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1 Figure# SPH Simulation of a Twin-Fluid Atomizer Operating with a High Viscosity Liquid

Abstract

A Smooth Particles Hydrodynamics (SPH) 2D simulation of a twin-fluid atomizer is presented and compared with experiments in the context of bio-fuel production. The configuration consists in an axial high viscosity liquid jet (µ l ≈ 0.5 Pa.s) atomized by a coflowing high-speed air stream (u g ≈ 100 m/s) at atmospheric conditions, and the experiment shows two types of jet instability (flapping or pulsating) depending on operating conditions and the nozzle geometry. In order to capture the 3D effects of the axial geometry with a 2D simulation, the surface tension force and the viscosity operator are modified. The mean and RMS velocity profiles of the single phase simulations show a good agreement with the experiment. For multiphase simulations, despite a qualitative good agreement, the type of instabilities as well as its frequency are rarely well captured, highlighting the limitation of 2D geometry in the prediction of 3D configurations.

1 Figures

ICLASS 2015, 13th Triennial International Conference on Liquid Atomization and Spray Systems, Tainan, Taiwan, August 23–27, 2015

SPH Simulation of a Twin-Fluid Atomizer Operating with a High Viscosity Liquid

G. Chaussonnet*,1, S. Braun1, L. Wieth1, R. Koch1, H.-J. Bauer1

A. Sänger2,3, T. Jakobs2, N. Djordjevic2, T. Kolb2,3

Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany

1Institute of Thermal Turbomachines, KIT Campus South

2Institute of Technical Chemistry, KIT Campus North

3Engler-Bunte-Institute, KIT Campus South

*geoffroy.chaussonnet@kit.edu

Abstract

A Smooth Particles Hydrodynamics (SPH) 2D simulation of a twin-ﬂuid atomizer is presented and compared with

experiments in the context of bio-fuel production. The conﬁguration consists in an axial high viscosity liquid jet

(µl≈0.5 Pa.s) atomized by a coﬂowing high-speed air stream (ug≈100 m/s) at atmospheric conditions, and the

experiment shows two types of jet instability (ﬂapping or pulsating) depending on operating conditions and the

nozzle geometry. In order to capture the 3D effects of the axial geometry with a 2D simulation, the surface tension

force and the viscosity operator are modiﬁed. The mean and RMS velocity proﬁles of the single phase simulations

show a good agreement with the experiment. For multiphase simulations, despite a qualitative good agreement, the

type of instabilities as well as its frequency are rarely well captured, highlighting the limitation of 2D geometry in

the prediction of 3D conﬁgurations.

Keywords: Numerical simulation, atomization, viscous ﬂuids, air-assisted atomizer, SPH

Introduction

Recent predictions estimate the depletion of coal, oil and gas to year 2040, 2112 and 2042, respectively [1],

which leaves approximately thirty years to ﬁnd alternative and sustainable energy sources. The Bioliq®process

(Fig. 1) proposes a solution based on the reﬁnement of agricultural residuals, decoupled in two steps: (i) a de-

centralized stage where the biomass (mainly straw) is reﬁned via fast pyrolysis in a compound of higher energy

density called bioSyncrude®, and (ii) a centralized stage that converts bioSyncrude®into synthetic gas (Syngas) at

high temperature (TG≈1500◦C) and pressure (PG= 80 bars). Syngas is ﬁnally transformed to methanol, motor

fuel [2] or even gas [3].

Biomass bioSyncrude®Syngas Synfuel

EFG

Gas

Gasoline

Diesel

Decentralized process Centralized process

Raw gas

Figure 1 The Bioliq®process

At the ﬁrst stage of the centralized process, bioSyncrude®is turned into a spray in a so-called Entrained Flow

Gasiﬁer [4] (EFG, depicted in Fig. 1) operating at PG, through an air-assisted atomizer: the axial liquid jet of

bioSyncrude®is sheared by a coaxial gas stream ﬂowing at high velocity. Due to chemical equilibrium require-

ments, the Gas-to-Liquid Ratio (GLR), expressed as ˙mg/˙ml, should be in the range 0.5 to 0.8 for a typical biomass

based fuel, which limits the amount of atomizing material. In addition, the bioSyncrude®has a very large viscosity

(≈10 Pa.s at low temperature in the low shear range) and a non-Newtonian behaviour. These constraints have a

strong inﬂuence on the atomizing process, and the nozzle must be carefully designed in order to optimize the spray

generation.

