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A Validated Numerical Simulation of Diesel Injector Flow Using a VOF Method

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  • Principia - La Ciotat France
Article

A Validated Numerical Simulation of Diesel Injector Flow Using a VOF Method

Abstract

Progress in Diesel spray modelling highly depends on a better knowledge of the instantaneous injection velocity and of the hydraulic section at the exit of each injection hole. Additionally a better identification of the mechanisms which cause fragmentation is needed. This necessitates to begin with a precise computation of the two-phase flow which arises due to the presence of cavitation within the injectors. For that aim, a VOF type interface tracking method has been developed and improved (Segment Lagrangian VOF method) which allows to describe numerically the onset and development of cavitation within Diesel injectors. Furthermore, experiments have been performed for validation purpose, on transparent one-hole injectors for high pressure injection conditions. Two different entrance geometries (straight and rounded) and various upstream and downstream pressure levels have been considered. This numerical approach allows to retrieve different cavitation regimes and a good agreement has been obtained for the discharge coefficients. Encouraging results have also been achieved concerning the emission frequency of the cavitation pockets at the injector exit. Then preliminary calculations have been performed on a VCO Diesel injector with needle displacement and an estimation of the injection velocity has been obtained for this configuration. Finally, the VOF method has been applied to calculate directly the three phase flow (liquid and vapour Diesel fuel, and external gas) downstream the injector exit. This method should give better insight, in a near future, into the mechanisms of fragmentation.
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SAE 2000-01-2932
A Validated Numerical Simulation of
Diesel Injector Flow Using a VOF Method
R. Marcer P. Le Cottier H. Chaves
Principia C.R.M.T I.F.F Freiberg
B. Argueyrolles C. Habchi B. Barbeau
Renault I.F.P P.S.A
ABSTRACT
Progress in Diesel spray modelling highly depends on a
better knowledge of the instantaneous injection velocity and
of the hydraulic section at the exit of each injection hole.
Additionally a better identification of the mechanisms
which cause fragmentation is needed. This necessitates to
begin with a precise computation of the two-phase flow
which arises due to the presence of cavitation within the
injectors.
For that aim, a VOF type interface tracking method has
been developed and improved (Segment Lagrangian VOF
method) which allows to describe numerically the onset and
development of cavitation within Diesel injectors.
Furthermore, experiments have been performed for
validation purpose, on transparent one-hole injectors for
high pressure injection conditions. Two different entrance
geometries (straight and rounded) and various upstream and
downstream pressure levels have been considered. This
numerical approach allows to retrieve different cavitation
regimes and a good agreement has been obtained for the
discharge coefficients. Encouraging results have also been
achieved concerning the emission frequency of the
cavitation pockets at the injector exit. Then preliminary
calculations have been performed on a VCO Diesel injector
with needle displacement and an estimation of the injection
velocity has been obtained for this configuration.
Finally, the VOF method has been applied to calculate
directly the three phase flow (liquid and vapour Diesel fuel,
and external gas) downstream the injector exit. This method
should give better insight, in a near future, into the
mechanisms of fragmentation.
INTRODUCTION
In recent years, the studies devoted to Diesel direct
injection have been considering the use of common rail
injectors with injection pressures as high as 135 MPa and
hole diameters in the range 130 to 200 microns. The present
tendency is to go towards higher pressures (200 MPa) and
smaller hole diameters. One of the essential aspects of this
type of injector is the occurence of cavitation in each
injection hole due to the fall of the static pressure at the hole
entrance and/or sometimes along the hole walls in the case
of hydroeroded injectors for instance. The location of
cavitation which depends on the relative position of the
holes and the needle, strongly affects the instantaneous
values of the injection velocity, the Diesel fuel hydraulic
section and the injection angle. It may also lead to different
injection rates from injector holes which can give rise to
asymmetric transient spray propagation in the combustion
chamber.
Furthermore, the cavitation arising in the injection holes is
convected towards the hole exit and when the life duration
of the cavities is higher than the convective time scale
which is usually the case for high injection pressure, the
cavities can survive in the very beginning of the jet outside
the hole and may play a major role in the spray break-up.
The mechanism frequently invoked is the collapse of the
cavitation bubbles due do the difference between the
ambient air pressure in the combustion chamber (typically 6
MPa) and the vapour pressure of Diesel fuel (some
thousand Pa) prevailing in the cavitation bubbles exiting the
nozzle, Soteriou et al. [1]. This mechanism leads to the
instantaneous break-up of the liquid on an extremely short
distance (some injector diameters) and corresponds to
recent observations of the liquid core length, Fath et al. [2],
Bruneaux [3]. Other mechanisms on which we will come
back later on have been proposed (amplification by Kelvin-
Helmholtz mode of perturbations issued in the liquid, Lee et
al. [4], influence of flow rate fluctuations, Chaves and
Obermeier [5], etc). Nevertheless, it remains much
uncertainty on the physical processes and on the relative
importance of each of these mechanisms, Dumont et al. [6].
When we turn our attention to the numerical modelling of
Diesel sprays at high injection pressure using the Kiva code
Copyright © 2000 Society of Automotive Engineers, Inc.
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for instance, we observe that the fuel injection velocity at
the holes exit which is the essential boundary condition of
any spray calculation is poorly known. Very few
measurements have been attempted up to now, and these
measurements have been performed mainly for steady state
flow conditions and/or at the spray edge, Chaves et al. [7] ,
Meingast et al. [8]. The injection velocity should be
comprised between the velocity obtained from the flow rate
(using the geometrical cross-section of the hole as
representative liquid section) and the Bernoulli velocity
()
2Pinj Pch l
ρ
which is the maximum allowable
velocity, Winklhofer [9]. For example, this gives an
uncertainty of the order of typically 150 m/s for an exit
Bernoulli velocity of 550 m/s. If for a given injection
velocity one takes care in the Kiva calculations to adapt the
size of the injected drops (blobs) to the fuel hydraulic
section deduced from the measurement of the injection flow
rate, one observes dramatic consequences in terms of
combustion behaviour, according to the value of the
injection velocity. This is due to a modification of the
droplet size distribution resulting from atomisation which
has a strong impact on the location of the fuel vapour
distribution.
The industrial practice is to work on a comparative basis
after having tuned (with the model constants and... the
injection velocity!) a first calculation using an experimental
pressure curve (which is not allways available for an
advanced engine project). Sometimes a preliminary study
has been done on a closed vessel experiment, anyhow for a
limited range of operating conditions. However, there are
situations, especially for the comparisons between full load
and part load, or for injections with preinjection, where the
value of the injection velocity relatively to the Bernoulli
velocity evolves (due to the more or less large extension of
cavitation within the holes) and for which comparisons are
not very reliable.
More generally speaking, spray calculations are based on a
set of submodels (primary and secondary atomisation,
collisions, drag, evaporation, wall impingement,
combustion etc) for which the coefficients have been
revisited due to the lack of representativity of the academic
configurations for which these models have initially been
developed. The tuning of these coefficients is generally
biased due to the poor knowledge of the boundary
conditions, in other terms the values of these coefficients
incorporate the uncertainty of the boundary conditions in
order to present acceptable comparisons between
calculations and measurements.
Furthermore, it is absolutely necessary to better know the
physical mechanisms of primary break-up for high pressure
injection and to implement submodels incorporating these
mechanisms in order to reproduce correctly the tendencies
observed in the engine. It follows from all this that the
knowledge of the boundary conditions and of the exact link
between the injector two-phase flow and the primary break-
up phenomena is really necessary to progress seriously in
the numerical description of Diesel sprays.
For this purpose, experiments have been conducted on a
simple configuration (transparent one hole injector at high
injection pressure) in order to better analyse the cavitation
characteristics and the type of induced break-up according
to the nozzle inlet geometry. Some quantitative values such
as the discharge coefficient, the exit velocity and the exit
frequency of the cavitation pockets (or bubbles) have been
measured and the main results will be presented in the first
part of this paper. Next, a numerical methodology which
has been developed in parallel is presented with first
comparisons with experimental values. Then we will discuss
calculation results obtained on a VCO type industrial
injector and we shall conclude showing a direct calculation
of the spray break-up at the nozzle exit (for the time being a
2D axisymmetrical calculation).
EXPERIMENTAL PART
Previous results on cavitation
Some thirty years ago, the occurrence of cavitation was put
into evidence within injection nozzles, Bergwerk [10].
Since then a large number of studies have shown the
extreme sensitivity of sprays on one hand to the
configuration of injection channels, and especially to the
radius of curvature at the hole entrance, the l/d ratio and the
inner wall roughness, Ohrn et al. [11] and on the other hand
to the needle position relatively to the injection holes. These
parameters affect the threshold of cavitation inception,
Genge [12], the injection velocity, the spray angle at the
injector exit and the spray droplet sizes, Karazawa et al.
