a mixing layer type instability (Climent and Magnaudet
(2006)).
Figure 3: Particle plume (black dots). Simulation has been
run on four processors solving each of the four subdomains
separated in the figure by solidlines. The color represents
the strength of the voriticity (blue showing negative
vorticity and red depicting positive) and black dots
represent point particles.
We now discuss a simulation of atomization using the
implementation of the Lagrangian particle tracking coupled
with the NavierStokes solver (Gerris). A liquid jet atomizes
into small droplets by a highspeed coaxial airjet flowing
over it forming thin liquid ligaments at the liquidgas
interface due to the instability. Subsequently, the ligament
breaks up into droplets. The prediction of the droplet size
distribution in an atomization process is of immense
importance in several industrial applications. For example,
the distribution of fuel in combustion chambers is critical to
fuel economy. We have performed an atomization
simulation, on a parallel machine using eight nodes, with
small droplets transformed into Lagrangian particles. In
order to perform simulations in parallel, special care needs
to be taken in prescribing consistent identification numbers
to the particles, thus enabling us to study the trajectory of
droplets. A planar simulation of a liquid jet destabilized by a
highspeed coflowing gas has been performed. A thin
tapered separator plate separates the liquid and gas flow
near the inlet. The taper angle used in the present simulation
is 7 the thickness of the separator plate at the inlet is
e=150ltm and the radius of the trailing edge is 20 jim. The
flow parameters of the liquid and gas used are given in
Table 1.
p /p, (Kg/m3) ug/u, (Pa.s) Sle (r m)
20/0.5 10/1000 1.7x105/103 175/150
Table 1: Properties of the liquid and the coaxially flowing
gas jet.
7th International Conference on Multiphase Flow
ICMF 2010, Tampa, FL USA, May 30June 4, 2010
distributions of the droplets in two different regions of the
computational domain (a) close to the nozzle inlet and (b)
far from it (marked in Fig. 4). Figure 5(a) shows a rather
sharp peak for a diameter (1020 pm) and another smoother
peak of the distribution for a diameter (4050pm). In
contrast, Fig. 5(b) shows a clear peak of the distribution at ~
125 pm suggesting formation of larger droplets away from
the nozzle. The above PDFs, thus, delineate the two
II ,
... .'l ,''. i ,.
Gas I
Liquid I
Figure 4: Break up of a liquid jet by a highspeed coaxially
flowing gasjet. A cloud of small droplets, modeled as
Lagrangian particles (shown in red), formed during the
atomization process are advected and spread by the high
speed gas. The two dash boxes mark regions where we
perform PDF of droplets.
The nondimensional parameters governing the process,
namely, liquidgas momentum ratio M = pU2 I pU, gas
and liquid Reynolds numbers
( Reg =pgUgS 1/g and Re, =p,U,, lu, ) and Weber
number We=pgPUg, oa are 16, 2060, 5000 and 10.2,
respectively. The boundary layer in the gas,
g = 6.05Reg1/2 Hg, at the nozzle inlet is a function of the
Reg and is prescribed at the nozzle inlet using an
experimental correlation (Rayana (2007)). Here, Hg is the
thickness of the gasjet. A similar correlation has been used
at the liquid inlet to obtain 8, The quadtree level
refinement used for computations is 11 thus the smallest
grid size is ~ 5pm and the interface is maintained at the
finest level in the computation. The size of the
computational domain is chosen to be 6cmX3cm. The
refinement criteria used in the simulations is described in
Fuster et al. (2009). The smallest droplets in the
computational domain that occupy less than 25 smallest
computational cells are transformed into particles. The local
computational grid upon transformation is coarsened. Figure
4 shows a snapshot of an atomization simulation performed
in parallel with eight nodes. The blue dots in the figure
represent droplets which have been modeled as Lagrangian
particles. Droplets once formed are entrained into the fast
moving airstream and are scattered in a wider region. The
particles formed from droplets are resolved again by VOF
when they approach a VOFresolved interface.
Figures 5(a) and 5(b) show the probability density
mechanisms of formation of droplets, namely, by primary
atomization of the jet and another by breakup of bigger
fragments of liquid downstream due to the interaction of the
smaller droplets formed near the nozzle during primary
atomization.
0 50 100 150 200
Diameter (pJm)
(b)
Figure 5: PDF of the diameter of the droplets formed at a
location (a) near the nozzleinlet (b) further downstream as
marked in the Figure 4. The abscissa of the plots is the
diameter of the droplet in meters.
Figure 6 shows the trajectory of droplets formed due to
primary atomization near the nozzle. The droplets travel
linearly through the coaxial gas jet all inclined at a nearly
same angle with it irrespective of the size of the droplets.
The black line in the figure shows the average angle of
droplet trajectories 180.
0 0.01 0.02 0.03 0.04 0.05
Figure 6: Trajectories of the particles formed near the
nozzleinlet (units in meters). The color of the trajectories
correspond to logarithm of the particle diameter with the
adjoining colorindex. The dark line shows the mean angle
of the trajectories (~ 18 ).
Conclusions
A multiscale algorithm has been presented for simulations
of atomization of a gas assisted liquid jet into droplets.
Smallest droplets are transformed into Lagrangian particles
moving under the fluid forces. Lagrangian particles are
transformed back into VOF resolved droplets upon their
proximity to a VOF interface. The algorithm has been
parallelized to allow simulation of atomization in a bigger
domain. The algorithm has been validated using different
test cases. We also present simulations of a high liquidgas
density and viscosity ratio (100) gas assisted atomization
using Gerris. Droplets formed during primary atomization
near the nozzle interact with the downstream jet and modify
the droplet size distribution. A probability density function
of the droplet size distribution shows smaller droplets near
the nozzle and droplet distribution shifts towards bigger
droplets in the downstream.
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