In this paper, the numerical simulation of a simpliﬁed nozzle supplied with a viscous Newtonian ﬂuid is compared

to experiments [5], in terms of gas velocity proﬁles and liquid instabilities frequency. The employed numerical

approach is Smooth Particle Hydrodynamics (SPH), a method originally developed for astrophysics [6] and later

adapted to free surface ﬂow [7], especially when the gas phase can be neglected. In the ﬁeld of air-assisted liquid

atomization, the gas/liquid momentum transfer is the driving phenomenon so that both phases (of large density and

viscosity ratio) must be accurately resolved. This particular context constitutes an original use of the SPH method

1

ICLASS 2015, 13th Triennial International Conference on Liquid Atomization and Spray Systems, Tainan, Taiwan, August 23–27, 2015

that was initiated by Höﬂer et al. [8, 9].

The experimental setup and its main observations are presented in the next part, followed by a description of

the SPH method and the modiﬁcations added in this particular conﬁguration. The results of the single phase and

the multiphase 2D simulations will be presented in the fourth and ﬁfth part, and a ﬁnal conclusion will end the

paper.

Experiment

In order to have a deeper insight into the atomization process occurring in the EFG, Sänger et al. [5] studied the

fragmentation mechanism of a viscous ﬂuid in a twin-ﬂuid external mixing atomizer at atmospheric pressure and

temperature conditions. It consists in an axial liquid jet sheared by a co-ﬂowing high speed gas stream discharging

into an open and quiescent atmosphere.

DlDg

Hg

es

Figure 2 Schematics of external mixing twin ﬂuid atomizer with parallel streams (left) and angle (right) nozzle.

Geometrical parameters (middle) are detailed.

Two different nozzles were studied: (i) both gaseous and liquid streams are parallel (Fig. 2 left) and (ii) the gas

stream has an incident angle of 30◦(Fig. 2 right). In both conﬁgurations, the liquid diameter Dland the gas height

Hgare equal to 2 and 1.6 mm respectively. In the parallel conﬁguration, the separator thickness esand the gas

diameter Dgare equal to 0.1 and 5.4 mm respectively and to 0.5 and 6.2 mm in the angle conﬁguration. Although

several types of liquid were experimentally investigated, only two types of Newtonian liquid (L1 and L2) of differ-

ent viscosity are discussed here. Their physical properties, as well as the gaseous ones, are summarized in Table 1.

Fluid type Density Dyn. viscosity Surface tension Mass ﬂow rate Bulk velocity

ρ[kg/m3]µ[Pa.s] σ[N/m] ˙m[kg/h] U [m/s]

L1 1233 0.2 0.0646 10 0.717

L2 1236 0.3 0.0643 10 0.713

Gas 1.2 1.73 10-5 - 4 - 8 54.9 - 107.8

Table 1 Physical properties of investigated ﬂuids in atmospheric conditions.

The non-dimensional numbers that characterize this conﬁguration are:

Re = DhU

ν,We = ρgDlU2

rel

σ,Oh = µl

√Dlρlσ,GLR = ˙mg

˙mg

,M = ρgU2

g

ρlU2

l

(1)

Equation (1) shows, in order of appearance: the Reynolds number where Dhis the hydraulic diameter equal to

Dliq for the liquid and 2Hgas for the gas, the Weber number where Urel is the liquid/gas relative velocity, the

Ohnesorge number, the Gas-To-Liquid ratio, and the momentum ﬂux ratio.

The test rig was instrumented with Laser Doppler Anemometry (LDA) for gaseous velocity measurement on ver-

tical line at x= 1.6 and 1.4 mm for the parallel and angle nozzle respectively, and the liquid instabilities were

recorded with a high-speed camera at a sampling frequency of 1200 Hz and with a pixel size of 143 µm. The time

series were then post-processed by Proper Orthogonal Decomposition (POD) in order to extract the frequency and

the wavelength of the instabilities. It was also observed that the jet undergoes different types of instability depend-

ing on the liquid viscosity and the GLR as illustrated in Fig. 3, leading to two different spray characteristics [5]. A

pulsating mode is observed and consists in a longitudinal and axisymetric mode triggered by a Kelvin-Helmholtz

instability. Approximately ﬁve diameters downstream, the jet disrupts into a non-axisymetric structure composed

of bags and ligaments (Fig. 3a). A ﬂapping mode is characterized by an deﬂection in the radial direction, in the

early stage of the jet at approximately two diameters after the nozzle exit (Fig. 3b). The objective of the numerical

simulation is to predict the onset of these modes at the right operating conditions.