[13]. To facilitate the experiments, some experimental
studies have been performed on enlarged injectors (scale
typically greater than 10), Soteriou [1], Afzal et al. [14],
using Reynolds and cavitation numbers representative of
situations encountered in engines. These studies enabled to
better understand the development of cavitation but present
some limitations. Even if the evolution of the discharge
coefficient as a function of the cavitation number
corresponds to the one observed on the real injector,
Arcoumanis et al. [15], the nature of cavitation is probably
different, rather of bubble type with homogeneous
nucleation within enlarged injectors and of film type,
breaking-up into cavities and bubbles, within real injectors.
Furthermore and especially, if the Reynolds number is kept,
the fuel velocity is very different from the one in the actual
injector, which modifies the ratio between the life duration
of the cavities (proportional to 1Pch ) and the convection
time within the injection channel (proportional
to1Pinj Pch
). For enlarged injectors, the cavities will
tend to collapse within the injection channel, which should
notably reinforce the turbulence of the liquid phase, He and
Ruiz [16], while for real size injectors the cavities will tend
to collapse beyond the nozzle exit and will participate to the
spray break-up. This tendency is reinforced due to the
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lengthening of the life duration of the cavities in the very
confined environment of the injection holes, Lush [17].
Other studies performed on transparent one hole axial
injectors and at moderate injection pressures (up to 30
MPa) have enabled to precise the nature of cavitation
encountered on real size injectors, Chaves et al. [7].
Cavitation films develop at the hole entrance when it is
straight and break-up partly as they are convected towards
the nozzle exit. For higher pressure differences, these films
may extend beyond the hole exit. This phenomenon is
called supercavitation. Moreover, direct measurements of
the injection velocity by the same authors have shown that it
is close to the Bernoulli velocity.
Other authors have preferred to work with 2D plane
configurations. Genge [18] used an axial feeding but the
wall profiles are asymmetrical in order to avoid the
characteristic instabilities of 2D symmetrical configurations.
This study shows a large number of cavitation regimes but
the injection pressures still remain moderate. Alajbegovic
[19] and Henry [20] (for low injection pressures in this
case), used feeding at right angle relatively to the injection
channel axis. This type of configuration allows to follow the
development of a large cavitation pocket but the need to
limit the influence of the lateral walls may result in low
Diesel fuel velocities.
Kim et al. [21] and Afzal et al. [14] observed on enlarged
injectors the formation of "string" cavitation originating in
the space surrounding the needle and located just ahead of
the injection holes. This string cavitation connects two
adjacent holes and may move in a chaotic way towards two
other adjacent holes. It penetrates within the injection holes
and interferes with the "geometrical" cavitation generated in
the recirculation zones induced at the upper walls of the
injection channels.
In a very recent study, Arcoumanis et al. [15] succeeded in
visualizing the development of cavitation within a real size
VCO injector. The injection pressures are for the time being
relatively low but the flow velocities are already much more
realistic than in an enlarged injector. This study shows that
string cavitation exists also in real size VCO injectors and
that cavitation bubbles exit the injection holes.
One can observe that the previous studies have been
performed for stationary flow conditions and for a fixed
needle lift. The periodic break-up of cavitation films
introduces however some flow unsteadiness although the
boundary conditions are fixed (Chaves et al. [7],
Arcoumanis et al. [22]). Some numerical and experimental
results show that the inertia of the liquid column plays a
significant role when the boundary conditions vary (needle
position or pressure fluctuations upstream the needle seat).
In particular, one does not obtain the same cavitating flow if
the needle is opening or closing, for the same injection
pressure at the hole entrance. In the same way a decrease of
cavitation is observed for a sharp increase of the injection
pressure just ahead the needle seat, Chaves and Kirmse
[23]. The quasi-steady hypothesis is therefore not valid.
Finally, when the needle is off-centered on a real size
injector, the cavitation zones may be located both on the
upper and the lower walls of the injection hole nearest to the
needle, or even only on the lower wall. Furthermore, the
flow in the injection hole may start rotating which leads to
hollow sprays with high injection cone angles (Arcoumanis
et al. [22], Soteriou et al. [1]).
Link between injector flow and spray break-up
One may observe at first the extreme diversity of the
injection conditions described in the literature. A high
injection pressure doesn't necessarily induce a fragmented
spray. For example for a 200 MPa injection pressure and a
cylindrical injection hole with a low l/d (~2.5) without a
needle located upstream, Tamaki [24] observes a narrow
smooth jet. Inversely for a low injection pressure (some
tenths of MPa) a cylindrical hole with l/d equal to 4 and a
straight entrance may produce a cavitating flow and a very
atomized jet, Hiroyasu et al. [25].
For various operating conditions giving rise to different full
conical sprays, several break-up modes have been proposed.
One has however to be cautious because the spray operating
conditions for which these break-up modes have been
proposed can be rather far from the conditions prevailing in
Diesel injectors.
So, the liquid phase turbulence can lead to spray break-up
(
ρ
lu'2 of the order of
ρ
ginj
U2 in some circumstances)
creating a surface perturbation whose amplitude may grow
due to a Kelvin-Helmholtz mechanism. This perturbation
may also originate from the collapse of cavitation zones
within the injector nozzle. This mechanism often described
in the literature is probably not that important considering
the range of injection pressures now used, the low values of
l/d and the low radii of curvature of hydroeroded injectors.
The mechanism of bubble collapse right at the jet departure
is supported since Prescher and Schaffitz [26] by a growing
number of studies. Prescher observed the erosion induced
by the collapse of cavitation bubbles on a small disc placed
right at the exit of a one hole injector. Eifler [27] put into
evidence by a Schlieren method the radial propagation of
pressure waves starting from collapsing zones. Fath et al.
[2] have shown, by performing tomography along the jet
axis, the survival of cavitation bubbles right at the exit of a
one hole injector leading to a very short dense core (some
nozzle hole diameters), decreasing when the injection
pressure or the chamber pressure increases. Chaves and
Obermeier [28] observed also a strong decorrelation of the
optical signals acquired inside and outside the nozzle.
Finally, Favennec and Fruman [29] observe that the
decrease of the discharge coefficient on an actual VCO
injector is correlated with the noise signal due to bubble
collapse.
Independently of the collapse of cavitation bubbles, the
injection velocity may be modulated due to the passage of
cavitation zones or bubbles through the hole exits and to the
fluctuations of the Diesel fuel pressure upstream the needle
seat (case of classical injectors when the needle hits the
4
back stop). This modulation may contribute significantly to
spray break-up (water hammer effect), Chaves et al.[5],[30].
Finally, for very high injection velocities, Nakahira [31]
observes the generation of shock waves in the ambient air,
centered on the injector exit. These waves could also have
an effect on spray break-up.
For each of the preceding mechanisms, models have been
proposed in the literature. One may quote the Huh and
Gosman model [32] [4] for which the liquid phase
turbulence is estimated (ignoring the presence of cavitation)
and constitutes the perturbation called to be amplified by a
Kelvin-Helmholtz mechanism. One may quote also the
phenomenological model of break-up through cavitation of
Arcoumanis and Gavaises [33] and the model of primary
break-up through axial collisions developed by Chaves and
Obermeier [5].
But before using such models it is necessary to compute at
best the injector flow to have a better estimate of the
boundary conditions. We need for that a better knowledge
of the physics of the cavitation development because this
will influence the choice of a numerical model.
Furthermore, basic experiments are needed for validating
the numerical results. The experiments performed at I.F.F
and then at C.R.M.T will be presented in the following
sections.
EXPERIMENTS WITH TRANSPARENT INJECTORS
Two different injection systems were applied at IFF for
visualisation purpose. For the "steady state" experiments the
fuel was injected by a quasi-steady injection pump
composed of two pistons. The larger one is driven by
compressed air through a magnetic valve. The smaller
piston coupled to the large one compresses the fuel. Figure
1 shows a schematic diagram of this set-up. Figure 2
displays a pressure trace of this pump. The pressure varies
much slower than in the case of a standard injection pump.
For the experiments presented here pressures of 36.1 MPa
and 100 MPa were chosen, although the peak pressure can
reach much higher values. It turned out that due to the
specifics of the facility, the fuel is first precompressed to an
intermediate pressure before it is compressed to the peak
pressure. This procedure lengthens the lifetime of the glass
nozzles. For the study of transients a distributor pump was
used. It allowed to perform stroboscopic observations of
cavitation phenomena, which is not possible with the single-
shot quasi-steady injection pump.
Figure 1: Schematic diagram of the quasi-steady injection
0.00 0.02 0.04 0.06 0.08 0.10
Time [s]
0
20
40
60
80
100
Pressure [M Pa]
Figure 2: Typical pressure trace of the quasi-steady
injection pump.