2

ICLASS 2015, 13th Triennial International Conference on Liquid Atomization and Spray Systems, Tainan, Taiwan, August 23–27, 2015

Figure 3 Side and oblique view of primary instabilities of a liquid jet with different dynamic viscosities at

GLR = 0.4. (a) pulsating mode at µl= 200 mPa.s and (b) ﬂapping mode at µl= 300 mPa.s, from [5]

Numerical Model

SPH Formalism

The SPH method is a mesh-free method that relies on a Lagrangian description of the ﬂuid through particles

moving at the ﬂuid velocity and carrying physical properties such as mass, volume, momentum and energy.

The starting point of the SPH discretization is the convolution of a ﬁeld f(r)by a Dirac function δ(r):

f(r) = Zf(r′)δ(r−r′) dr′(2)

In order to apply Eq. (2) to discrete particles, the Dirac function is replaced by a smooth interpolation function

W(r−r′,h)called the kernel and depicted in Fig. 4 (top). This function is deﬁned on a compact support, the

so-called sphere of inﬂuence that depends on the smoothing length h, and must fulﬁll mathematical properties such

as the unity integral (RW(r−r′,h)dr′= 1) and the convergence to δwhen h→0. The kernel is chosen here

as a quintic spline and h= 3∆xwhere ∆xis the mean particle spacing. The function fis thus expressed at the

particle location raby:

f(ra) = X

b∈Ω

Vbf(rb)W(rb−ra, h)(3)

where Vbis the volume of particle b. The index brefers to neighbour particles located in Ω, the sphere of inﬂuence

of the particle a, as illustrated in Fig. 4 (bottom). For the sake of clarity, in the following, f(ra),f(rb)and

W(rb−ra, h)are shortened to fa,fband Wab, respectively.

Figure 4 Bottom part: particle distribution superimposed with the kernel value and illustration of the sphere of

inﬂuence. Top part: surface of a 2D kernel

The differential operators needed to evaluate the contact forces such as pressure, viscosity and surface tension

are computed with the gradient of the kernel. The gradient [10] ∇faand the Laplacian [11] ∆faare expressed as:

∇fa=X

b∈Ω

Vb(fb+fa)∇Wab and ∆fa= 2 X

b∈Ω

Vb(fb−fa)∂Wab

∂r (4)

Governing Equations

The Navier-Stokes equations are turned to a SPH form. The ﬂow is considered isothermal so that the energy

equation can be neglected. The continuity equation is solved algebrically by computing the particle volume and

density:

Va= 1/X

b∈Ω

Wab and ρa=ma/Va(5)

3

where mais the constant mass of particle a. Equations (5) exactly conserve mass and as it relies only on the

particle volume, the density expression avoids numerical diffusion of density near the liquid/gas interface. The

momentum equation is given by:

ρa

du

dt

a

=fa,p +fa,v +fa,st (6)

where uis the particle velocity and the terms fp,a,fv,a and fst,a are the forces due to pressure, viscosity and

surface tension, respectively. They write:

fa,p =−X

b∈Ω

Vb(pb+pa)∇Wab ,fa,v =KX

b∈Ω

Vbµuab ·rab

r2

ab +η2∇Wab and fa,st =−σaκ(∇·n)n(7)

The expression of the pressure term conserves the linear momentum locally by ensuring Fab =−Fba where

Fab =VaVb(pb+pa)∇Wab is the elementary force that particle bexerts on particle a. Although the accuracy

of this expression is signiﬁcantly reduced by particle disorder [12], it artiﬁcially creates a term proportional to the

background pressure that avoid "holes" of particle during the simulation.

The expression of the viscous term is a SPH form of the velocity Laplacian ∂(µ∂u/∂x)/∂x. It exhibits the

prefactor Kequal to 8 in 2D [13], the inter-particle viscosity µderiving from a density-based average µ=

(ρaµb+ρbµa)/(ρa+ρb). It also invloves the scalar product of velocity difference uab =ua−ubby the

inter-particle distance vector rab =ra−rb. The term η= 0.1havoids the singularity when r2

ab = 0.

In the surface tension force, σais the surface tension at particle aand κis the interface curvature. The terms nand

(∇·n)are the interface normal and its divergence, respectively, computed with the method proposed by Adami

et al. [14].

To close the system, the pressure is expressed through a Tait state equation that depends on the particle density

only:

pa=ρ0c2

γρa

ρ0γ

−1+pback (8)

where ρ0is the nominal particle density and γis the polytropic ratio that controls the stiffness of the pressure: due

to the term (ρa/ρ0)γ, the pressure varies more intensively with regards to density variations when γis larger. Note

that γintervenes directly in the compressibility βof the ﬂuid, deﬁned by β= (∂ρ/∂p)/ρ at constant entropy:

inverting Eq. (8) leads to β= [γ(p−pback) + ρ0c2]−1so the larger γthe lower the compressibility.

The term cin Eq. (8) is the ﬁctive speed of sound and must be chosen to verify c>10 umax in order to fulﬁll the

weakly compressible condition by ensuring that density variation is lower than 1% [15], the ﬁnal purpose being to

increase the time step through the CFL condition. In the present conﬁguration, the liquid and gaseous real Mach

numbers are of different orders of magnitude, inducing different type of ﬂow. With a velocity of the order of

magnitude of 1 m/s, the liquid Mach number is ≈10-3 so that setting the ﬁctive liquid speed of sound clto 50 m/s

is appropriate. As a lower speed of sound increases the compressibility, it is necessary to increase γfor the liquid,

so γlis set to 7. On the contrary, in the gas phase, the Mach number is ≈0.3 and the ﬂow is compressible. The

gaseous sound speed is therefore set to its real value (340 m/s in the experiment conditions) and γgis set to 1, so

that Eq. (8) yields pa,g =c2

g(ρa−ρ0) + pback.

Finally the background pressure pback is added in Eq. (8) to ensure that pressure is always positive. Additionally,

pback controls the stiffness of the pressure gradient: rewriting Eq. (8) as p=δp +pback and injecting it in Eq. (7)

leads to:

fa,p =−X

b∈Ω

Vb(δpb+δpa)∇Wab −2pback X

b∈Ω

Vb∇Wab (9)

where the second term of the RHS, proportional to pback, is the artiﬁcial part of the pressure gradient that avoids

holes formation in the lattice.

Three Dimensional Terms

As the simulations conducted in this paper are 2D, two modiﬁcations are added to the numerical method to

represent the 3D effects of the experiment. First, in order to take into account the curvature of the round jet

interface, an artiﬁcial surface tension force fa,curv is added to the momentum equation:

fa,curv =−ǫxǫy

σa

|y|(∇·n)nwith ǫy= 1 −exp −y2

h2and ǫx=1−x

Lc0.2

(10)

where yis the radial coordinate and ǫya damping function to avoid the singularity at y= 0. The function ǫx

ensures that far from the nozzle, when the jet is fragmented in small droplets, the artiﬁcial surface tension force is

4

zero. It is deﬁned between 0 and Lc= 4 Dl, and is equal to zero otherwise.

The second modiﬁcation is to adapt the particularities of vectorial operators expressed in a cylindrical system to

their Cartesian expressions, in order to take into account the divergence of the ﬁelds with y(rin the cylindrical

system). The comparison between Cartesian (x,y)and cylindrical (r,z)coordinates show no difference for the

gradient operator whereas the Laplacian differences yield, for the velocity:

∆CYL(Ur) = ∆CART (Uy) + 1

y

∂Uy

∂y −Uy

y2(11)

∆CYL(Uz) = ∆CART (Ux) + 1

y

∂Ux

∂y (12)

The Laplacian operator, used for viscosity, is thus modiﬁed according to Eqs (11) and (12), the additional terms

being also multiplied by ǫy(Eq. 10) to avoid the singularity at y= 0. Physically, these modiﬁcations render the

difference of the shearing surface between the inner and the outer radius of an inﬁnitesimal element. This differ-

ence is particularly signiﬁcant at small radius (terms in "1/y").

Note that the modiﬁcations do not aim to model an axisymmetric domain but only the center slice of an axial

conﬁguration, so that it is not necessary to modify the computation of density.