The flow in the nozzle holes was observed by using
transparent nozzles. These nozzles are glass wafers of 1 mm
thickness with a 0.2 mm hole. They replace the tip of a
single hole sac type injector. The refractive index of the
glass matches that of Diesel fuel, therefore even though the
holes are cylindrical the view is undisturbed by the surface
of the glass. The outer surfaces of the wafer are plane
surfaces. Three different nozzle geometries were used.
Illumination is obtained by back lightening with either the
short pulse (200ns) of a high efficiency LED diode or by the
flash from a Nanolight (8ns). The pictures are recorded with
a CCD camera with apropriate lenses for the different fields
of view. The injectors were also equipped with a pressure
tap for a Kistler piezoresistive transducer. The tap is located
15 mm from the inlet of the nozzle hole. The signal from
this transducer was also adapted for triggering. The
injectors are mounted in a high pressure chamber which can
either be filled with nitrogen or with Diesel fuel.
Pictures of the nozzle flow at 36.1 MPa injection
pressure
Due to the refractive index matching and back illumination,
fuel appears white on the pictures whilst cavitation is black.
One picture was taken per shot of the injection pump. The
results are shown Figure 3. They demonstrate how
remarkably reproducible the cavitation structures are. The
main variation occurs at the tip of the elongated film visible
at the bottom of the pictures. From this tip smaller bubbles
separate. These structures change from one picture to the
next.
In order to check the hypothesis that the well defined
cavitation structures are determined by minute
imperfections of the nozzle inlet, the nozzle was rotated by
180° around its axis, Figure 4. The structures are obviously
linked to the nozzle.
5
Figure 3: Pictures of the cavitating nozzle flow taken from
three different runs at an injection pressure of 36.1 MPa
into 0.1 MPa. Flow from left to right. The right side of the
picture corresponds to the nozzle exit. The inlet corner is
not visible and just outside of the field of view. The nozzle
has a diameter of 0.2 mm and a sharp inlet corner.
Figure 4: Pictures of the nozzle rotated by 180°. Injection
pressure 36.1 MPa, chamber pressure 0.1 MPa.
Pictures of the nozzle flow at 100 MPa injection
pressure
First experiments revealed that at 100 MPa cavitation
extends across the whole nozzle hole, Figure 5. Therefore, it
made little sense to take additionally close-up pictures of
the nozzle hole alone.
Figure 5: Cavitating nozzle flow at 100 MPa injection
pressure into the atmosphere.
A larger field of view was chosen for further experiments,
which were performed by injecting Diesel fuel into Diesel
fuel that had been pressurized by nitrogen. Figure 6 shows a
series of pictures taken for increasing chamber pressure up
to 6 MPa. The pictures were illuminated by a 20 ns pulse
from a frequency doubled Nd-Yag laser at 512 nm.
Chamber pressure : 0.1 MPa
0.5 MPa
6 MPa
6 MPa, field of view is shifted
Figure 6: Cavitating nozzle and jet flow for 100 MPa
injection pressure and various chamber pressures, injection
of Diesel fuel into Diesel fuel, straight nozzle d=0.2 mm,
l=1 mm
6
The granularity of the background is produced by speckles
due to the use of coherent light. Cavitation was observed
within the nozzle as well as downstream of the nozzle exit.
The jet structure observed downstream in the pressure
chamber is caused by cavitation due to fluctuations caused
by high shear velocities of the jet against the surrounding
fuel. As one can see the increase of the chamber pressure
reduces this kind of cavitation.
At high chamber pressures, the distance at which the
cavitation is fully collapsed is already so short that one
could conjecture that it is the collapse of the cavitation
exiting the nozzle. For injection into a gas the collapse of
cavitation is likely to occur even closer to the nozzle
because smaller fuel masses have to be accelerated to
permit it.
Direct measurements of injection velocity
When a reasonable amount of cavitation exits the nozzle,
the actual jet velocity does no longer correspond to a mean
velocity calculated from volumetric discharge
measurements by dividing the volumetric flux by the
geometric nozzle area. In reality it is much higher which
also implies that the jet thrust is higher. So it becomes
necessary to measure directly the injection velocity. For that
purpose this velocity was measured within and immediately
outside of the nozzle in the very dense spray region
applying what might be called an imaging correlation
velocimeter. The set-up is shown in Figure 7.
Figure 7 : Set-up of the correlation velocimeter.
The basic idea is to project an image of the nozzle which is
illuminated by a He-Ne laser onto a plane in which plastic
fibres are positioned at a preset distance apart. The inverse
magnification of the optics defines measuring volumes in
the object plane of the nozzle which correspond to the light
collecting areas of the fibres in the image plane. Although
the method is a line of sight integrating method, the depth of
field defined by the high aperture optics is very small. In
fact the dimensions of each of the measuring volumes
corresponds to that of a laser doppler velocimeter. If a
bubble in the nozzle or a ligament or drop in the spray
crosses the two measuring volumes which are aligned with
the main flow direction, then the photomultipliers connected
to the fibres will detect a signal one after the other. In
reality not only one but many bubbles or drops cross the
volumes producing a signal or signature in each of the
multipliers. A time of flight analysis is not possible in such
a case. Therefore the two signals are cross-correlated to
obtain the mean time of flight between the two measuring
volumes for a time window used for the correlation. The
data from the multipliers was sampled at 100 MHz. The
bandwidth of the multipliers was adjusted to about 20 MHz.
In cases where the information contained in the data is high
enough, i. e. the edge of the spray near the nozzle, data
windows of 10 microseconds contain enough information to
give a well defined correlation peak with values as high as
0.9. This type of measurement method works best where
LDV gives no results. The diameter of the measuring
volumes (30 µm) was chosen to be smaller than the diameter
of the nozzle (0.2 mm) but still large enough to collect
enough light to keep the signal to noise ratio at a
satisfactorily high level. The distance between the
measuring volumes used was 60 µm. This is a trade off
between velocity resolution and the deterioration of
correlation between the two signals. The larger the distance
between the two volumes the better the velocity resolution
for a given sample rate, however if the two volumes are too
far apart then the signatures contained in the two signals are
also too different giving a lower correlation peak. One
advantage of the imaging of the object plane is that the
relative position of the nozzle and the measuring volumes
are visible. This allowed to measure velocity profiles across
the nozzle hole and in the spray. The results for the case of
injection into atmospheric conditions will be shown in a
comparison with the numerical simulations (see Figure 27).
Measurement of frequencies of cavitation pockets
exiting the nozzle
Coming back to Figure 3 one realizes that the film
cavitation observed close to the inlet of the nozzle tends to
break up into smaller pockets further downstream.
Therefore it seemed appropiate to measure the frequency of
the detachment of these pockets for comparison with the
simulations. The same optical set-up as for the velocity
measurements was used for this purpose using only one
fibre. However, a smaller sampling frequency was used to
obtain a more representative set of data. The signals of the
photomultipliers were analysed using an FFT algorithm.
The results will be shown together with the numerical
results in Figure 26.
Influence of the radius of curvature
As the VCO injectors are hydroeroded at the hole inlets
[34], it is important to see how the flow behaves for
rounded entrances. The aim is also to acquire quantitative
data to test the representativity of the numerical calculations
against the effect of the inlet radius of curvature R.
The following experiments have been performed at CRMT,
with a similar one stroke injection system set up through a
cooperation with I.F.F. The transparent injectors are built in
the same way. The visualisations have been carried out with
7
a CCD camera associated with a nanolite lamp whose flash
duration is equal to 10 ns. As previously, the cavitation
zones appear dark in the white background for the liquid.
As before, the injection channel has a diameter equal to 0.2
mm and three radii of curvature are considered R=0, 50 and
150 µm. The needle is positioned at a lift equal to 0.45 mm.
The injection pressure is 50 MPa and the chamber pressure
6 MPa. The cavitation number is defined in the following
as: XPPch
==
()
Pinj Pch Pch
. As evidenced in Figure
8, the hole entrance geometry affects notably the length of
the cavitation zone in the channel. The highest radius of
curvature can even suppress any cavitation occurence, for
this injection pressure.
R0 R50 R150
Figure 8 : Entrance geometry effect on injector flow
Pinj=50 MPa, Pch=6 MPa (X=7.3)
The discharge coefficients CD, for these operating
conditions, vary between 0.813 for R=0 and 1 for R=150
µm. These discharge coefficients include geometrical
pressure losses and those eventually due to cavitation. The
examples of Figure 8 correspond to two extreme situations
where on one side (R0) all the conditions of a low Cd are
met and on the other side (R150) all the conditions of a high
Cd are achieved.