Geometry and Boundary Conditions

The numerical domain depicted in Fig. 5 is composed of the inlet ducts and the cavity (length of 40 mm)

where atomization takes place. The length of inlet ducts are seven diameters for the gas and three diameters for

the liquid and the inlet velocity proﬁles are turbulent and laminar, respectively. A no-slip boundary condition is

imposed at the walls of the inlets duct and the nozzle. In order to reproduce the entrainment rate ˙

Qeinduced by

the gaseous jets, the cavity is fed with a coaxial gas stream with a bulk velocity ueover a slit of height He= 6.67

mm. This ensures a proper entrained volume ﬂow rate up to x= 10 mm such as (ueHe) = ˙

Qe(x=10 mm), and

lead to ue≈0.15 U. The restriction of the distance to 10 mm is to avoid a too large uethat could perturb the

atomizing gas streams. On the sides of the cavity, the velocity is mainly axial due the entrainment ﬂow, and the use

of an outﬂow boundary condition would generate a strong numerical noise. Therefore, the cavity sides are set to

slipping walls and they slightly open (semi-angle of 10◦) to mimic the free jet opening after the nozzle. The outlet

is set to a constant pressure equal to pback.

Gas inlet

Liquid inlet

Entrainment inlet

Slip wall

No-slip wall

Outlet

Probes location Velocity proﬁle location Probes location

Velocity proﬁle location

Figure 5 Sketch of the numerical domains and zoom on the nozzle exit. Left: parallel nozzle. Right: angle nozzle

Test Cases

Single-phase simulations assess for the accuracy of the SPH method when modeling low viscosity ﬂuids such

as air. Two different mass ﬂows are tested on both nozzles. With the parallel nozzle, multiphase simulations are

conducted to verify that 2D SPH can capture the pulsating mode at two different mass ﬂows, and with the angle

nozzle, the transition pulsating/ﬂapping is studied by varying the gaseous mass ﬂow and the liquid viscosity. All

test cases are summarized in Table 2 where bold letters indicate how the case names are constructed.

Results and Discussion of Single-Phase Simulations

Figure 6 shows a velocity proﬁle comparison, for the angle and parallel nozzle at two gaseous mass ﬂow rates.

The mean and RMS values are displayed for the axial (u) and radial (v) component. Note that as the radial compo-

nent is Cartesian, it can have negative values. The simulations are averaged over 1000 snapshots regularly sampled

over 18 convective times based on the cavity length.

5

Name [-] PG6 PG8 PL4 PL6 PL8 AG4 AG6 AL1M4 AL1M5 AL1M6 AL2M4 AL2M5 AL2M6

Nozzle [-] PAR PAR PAR PAR PAR ANG ANG ANG ANG ANG ANG ANG ANG

Liq. type [-] Gas Gas L1L1L1Gas Gas L1 L1 L1 L2 L2 L2

˙mg[kg/h] 6 8 4 6 8 4 6 4 5 6 4 5 6

Ug[m/s] 81.9 107.8 54.9 81.9 107.8 54.9 81.9 54.9 68.5 81.9 54.9 68.5 81.9

GLR [-] - - 0.4 0.6 0.8 - - 0.4 0.5 0.6 0.4 0.5 0.6

Reg[×1000] 18.3 24.1 12.3 18.3 24.1 12.3 18.3 12.3 15.3 18.3 12.3 15.3 18.3

We [-] - - 175 393 684 - - 175 274 393 176 274 395

M[-] - - 5.72 12.7 22.1 - - 5.72 8.91 12.74 5.76 8.96 12.81

Table 2 Test-case matrix

For the angle nozzle, the mean values are in good agreement with the experiment for the two mass ﬂows but

(i) ushows a too smooth proﬁle, possibly due to the lack of turbulence model, and (ii) the extrema of vare slightly

over-predicted, resulting in too opened jet. The predicted uRMS show peaks at the right yposition but they are

smoother than experimental results, highlighting more widespread ﬂuctuations. The model cannot capture the cen-

tral peak, probably due to the too large jet opening. The radial RMS velocity is slightly unsymmetrical and predicts

a too large peak in the center for AG6. The same comments are valid for the parallel nozzle with the difference

that (i) uextrema are slightly over-predicted, (ii) the jet is even more opened and (iii) uRMS is predicted with the

right intensity, except for the central peak.