The evolution of Cd has been measured as a function of the
cavitation number, for a fixed P=44 MPa (Bernoulli
velocity 325 m/s) (Figure 9). The error bars superimposed
on the figure include a geometrical uncertainty of 2.5% on
the geometrical section. One observes for the straight
entrance the classical decrease of Cd towards a plateau as
the cavitation number increases. This behaviour
corresponds to the growing contraction of the liquid section
due to the presence of cavitation. The fact that the same
tendency exists for the rounded entrance R150 seems to
indicate that cavitation appears also in this case for high
cavitation numbers.
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1101001000
cavitation number
Cd
R150
R0
Figure 9 : Cd versus cavitation number for R0 and R150
constant P=44 MPa
The visualisations for the rounded entrance in Figure 10
effectively show the presence of cavitation films which
appear towards the hole exit as soon as the cavitation
number is greater than 11.5. The cavitation onset seems
here to be caused by wall defects.
50 MPa 75 MPa
Figure 10 : Cavitation films set-up towards exit (injector
R150) Injection of Diesel fuel into Diesel fuel at 6 MPa
(X=7.3 left, X=11.5 right).
Afterwards visualisations have been performed to see the
effect of a cavitating flow on the spray break-up. For the
same operating conditions: Pinj=30 MPa, Pch=6 MPa,
Figure 11 shows that the jet issued from an injection hole
with a straight entrance, where cavitation is already present,
breaks up right at the exit while the one issued from an
injection hole with a rounded entrance R50 is connected to
the injector exit through a smooth cylindrical section
equivalent to one hole diameter.
This shows that cavitation is beneficial for spray break-up
and that too large inlet radii obtained with hydroerosion
techniques should be avoided to favour spray atomization
even at low injection pressures.
8
R0 R50
Figure 11 : Break-up potential for R0 versus R50
Pinj=30 MPa, Pch=6 MPa (X=4)
Indirect measurements of injection velocity
The injection velocity Vinj can be deduced from the
measurement of the mass flow rate and of the jet thrust
(Vinj=thrust/(mass flow rate)). This type of measurement
gives rather a core velocity at the hole exit compared to the
IFF measurement which gives in essence a local velocity.
The measurements are performed for the time being for
stationary conditions i.e fixed needle lift and integrate the
whole flow rate of momentum on the impact surface of the
jet (Figure 12).
Figure 12 : Spray thrust measurements into a pressurized
chamber under steady flow conditions
The following table summarizes the exit velocities obtained
at CRMT and at IFF for the R0 configuration.
Operating
conditions
(MPa) Cd Vinj
m/s Vinj/VBern
CRMT
IFF 44/0.1
36/0.1 0.77
0.77 313
280 0.96
0.95
CRMT
IFF 100/6
100/0.1 0.81
0.79 463
480 0.98
0.99
It is possible to refer the results to the same experimental
conditions by using a scaling factor of type
(
)
Pinj Pch
.
Then one would obtain 310 m/s at IFF for a case 44/0.1
MPa and 465 m/s for a case 100/6 MPa. This shows a good
convergence of both methods for a large range of operating
conditions.
Vinj/VBern appearing in this table is the ratio between the
injection velocity and the Bernoulli velocity. One sees that
the measured injection velocities tend towards the Bernoulli
velocity when the injection pressure is increased. This is
illustrated in Figure 13 for a given cavitation number and
Bernoulli velocities ranging from 280 to 450 m/s.
0.85
0.9
0.95
1
1.05
200 250 300 350 400 450 500
flow velocity (m/s)
Vinj / VBern
R150
R0
Figure 13 : Vinj/VBern versus Bernoulli velocity at
constant cavitation number X=15.4 for R0 and R150.
This velocity dependance of the ratio Vinj/VBern can be
explained by the fact that even for the same cavitation
number, cavitation extends less in the hole at low Bernoulli
velocity than at high Bernoulli velocity. Finally, one
observes in this figure that for a same P, the injection
velocity is higher for a rounded entrance compared to a
straight entrance.
Now one considers the effect of the cavitation number at a
fixed pressure difference of 44 MPa on the injectors R0 and
R150. One observes in Figure 14 that the ratio of the
injection velocity to the Bernoulli velocity doesn't evolve
significantly in both cases.
0.9
0.925
0.95
0.975
1
1.025
1.05
1 10 100 1000
cavitation number
Vinj / VBern
R150
R0
Figure 14 : Vinj/VBern versus cavitation number for R0
and R150, at a fixed Bernoulli velocity equal to 325 m/s.
The liquid exit section Sinj deduced from the measured
flow rate and injection velocity (Figure 15) displays the
same tendency as Cd (see Figure 9) when the cavitation
number increases. The fact that the liquid exit section is
smaller than the geometrical section for cavitation numbers
9
larger than ~12 for the injector R150 confirms the
occurence of cavitation.
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1 10 100 1000
cavitation number
Sinj / Sgeom
R0
R150
Figure 15 : Sinj/Sgeom versus cavitation number for R0 and
R150, at a fixed Bernoulli velocity equal to 325 m/s.
NUMERICAL MODELLING PART
Introduction
As we have seen previously, cavitation zones are created in
the upstream part of the nozzle where the liquid pressure
reaches the vapour pressure. They move inside the nozzle
and some of them spread outside the hole in a higher
pressure gas medium leading to very quick collapse of the
cavities, to disturbances of the pressure field induced in the
liquid jet and initiation of jet interface instabilities close to
the hole exit. Downstream, theses instabilities grow-up
under the dynamical interaction between the liquid jet and
the gas flow, until atomisation happens.
Numerical simulation of such coupled physical mechanisms
turns out to be difficult but new numerical approaches might
allow significant progress.
Several numerical approaches exist to simulate unsteady
cavitating flows : the two phase one-fluid models (mixture
models), the two-fluid models and the free surface tracking
models.
For the first, one considers the fluid as a continuous
medium in which the density varies strongly on the liquid-
vapour interfaces. The state equation of the mixture can be
an exact one (with the energy equation added to the system
[35]), but most of the authors have used a barotropic
pseudo-state law, artificially smoothed (for example
[36],[37],[18]) or based upon bubble dynamics governed by
the Rayleigh-Plesset equation ([38], [39]). But one of the
problems of the mixture models remains the difficulty to
choose a state equation representative of the whole
cavitation physics depending on the flow configuration.
In the two-fluid models (see for example [40],[41]), the
problem is the need of empirical source terms in each of the
two sets of Navier-Stokes equations, to model mass and
eventually energy exchanges, and drag law of bubbles… As
for the mixture models, the two-fluid models are not well
adapted to two-phase flows separated by sharp interfaces, as
in pocket cavitation, where dynamical effects are
preponderant. However, first results have been obtained on
industrial injectors in [42].
The free surface tracking models are able to preserve and
transport (by the liquid phase velocity) liquid-vapour
discontinuities and are well-adapted to unsteady pocket
cavitation problems. One problem remains concerning mass
transfer between phases which cannot be taken into account
easily because of the purely kinematic nature of these
models. Then, the introduction of a mass transfer model,
even simplified, is needed in order to simulate vaporization
effects on liquid-vapour interfaces.
In this context, Principia has been working on the
development of a multi separated phase Navier-Stokes
model (code EOLE) using a new VOF concept called
Segment Lagrangian – Volume Of Fluid (SL-VOF) [43] in
order to calculate the interface evolution between phases. A
simplified semi-empirical model allows to take into account
the mass-transfer process [44].
Equations
The unsteady 3D Navier-Stokes equations for the
multiphase flows are written in the following semi-
conservative form, in curvilinear formulation :
T
HGF
t
W
J=+++
∂χ
∂η
∂ξ
1 (1)
where F, G and H are the flux terms, and T the surface
tension source terms [45]:
()
()
()
+
+
+
===
z
p
z
wu
y
p
y
vu
x
p
x
uu
u
J
F
z
Kn
y
Knx
Kn
T
w
v
u
W
τξξρ
τξξρ
τξξρ
ρ
σ
σ
σ
ρ
ρ
ρ
!
!
!
!
!
!
.
~
.
~
.
~
~
1
;
0
;
0
()
()
()
()
()
()
+
+
+
=
+
+
+
=
z
p
z
ww
y
p
z
vw
x
p
z
uw
w
J
H
z
p
y
wv
y
p
y
vv
x
p
y
uv
v
J
G
τχχρ
τχχρ
τχχρ
ρ
τηηρ
τηηρ
τηηρ
ρ
!
!
!
!
!
!
!
!
!
!
!
!
.
~
.
~
.
~
~
1
;
.
~
.
~
.
~
~
1
()
()
zyx
Jw
z
v
y
u
x
w
w
z
v
y
u
x
vw
z
v
y
u
x
u
,,
,,
;
~
~
~
χηξ
χχχ
ηηηξξξ
=++=
++=++=
()
.. . . .U
t
U=
z
e
zy
e
yx
e
x
!!!!
!
!!
!
!!!
!
!!!
!
!! +===
µτττττττ
with
()
χηξ
,, the curvilinear coordinates, Jthe Jacobian
of the coordinates transformation,
σ
the surface tension
coefficient, K the surface curvature and n
! the normal to the
10
interface. Additionnally,
()
wvu ,, are the cartesian velocity
components for each phase,
()
wvu ~
,
~
,
~ the contravariant
velocity components,
ρ
the density,
µ
the molecular
viscosity and
τ
!
! the viscous stress tensor.
Vapour (and external gaseous) phases are assumed to be
incompressible, but the liquid flow is considered as
compressible because of the high velocity and great
pressure gradients existing in the injector cavitating flow.
The density variation being related essentially with
dynamical effects (and not thermal), the state equation can
be writen [46]:
=p
pdP
p
Ln 0)(
1
)( 0
βρ
ρ
(2)
where 1/β(p) is the isothermal compressibility coefficient.
Pseudo-compressibility method
Time discretization is ensured using a fully implicit second
order scheme. The solution of the non-linear system for the
unknown values at step n+1 is based on the pseudo -
compressibility method [47],[48].
Considering the semi-discretized equations at the time level
n+1 and introducing a time-like variable
τ
, called pseudo-
time, one adds pseudo-unsteady terms which are derivatives
of the unknowns at time level n+1 with respect to
τ
:
1
11
1
2
1
4
1
31
1~
1
+
=
+
+
+
+
+
+
+
+
+
+
n
T
n
H
n
G
n
F
t
n
W
n
W
n
W
J
n
W
J
∂χ
∂η
∂ξ
∂τ
(3)
with:
=
w
v
u
W
ρ
ρ
ρ
ρ
~
~
~
~
~
The pseudo-unsteady terms involve a new unknown
variable
ρ
~ called pseudo-density and subject to remain
positive. The pressure is calculated as a function of
ρ
~
through an additional pseudo-state equation :
()
ρ
~
fp =(4)
This numerical relation can be selected in different ways as
discussed in Viviand [47]. The system (3,4) is integrated
step-by-step in pseudo-time until convergence towards a
solution independent of
τ
which is then the numerical
solution at time level n+1. The liquid density is then
obtained from the pressure using equation (2).
The system is hyperbolic with respect to
τ
and it is formally
very close to Navier-Stokes equations for compressible
flow, due to the presence of the same
ρ
~ factor in the
continuity and momentum equations in
τ
. The real time term
is a source term for the evolution in
τ
. This property makes
it possible to directly apply existing and efficient algorithms
which have been developed for compressible flows. We
have used an adaptation of the finite volume method on
multi-block curvilinear deforming grids, using a centered
scheme with artificial viscosity, originally developed by
Jameson and al. [49].
Discretization in space is of the centered type for the
variables pvu ,
~
,
~
,
~
ρ
. Artificial viscosity, which is necessary
in the case of a centered scheme to ensure stability and
convergence, includes second order derivatives and fourth
order derivatives and is adjustable using tunable
coefficients.
The scheme used in pseudo-time is the explicit 5 step
Runge-Kutta scheme, associated with an implicit residual
smoothing technique. The basic Runge-Kutta scheme is
explicit, but one introduces an implicit treatment of the
unsteady source term tW
/, which reinforces stability
while leaving the calculation effectively explicit. The
maximum value of the pseudo-time step
τ
is fixed by the
local CFL stability criterion. For each cell, one uses the
maximum local value (local time step technique). The
method is unconditionally stable with respect to the physical
time step. Finally this method is particularly convenient to
deal with 2-phase flows having high liquid-gas density
ratios.
SL-VOF method
The interface and its movement are obtained for each time
step of the simulation by an original method, called SL-
VOF [43], using the two well known concepts of VOF [50]
and PLIC (Peacewise Linear Interface Calculation) [51].
The interface is calculated in each cell thanks to a discrete
function C whose value in each cell is the cell fraction
occupied by the denser fluid (VOF concept). The original
SOLAVOF method [50] assumes the interface to be parallel
to the grid faces, so the accuracy of this method is low,
whereas the PLIC method allows the interface to be
represented by a segment of any orientation. As shown in
Figure 16, the normal n
!to the interface within each cell is
defined as the opposite of C
!. Thus is it possible to
represent the interface within each cell by a segment normal
to n
!, whose position is evaluated so that the fraction of the
cell delimited on the opposite side of the normal is equal to
the value of C within this cell [51].
The velocity at the ends of each segment is deduced from
the general velocity field by bilinear interpolation. The ends
of the segments are then advected in a lagrangian way using
a first order scheme (Figure 17). The new positions of the
segments allow to calculate new values of C.
Due to the lagrangian nature of the SL-VOF method, there
is no need to solve a conservation equation of the VOF
function. So, one of the advantages of this method
compared to the former PLIC method and to the original
SOLAVOF method is to be able to use larger time steps.
11
The quantity vof
CFL being the maximun value of the ratio
of the displacement of a fluid element of the interface
during a time step to the maximum size of the cell, earlier
VOF methods were limited by the stability criterion
5.0
<
vof
CFL [50],[51]. The SL-VOF method has no
theoretical constraint about the vof
CFL criterion (the mean
value imposed in all the computations is about 2), which
allows for a significant gain in computational time.
1
0
0.9
0.40.8
0.10.3
1
1
Values of C in each cell.The initial interface The PLIC modelisation
Figure 16 : PLIC modelling of the interface
Figure 17 : SL-VOF principle
Cavitation criterion
The passage of noncavitating flow to cavitating flow is
realized when the pressure in the liquid flow reaches locally
a critical threshold corresponding to the vapour pressure of
the liquid phase. In these conditions, the cavity is initialized
in cells of the mesh which verify the vaporization criterion.
These cells are given a VOF value equal to 0
(corresponding to the vapour phase) and a pressure equal to
the vapour pressure.
During the computation, the kinematic displacement of the
interface is achieved by the SL-VOF algorithm with a
velocity equal to the normal velocity of the liquid. The mass
transfer is taken into account from a semi-empirical
cavitation criterion. At each time step, an instantaneous
mass transfer is imposed for all liquid and partially liquid
cells of the mesh having a pressure lower than the cavitation
pressure. It means a correction of the liquid-vapour
interface position computed by the SL-VOF method (and
representative of the purely kinematic part of the cavitating
flow) in order to take into account thermodynamical effects.
So the "kinematic" VOF field is modified in such a way that
the interface is constrained to fit the vapour pressure isobar
and therefore the pressure of the liquid phase is higher than
the vapour pressure.
VALIDATION OF THE SIMULATIONS WITH
GLOBAL MEASUREMENTS
The first phase of validation has been performed starting
from global measurements provided by C.R.M.T. These
measurements concern the flow rate within the injector, the
discharge coefficients which encompass flow reductions
due to singular pressure losses and cavitation, and the
injection velocities.
Geometries considered
Calculations have been performed on an injector of 1 mm
length and 0.2 mm diameter. The needle is in place at its
maximum lift equal to 0.45 mm. Two entrance shapes have
been considered : a straight entrance with a radius of
curvature R=0 and a round entrance with a radius of
curvature R=150 µm.
An uncertainty remains concerning the exact shape of the
straight entrance, related to the machining precision of the
injection hole. So the radius of curvature R0 is certainly not
equal to zero but is probably in the range 0<R<10 µm.
Knowing the significant sensitivity of the inception and
development of cavitation to the entrance shape (Ohrn et al.
[11]), we decided to explore several shapes of the entrance,
acting on the inlet radius of curvature.
So the calculations have been performed for 4 different inlet
shapes (see Figure 18) denoted as:
- R0: perfect straight entrance
- R10: very slightly rounded entrance with R=10 µm
- R50: moderately rounded entrance with R=50 µm
- R150: strongly rounded entrance with R=150 µm
0.0006 0.0008 0.001
-4.000E-04 -2.000E-04 0.000E+00
R150
R50
R0= perfect straight entrance
R10
symmetry axis
Figure 18 : The different profiles of the "numerical
injectors" inlet edges.
The Euler equations have been used in the following
calculations although some tests have been already done
with the full Navier-Stokes equations.
Influence of the injection pressure
The behaviour of the cavitating flow has been studied for
various injection pressures Pinj (assumed steady) for a fixed
chamber pressure Pch. The operating conditions are the
12
following, with X the cavitation number as previously
defined:
XPinj (MPa) Pch (MPa)
4.0 30 6
7.3 50 6
15.6 100 6
The characteristic time scales are very different for the
measurements (integration during several tenths of seconds)
and the simulations which allow to follow the actual
transient phenomena characterized by very short time
scales.
So the nature of the results is different, global for the
measurements and local for the calculations, which doesn't
always allow to perform easy comparisons from the results.