Concerning the axial velocity, the good predictions of its mean and the acceptable levels of its RMS in the air

−20

0

20

40

60

80

100 AG6

−40

−20

0

20

40

0

5

10

15

20

25

30

35

40

−3−2−10 1 2 3

y [mm]

0

5

10

15

20

25

30

−40

−20

0

20

40

60

80

100 PG6

−20

−15

−10

−5

0

5

10

15

20

25

0

5

10

15

20

25

30

35

40

−3−2−10 1 2 3

y [mm]

6

8

10

12

14

16

18

20

22

−20

−10

0

10

20

30

40

50

60

70

u[m/s]

AG4

−30

−20

−10

0

10

20

30

v[m/s]

Exp.

0

5

10

15

20

25

30

uRMS [m/s]

−3−2−10 1 2 3

y [mm]

3

4

5

6

7

8

9

10

11

vRMS [m/s]

−50

0

50

100

150 PG8

−40

−20

0

20

40

Sim.

5

10

15

20

25

30

35

40

−3−2−10 1 2 3

y [mm]

5

10

15

20

25

Figure 6 Mean and RMS proﬁles of axial and radial velocity, at x= 1.6 mm and 1.4 mm for the parallel and angle

nozzle, respectively. Experiment: ◦, simulation: −

stream region ensure that the gas will shear the liquid with the right average and ﬂuctuating intensity. As the radial

velocity is more difﬁcult to capture, it can be expected that the dynamic pressure felt by the liquid interface (ρgv2)

6

will slightly deviate from reality, especially in terms of ﬂuctuations, possibly leading to an earlier destabilization

of the jet.

Results and Discussion of Multiphase Simulations

Table 3 summarizes the instability modes observed in experiment and simulation. For the parallel nozzle, the

simulation capture a transition between a sinusoidal pulsating mode (phase shift ϕ=πbetween top and bottom

probes) and a ﬂapping mode, while experiments only show a varicose pulsating mode (ϕ= 0). Cases PL4 and

PL6 can be compared to the experiment in terms of frequency while PL8 simulation is purely ﬂapping. With the

angle nozzle and liquid L1, experiment show a pulsating mode while simulations only predict a ﬂapping mode,

preventing further quantitative comparisons. With the liquid L2, ﬂapping modes in both the experiment and the

simulation are observed, but with an opposite trend with increasing gas velocity. Nevertheless, a quantitative

comparison is achieved for L2 in the following.

Case PL4 PL6 PL8 AL1M4 AL1M5 AL1M6 AL2M4 AL2M5 AL2M6

Exp. Puls. Puls. Puls. Puls. Puls. Puls. Flap. Flap. Puls./Flap.

Sim. Puls. Puls./Flap Flap. Flap. Flap. Flap. Puls./Flap. Flap. Flap.

Table 3 Observation of instability modes

Figure 7 compares experimental and simulated time series, for one case of each geometry. For PL6 (Fig. 7a-

c), similar structures are observed: a central liquid core right after injection of length ≈2Dlthat further disrupts

into ﬁnner ligaments. However the time series of PL6 shows a behaviour closer to ﬂapping than pulsating, and

the second instability stage (p1) is tilted due to ﬂapping while it remains axially oriented in the experiment. In

addition, when the liquid bubble bursts (bag breakup), the atomized droplets are ejected axially in the experiment

whereas radially in the simulation, due to the ﬂapping motion (arrow in Fig. 7c). The AL2M5 case (Fig. 7d-f )

shows a proper ﬂapping mode with a predicted ﬂapping frequency qualitatively close to the experiment. In this

conﬁguration, the second stage of the liquid core (p2and p3) is tilted in the simulation.

a) time = 0 ms b) time = 0.922 ms c) time = 1.923 ms

EXP. SIM.

d) time = 0 ms e) time = 0.587 ms f) time = 1.161 ms

p1p1

p2p2p3

p3

5 mm

Figure 7 Comparison of experiment/simulation time series for PL6 (a,b,c) and AL2M5 (d,e,f)

Figures 8aand 8bdepict the frequency fversus the gas mass ﬂow rate, for the pulsating mode with the parallel

nozzle (a) and for the ﬂapping mode with the angle nozzle with liquid L2 (b). Experimentally, the frequencies were

calculated by POD while in the simulation, they were computed from probes recording the liquid presence at the

location deﬁned in Fig. 5. Several interspectra were computed via FFT and the frequency of largest intensity was

kept. The error bars in Fig. 8 correspond to the frequency resolution of the FFT. For the parallel nozzle (Fig.