For example, for the exit velocity, the calculation can get
quasi-instantaneous velocity fluctuations due to the
presence of cavitation within the injection channel and
consequently to the induced contraction of the liquid
section, when the experiments give an average exit velocity
deduced from integral thrust measurements. It is however
possible, while integrating on a sufficient duration, to
display average quantities whose comparison with
measurements allow to check the validity of the tendencies
and of the orders of magnitude given by the simulation.
In this paper, the following numerical results will be
considered:
The temporal evolution of the flow rate q(t) at the hole
entrance
The average flow rate given by dttqT )(/1 (where T
represents a simulation time during which the flow is
fully established), from which the discharge coefficient
is calculated.
The exit velocity Vinj integrated on the exit section of
the liquid phase.
Figure 19 shows the transient phase of cavitation set-up in
the channel up to its fully developed regime (low part of
Figure 19), for the R0 injector and an injection pressure
equal to 30 MPa (X=4). (In this figure as in Figures 20 and
21, the velocities have been represented on one cell over
two for a better visibility).
Figure 20 shows the typology of the fully developed
cavitating flow for higher injection pressures equal to 50
and 100 MPa. For moderate injection pressures (30 MPa),
one observes a partial cavitation regime where the vapour
films occupy only a fraction of the length of the channel.
The unstationarity of the phenomena is governed by the
well known mechanisms of the pocket type cavitation: a
liquid reentrant jet is formed in the wake of the vapour film,
detaches this film while flowing backwards along the wall
and periodically cuts it upstream (Figure 19). As a
consequence small cavitation zones are periodically
shedded and convected along the channel, generating
0.0007 0.0 008 0.0009 0.001 0.0011 0.0012 0.0013
-0.0001
0
0.0001
t=2.42 µs
0.0007 0.0 008 0.0009 0.001 0.0011 0.0012 0.0013
-0.0001
0
0.0001
t=6.63 sµ
0.0007 0.0 008 0.0009 0.001 0.0011 0.0012 0.0013
-0.0001
0
0.0001
350m/s
t=5.23 sµ
Figure 19 : Set-up of cavitation for the straight entrance
injector R0 – Pinj=30 MPa, Pch=6 MPa and X=4. The
occurence of reentrant jets along the walls gives rise to
periodic cutting of the cavitation films, so the phenomenon
is unsteady although the boundary conditions are fixed.
periodic contractions of the liquid section which affect
directly the flow rate. This behaviour corresponds
qualitatively to the visualisations of Figure 3 whose
injection pressure is in the same range.
When the injection pressure increases, the effects of the
reentrant jets diminish and the vapour films have a tendency
to elongate and to cover the whole length of the channel
(Figure 20). The flow tends towards a regime of
supercavitation.
Figure 21 shows, for an injection pressure equal to 30 MPa
(X=4), the different aspects that cavitation can take within
the injector according to the inlet shape of the channel. As
soon as the entrance is slightly rounded (R=10 µm), the
recirculation phenomena (induced by the straight edge for
the injector R0) are suppressed as also the effects of
reentrant jets responsible of the detachment and periodic
break-up of the vapour films. Furthermore, for a still larger
radius of curvature (R=50 µm), one observes a decrease of
the film thickness going up to some cuts along it. This could
explain that for real geometries, according to the local
radius of curvature at the entrance, the cavitation film length
may vary from one inception site to the other (see Figure 3).
The thinning of the cavitation films when the radius of
curvature increases has also been observed experimentally
by Badock et al. [52].
13
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
X=4 - Pinj= 30MPa
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
X=7.3 - Pinj=50MPa
350m/s
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
X=15.6 - Pinj=100MPa
Figure 20 : Effect of the injection pressure for the straight
entrance injector R0, Pch=6 MPa – supercavitation at the
higher injection pressures.
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
Injector R0
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
Injector R50
0.001 0.00 15
-0.0002
-0.0001
0
0.0001
0.0002
250m/s
Injector R10
Figure 21 : Influence of the inlet shape on cavitation –
Pinj=30 MPa and X=4
For X=4 and R=150 µm the pressure losses at the entrance
of the channel are not sufficient to get cavitation and this
confirms the observations of Figure 15. For X=16 (not
represented here), one observes cavitation films going along
the inner wall from the inlet to the outlet (supercavitation)
and a continuous decrease of the film thickness when the
radius of curvature increases. Furthermore, cavitation is
present along the walls even for R0=150 µm.
The quantitative comparisons concerning the flow rates, the
discharge coefficients and the exit velocities as previously
defined are presented in Figure 22. These comparisons
should be analysed while taking into account the value of
the inlet radius of curvature. So one compares directly :
For the straight entrance injector: the numerical
injectors R0 and R10 (slightly rounded) with the
injector R0 of CRMT.
For the strongly rounded entrance, the numerical
injector R150 with injector R150 of CRMT.
The calculations on injector R50 with a moderately
rounded entrance constitute an intermediate case
between the two preceding configurations.
0 5 10 15 20
X
5E-06
6E-06
7E-06
8E-06
9E-06
1E-05
1.1E-05
1.2E-05
1.3E-05
1.4E-05
1.5E-05
Eole R0
Eole R10
CRMT R0
Eole R50
Eole R150
CRMT R150
Flow rate (m3/s)
0 5 10 15 20
X
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
Eole R0
Eole R10
CRMT R0
Eole R50
Eole R150
CRMT R150
Cd
14
0 50 100 150 200
Radius of curvature (
100
200
300
400
500
600
Vinj(m/s)
m)µ
Eole
CRMT
Bernouilli
X=15.6
X=7.3
X=4
Figure 22 : Comparisons calculations-measurements for the
flow rates, Cd and injection velocities
For the straight inlet configuration, the calculation gives
values of the flow rate and of the discharge coefficient very
close to the measurements with maximum discrepancies (for
X=15.6) lower than 10%. The numerical injector R10 is the
one which appears to have the closest characteristics to the
experimental injector R0. This confirms that the
experimental injector R0 doesn't possess a perfectly sharp
inlet edge.
The results provided by the model for the rounded inlet
configurations are in agreement with the measurements
since they indicate that the flow rates and discharge
coefficients increase when the radius of curvature increases.
The increase of the radius of curvature reduces the pressure
losses due on one hand to the geometrical effects and on the
other hand to a limited development of cavitation, which
appears as thinner vapour films. So a gain in the order of 20
% of the flow rate is evidenced from the calculations, with
the injector R150 as compared to the injector R0, at all the
operating conditions. The measurements give similar gains
comprised between 15 and 20% from one injector to the
other.
Finally, the comparisons in Figure 22 for the calculated and
experimental injector R150 show very close results (less
than 3% discrepancy). For all the injector configurations
(straigth and rounded entrances), the computed velocities
are also very close to the measurements (less than 10%
discrepancy) and for the three injection pressures slightly
less than the corresponding Bernoulli velocity.
Influence of the ambient pressure
For a straight entrance injector and a slightly rounded
injector R10 and a fixed P equal to 44 MPa, that is a fixed
Reynolds number, the effect of the cavitation number has
been explored as shown in the following table:
X 7.3 8.6 10.7 14 21 40 440
Pinj 50 49 48 47 46 45 44
Pch 6543210.1
The comparisons calculations-experiments concerning the
variation of the flow rate and the discharge coefficient Cd
as a function of X are given in Figure 23.
The results show a correct global coherence of the
numerical results, in particular a decrease of the flow rate
and of the discharge coefficient when X increases, that is
when the chamber pressure diminishes and the pressure
within the injection channel becomes closer to the vapour
pressure, and so the cavitation regime is more pronounced.
These results confirm also that the performances in terms of
flow rate and Cd of the experimental injector R0 are
comprised between those of the numerical injectors R0 and
R10
10
0
10
1
10
2
10
3
X
6.5E-06
7E-06
7.5E-06
8E-06
8.5E-06
9E-06
9.5E-06
1E-05
Eole R0
Eole R10
exp. CRMT R0
Flow rate (m3/s)
10
0
10
1
10
2
10
3
X
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
Eole R0
Eole R10
exp. CRMT R0
Cd
Figure 23 : Comparisons for flow rates and Cd
versus cavitation number
(P=44 MPa=cst but Pinj and Pch variable)
15
VALIDATION OF THE SIMULATIONS WITH
LOCAL MEASUREMENTS WITHIN THE
INJECTOR
Study of the pockets emission frequency
The aim here is to focus on the unsteady phenomena of the
injector flow characterized more specifically by the periodic
emission of pockets (or bubbles), originating from the
break-up of the cavitation films by liquid reentrant jets. As
we said in the introduction, the modulation of the flow rate
itself can give a significant contribution to spray break-up
and it is important to have access to the amplitude and
frequency of these fluctuations.
A low injection pressure case (Pinj=1.65 MPa into ambient
pressure, X=15.5) has been considered. For this case, one
can be sure that partial cavitation will take place (of the
same nature as the one observed in Figure 19).
The transient evolution of the liquid section at a distance of
1.7 diameters upstream of the hole exit is now presented in
Figure 24. The peaks directed towards low values are
representative of periodic reductions of the liquid section
due to the passage of gaseous cavities and inversely the
values equal to the orifice cross section, 3.14e-08 m2,
indicate the absence of vapour.
0.0008 0.0009 0.001 0.0011
t(s)
1.5E-08
2E-08
2.5E-08
3E-08
3.5E-08
Liquid
section (m2)
Figure 24 : Evolution of the liquid section situated at 1.7D
of the exit
This raw temporal signal is not easy to analyse but it shows
however that the passage of cavities at this location is not
chaotic but has a quasi-periodical behaviour. In order to
identify characteristic frequencies, a Fourier decomposition
with sliding windows has been applied.
In the frequential domain, the signal shows several peaks
but a characteristic frequency of the phenomenon appears at
29 KHz (Figure 25). This value was compared to the
measurements of I.F.F, obtained by an optical method
previously described. These measurements give in the mean
frequencies in the range of 20 to 30 KHz (Figure 26), close
to the numerical values. These first results of the validation
of the transient aspects of the flow look very promissing.
0 100000 200000 300000
Frequency (Hz)
5E-10
1E-09
1.5E-09
2E-09
2.5E-09
3E-09
3.5E-09
29000 Hz
Signal
Figure 25 : Calculated emission frequencies of the cavities
-0.4 -0.2 0 0.2 0.4
r/D
0
5
10
15
20
25
30
35
40
45
50
Occurence frequency (kHz) of the cavitation bubbles
Mean frequency = 24.9 kHz, standard deviation = 6.2kHz
Numerical results
= 29 kHz
Figure 26 : Measurements of the emission frequencies of
the cavities at various radial positions.
Velocity profile in the injection channel
Using the optical method previously described (see Figure
7), velocity profiles have been measured radially within the
injection channel. The studied case corresponds to an
injection pressure equal to 36.1 MPa in a chamber at
ambient pressure and corresponds to the visualisations of
Figure 3.
Figure 27 shows the calculated velocity profiles compared
to the measurements. The experimental profile is not
symmetrical and this confirms the doubts already evoked
concerning the precision of machining of the inlet (non
uniformity of the edge at the channel inlet leading to
unsymmetry of the flow).
The levels of the computed axial velocities in the central
part of the channel are similar to the measured ones with
discrepancies smaller than 10 % according to the radial
position. Near the walls, the conclusions are more delicate
to draw due to the experimental uncertainties inherent to the
16
optical methodology used in a part essentially covered by
the vapour film.
-0.5 -0.25 0 0.25 0.5
r/D
0
50
100
150
200
250
300
350
400
450
500
axial velocity (m/s)
EOLE
I.F.F. measurements
thickness of the
cavitation film
(EOLE)
Figure 27 : Comparisons of the calculated velocity profile at
a distance equal to 0.6D from the hole exit with the
measurements at various radial positions.
3D TWO-PHASE FLOW IN A VCO INJECTOR
We will present here a calculation of the cavitating flow
within a centered VCO injector having 5 holes, of 1 mm
length and 0.2 mm diameter. Due to symmetry conditions,
the calculation is done on 1/5th of the geometry. The mesh
is of the multi-domain type (here comprising 4 sub-
domains) and the needle displacement is taken into account
thanks to a technique of mesh deformation. The following
example corresponds to an injection pressure equal to 30
MPa and a chamber pressure equal to 6 MPa (cavitation
number X=4).
A slight swirl flow has been given at the injector entrance in
order to avoid generating two unphysical counter rotating
vortices in the orthogonal cuts of the injector channels.
Figure 28 shows the pressure and velocity fields during the
opening phase and at full lift. The zone of strongest
depression where cavitation is going to start is located in the
separation zone at the upper side of the injection channels.
Then the cavitation zones develop as pockets or films in a
regime of partial cavitation, in an unsymmetrical way along
the upper walls of the injection channels (Figure 29). This
behaviour resembles the "canopy" shape observed by
Arcoumanis et al. [15] on a similar VCO injector. These
films break-up quasi-periodically as in a one hole injector,
which leads to fluctuations of the injection velocity. Intense
vortices are observed in the channels at the same time.
Beginning of the nee dle lift Maximum needle lift
20 58 97 135 173 212 250
mod U
(m/s)
0 3 6 9 12 16 19 22 25 28
P(Mpa)
Pressure
Velocity
250m/s
Figure 28 : Pressure and velocity fields at the beginning of
needle lift and at full lift (Pinj=30 MPa, Pch=6 MPa, X=4).
r=0
r= -R / 2 r=R/2
cavitation
Fragmentatio n of the
cavitat ion film
Entry sectio n of
the nozzle cavitation
Figure 29 : Location of the cavitation pockets a few time
after needle opening (Pinj=30 MPa, Pch=6 MPa, X=4).
17
Furthermore the distribution of the radial velocity
component on the hole exit is strongly heterogeneous (see
Figure 30) and associated with an heterogeneity of the
injection velocity modulus. In this particular case the axis of
the injected spray will depart from the injection channel
axis. Then the spray angle will have different values
according to the observation direction as noted in [53].
exit section
mod U
(m/s)
10
8
6
4
1
-1
-3
-6
-8
-10
V: Radial
component (m/s)
130
126
121
117
112
108
103
99
94
90
exit section
inlet of the
channel
Figure 30: Velocity field and distributions of the velocity
modulus in three cuts of the injection channel. Distribution
of the radial velocity component in the exit section.
(Pinj=30 MPa, Pch=6 MPa, X=4).
THREE-PHASE MODELLING OF SPRAY BREAK-
UP
The interest of the EOLE code methodology is to be
adapted to the calculation of the two-phase flow in both the
injector itself and the region downstream from the injector
exit. The direct calculation of the spray break-up at the exit
of a one hole injector was performed for an injection
pressure equal to 100 MPa and a chamber pressure equal to
6 MPa. For the time being this calculation is 2D
axisymmetrical and the aim is more specifically to put into
evidence some complex mechanisms related to spray
atomization under the effect of the convection of cavitation
pockets within the injection channel.
The modelling is here of 3-phase type involving the
calculation of the flow into the three following phases
(liquid and vapour Diesel fuel, and external gas) and the
tracking of the liquid-vapour Diesel fuel and liquid-gas
interfaces. Two VOF functions are taken into consideration
in each cell, and the tracking of all the interfaces is realized
with the SL-VOF method as described previously.
As long as the cavitation pockets travel through the
injection channel, low perturbations of the jet interface are
observed except the thickening of the jet tip under
dynamical effects against the gas (Figure 31). This is a well
known effect observed by many authors (Eifler [27],
Stojanoff [54]).
Figure 31: 3-phase modelling (liquid, Diesel fuel vapour,
external gas) of injection (Pinj=100 MPa, Pch=6 MPa,
X=15.6).
When the cavitation pockets (whose internal pressure is
close to Diesel fuel vapour pressure) arrive beyond the
injector exit, they collapse when they are submitted to the
strong pressure of the external gas within the chamber and
then they corrugate strongly the interface (Figure 32).
Fragments and ligaments are also emitted. The external gas
entrainment is increased and gaseous pockets may be
observed within the spray which is now a 3-phase flow.
This increase of air entrainment for transient injection has
been also observed by Fath et al. [2], Cossali et al. [55].
This type of calculation still keeps the limitations due to the
2D hypothesis (toroidal shapes of the emitted fragments)
and will have to be undertaken again in 3D conditions to
draw more reliable conclusions.
Figure 32 : Direct calculation of spray break-up at the
injector exit (Pinj=100 MPa, Pch=6 MPa, X=15.6)
18
CONCLUSION
The experiments and calculations discussed in this paper
allow a better understanding of cavitation features and links
with atomization.
(1) The experimental pictures demonstrate how remarkably
reproducible the cavitation structures are.
(2) Cavitation was observed within the hole as well as
downstream of the nozzle exit in the case of injection of
Diesel fuel into Diesel fuel.
(3) For injection of Diesel fuel into a gaseous atmosphere,
the collapse of cavitation is likely to occur very close to the
nozzle exit, because smaller fuel masses are easier to be
accelerated outside the channel.
(4) The injection velocity was measured within and
immediately outside of the nozzle in the very dense spray
region applying a new experimental technique called
" Imaging correlation velocimetry". This technique was also
used to measure the frequencies of detachment of the
cavitation pockets within the nozzle channel.
(5) Numerical and experimental investigations show that the
cavitation structures are highly linked to the nozzle inlet
imperfections.