8a), the frequency fpincreases with the gas mass ﬂow rate as in experiments, but the linearity (fp∝˙m) is not

recovered, and two points may be not sufﬁcient to endorse the validaty of the present linear regression. With the

angle nozzle (Fig. 8b), the trend is also recovered but the linear regression does not show a purely proportional

depency, due to a non-zero y-intercept (y=a x +bwith b6= 0). Figure 8cshows the velocity ucof the wave

7

propagation on the interface directly downstream the injection and the associated wave length λfor PL6. The

velocity ucis computed by measuring the time difference ∆tbetween two consecutive probes separated by ∆x.

The velocity is then ∆x/∆tand the wavelength λis equal to the ratio of the velocity by the peak frequency fp

according to λ=uc/fp. The prediction of ucshows an appropriate order of magnitude but its dependence on ˙m

is not clearly deﬁned due to a strong uncertainty, while λis acceptably predicted with PL4 and underpredicted for

PL6.

4 5 6 7 8

˙m[kg/h]

200

300

400

500

600

700

f [Hz]

a)

Experiment

Simulation

4.0 4.5 5.0 5.5 6.0

˙m[kg/h]

200

300

400

500

600

700

800

900

f [Hz]

b)

Experiment

Simulation

45678

˙m[kg/h]

1.0

1.5

2.0

2.5

3.0

3.5

4.0

uc[m/s]

c)

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

λ[mm]

Figure 8 Comparison of experiment/simulation, a) pulsating frequency with the parallel nozzle, b) ﬂapping fre-

quency with the angle nozzle and L2. On plot c), and represent ucfor experiment and simulation respectively,

while #and

represents λfor experiment and simulation respectively.

Conclusion

The 2D numerical simulation of a twin ﬂuid atomizer with SPH method showed that the gaseous ﬂow was

acceptably predicted on a two-dimensional domain, in terms of mean and RMS velocity proﬁle. When considering

the multiphase conﬁguration, the simpliﬁed 2D approach has shown strong limitations, despite the 3D modiﬁ-

cations of the surface tension force and the viscosity. This emphasizes that in this particular conﬁguration of

air-assisted atomization, the prediction of an appropriate gaseous shear stress is not sufﬁcient to guarantee a proper

liquid behaviour. The modes of instability are not well captured and predictions even show an inverted pulsat-

ing/ﬂapping transition with gas mass ﬂow rate in case of angle nozzle with liquid L2. However when simulations

and experiments have the same mode, the time series show a qualitatively good agreement in terms of shape and

structure of the ligament, and the predicted frequencies have comparable values with the experiment. Finally the

wave velocities and the associated wavelengthes show the right order of magnitude but the trends are not well cap-

tured. In the optic of accurately capturing the transition of instability modes and their associated values (frequency

and wavelength) with SPH method, it is thus necessary to compute a 3D domain.

Acknowledgement

This work was performed on the computational resource ForHLR Phase I funded by the Ministry of Science,

Research and the Arts Baden-Württemberg and DFG ("Deutsche Forschungsgemeinschaft").

References

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[5] Sänger, A.; Jakobs, T.; Djordjevic, N. and Kolb, T., ILASS Europe (2014).

[6] Gingold, R. and Monaghan, J., Monthly Notices of the Royal Astronomical Society, 181:375 – 389 (1977).

[7] Monaghan, J. and Kocharyan, A., Computer Physics Communications, 87:225 – 235 (1995).

[8] Höﬂer, C.; Braun, S.; Koch, R. and Bauer, H.-J., ILASS - Europe 2011, 24th European Conference on Liquid Atomization

and Spray Systems, Estoril, Portugal, September 2011 (2011).

[9] Höﬂer, C.; Braun, S.; Koch, R. and Bauer, H.-J., Proceedings of ASME Turbo Expo 2012 (2012).

[10] Monaghan, J., Reports on Progress in Physics, 68:1703 – 1759 (2005).

[11] Cleary, P. and Monaghan, J., Journal of Computational Physics, 148:227 – 264 (1999).

[12] Chaussonnet, G.; Braun, S.; Wieth, L.; Koch, R. and Bauer, H.-J., Proceedings of 10th SPHERIC International Workshop

(2015).

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8

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**Progress of theoretical physics**. . 2011.1091-1121.- 2009.181-189..
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