(6) A high hole inlet edge curvature decreases notably the
length of the cavitation film in the channel. Under a large
hydroerosion rate, cavitation onset seems to be caused by
wall defects and appears rather towards the channel exit.
For moderate injection pressure, cavitation may even be
suppressed by a large hydroerosion rate.
(7) The hydroerosion effects on the discharge coefficient
were quantified as function of the cavitation number. When
the cavitation number increases, it was shown that the
discharge coefficient decreases and reaches a limiting value,
that the injection velocity tends towards the Bernoulli
velocity and that the hydraulic section at the exit displays
the same behaviour as the discharge coefficient.
(8) Other things being equal, the injection velocity increases
for a rounded entrance compared to a straight entrance,
even for cavitating conditions.
(9) A Multiphase flow code EOLE has been adapted to the
Diesel injection cavitating problems. The VOF interface
tracking method has been improved by using a Segment
Lagrangian Volume of Fluid concept (SL-VOF). Also, a
simplified semi-empirical mass transfer model was
implemented into the EOLE code to take into account phase
equilibrium at the liquid-vapour interface.
(10) Numerical result visualisations correspond
qualitatively to the experimental images of cavitation.
(11) Numerical tests confirm that the cavitation rate
depends strongly on the local inlet edges state.
(12) Results provided by the two phase flow model well
agree with the experiments in terms of mass flow rate,
discharge coefficient and exit velocity. As well, both the
emission frequency of the cavitation and the velocity profile
in the channel are relatively well predicted.
(13) Finally, the 3D VCO injector calculation and the three
phase modelling of the spray atomization show that the
model exhibits a good behaviour and may help engineers to
better estimate boundary conditions for CFD engine
calculations and to better understand the link between
cavitation and spray atomization.
AKNOWLEDGEMENTS
This work was supported by the Groupement Scientifique
Moteurs (Renault, PSA, IFP). The authors thank Pr.
Obermeier and Pr. Fruman for many fruitful discussions.
We aknowledge the help of C. Kirmse at IFF and B.
Compagnon at CRMT who performed the measurements.
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... An alternative approach for modeling the coexistence of three phases is by employing the VOF, with a high-resolution interface capturing scheme such as the one of [35]; this approach can be advantageous for modeling atomization. We are aware of five studies available in the literature that attempted to link a two-phase VOF model with a cavitation model for studying the in-nozzle effects on atomization [36][37][38][39][40]. These models differ in the way cavitation is resolved. ...
... A comparative study between two transport-based cavitation models [10,11] and employing VOF can be found in [40] for a single-hole solid cone injector. Further studies that assume the phases to be incompressible can be found in [36,39]. An Eulerian-Eulerian cavitation model with VOF was used to study cavitation and liquid jet breakup in a step nozzle in [36]. ...
Preprint
Full-text available
The aim of this paper is to present a fully compressible three-phase (liquid, vapor, and air) model and its application to the simulation of in-nozzle cavitation effects on liquid atomization. The model employs a combination of the homogeneous equilibrium barotropic cavitation model with an implicit sharp interface capturing volume of fluid (VOF) approximation. The numerical predictions are validated against the experimental results obtained for injection of water into the air from a step nozzle, which is designed to produce asymmetric cavitation along its two sides. Simulations are performed for three injection pressures, corresponding to three different cavitation regimes, referred to as cavitation inception, developing cavitation, and hydraulic flip. Model validation is achieved by qualitative comparison of the cavitation, spray pattern, and spray cone angles. The flow turbulence in this study is resolved using the large-eddy simulation approach. The simulation results indicate that the major parameters that influence the primary atomization are cavitation, liquid turbulence, and, to a smaller extent, the Rayleigh-Taylor and Kelvin-Helmholtz aerodynamic instabilities developing on the liquid-air interface. Moreover, the simulations performed indicate that periodic entrainment of air into the nozzle occurs at intermediate cavitation numbers, corresponding to developing cavitation (as opposed to incipient and fully developed cavitation regimes); this transient effect causes a periodic shedding of the cavitation and air clouds and contributes to improved primary atomization. Finally, the cone angle of the spray is found to increase with increased injection pressure but drops drastically when hydraulic flip occurs, in agreement with the relevant experiments.
... An alternative approach for modeling the coexistence of three phases is by employing the VOF, with a high-resolution interface capturing scheme such as the one of [35]; this approach can be advantageous for modeling atomization. We are aware of five studies available in the literature that attempted to link a two-phase VOF model with a cavitation model for studying the in-nozzle effects on atomization [36][37][38][39][40]. These models differ in the way cavitation is resolved. ...
... A comparative study between two transport-based cavitation models [10,11] and employing VOF can be found in [40] for a single-hole solid cone injector. Further studies that assume the phases to be incompressible can be found in [36,39]. An Eulerian-Eulerian cavitation model with VOF was used to study cavitation and liquid jet breakup in a step nozzle in [36]. ...
Article
The aim of this paper is to present a fully compressible three-phase (liquid, vapor, and air) model and its application to the simulation of in-nozzle cavitation effects on liquid atomization. The model employs a combination of the homogeneous equilibrium barotropic cavitation model with an implicit sharp interface capturing volume of fluid (VOF) approximation. The numerical predictions are validated against the experimental results obtained for injection of water into the air from a step nozzle, which is designed to produce asymmetric cavitation along its two sides. Simulations are performed for three injection pressures, corresponding to three different cavitation regimes, referred to as cavitation inception, developing cavitation, and hydraulic flip. Model validation is achieved by qualitative comparison of the cavitation, spray pattern, and spray cone angles. The flow turbulence in this study is resolved using the large-eddy simulation approach. The simulation results indicate that the major parameters that influence the primary atomization are cavitation, liquid turbulence, and, to a smaller extent, the Rayleigh-Taylor and Kelvin-Helmholtz aerodynamic instabilities developing on the liquid-air interface. Moreover, the simulations performed indicate that periodic entrainment of air into the nozzle occurs at intermediate cavitation numbers, corresponding to developing cavitation (as opposed to incipient and fully developed cavitation regimes); this transient effect causes a periodic shedding of the cavitation and air clouds and contributes to improved primary atomization. Finally, the cone angle of the spray is found to increase with increased injection pressure but drops drastically when hydraulic flip occurs, in agreement with the relevant experiments.
... An alternative approach for modelling the co-existence of three-phases is by employing the Volume of Fluid (VoF). To the author's knowledge, there are five studies available in literature, that attempted to link a two-phase VoF model with a cavitation model for studying the in-nozzle effects on atomization (2,3,(17)(18)(19). These models differ in the way cavitation and compressibility is treated. ...
... A comparative study of two transport-based cavitation models with application of VoF can be found in (19) for atomization occurring from a single solid cone injector. Further studies that assume the phases to be incompressible can be found in (3,17). In this study, we present a compressible three-phase model which considers compressibility of both the mixture and pure phases, using non-linear isentropic relations. ...
... The widely-applied sharp interface capturing method Volume of Fluid (VOF) technique [36,43,46,47] has been chosen to capture the liquid/gas interface. The model is similar to the homogeneous model, where a single momentum equation is calculated for all phases that interact using the VOF model. ...
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... However, compared with the multi-phase flow model, less flow field information is obtained. Thus, Marcer [27] used the improved volume of fluid (VOF) method to study the vapor-liquid two-phase flow, and he found that the simulation result provided a good fit to that of the experimental. Then, Alajbegovic [28] verified that cavitation was caused by the decrease of fuel pressure via the two-fluid model and believed that the two-fluid model was an engineering tool suitable for large-scale processes. ...
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The fuel flow in the diesel engine nozzle has a vital impact on the fuel atomization and spray, and the fuel mass flux affects the internal flow of the nozzle. The visual experimental platform for a transparent nozzle was built to obtain the image of fuel flow in a nozzle with a small sac combining the back-light imaging technology and a high-speed framing camera. A two-phase three-component numerical model, based on the OpenFOAM solver, was calculated to quantitatively analyze gas ingestion and cavitation in the nozzle. The results indicate that at the end of injection (EOI), fuel cavitation and external air backflow (gas ingestion) occur successively in the nozzle, and both phenomena first appear in the orifice and then transition to the sac. Cavitation collapse is the major factor of gas ingestion, and the total amount of gas ingestion and cavitation mainly depends on the sac. The outflow of fuel largely depends on the total amount of cavitation and the inertial outflow of fuel at the EOI. The type of cavitation in the nozzle mainly presents annular and bulk cavitation, the former primarily exists in the sac, while the latter is established within the orifice. Therefore, larger mass flows will contribute to stronger cavitation and gas ingestion.
... Even though VOF model is computationally expensive compared to the mixture model, 32 the former is employed here to accurately capture sharp interface between different phases during the purging process. [33][34][35] The volume fraction for a phase is solved using mass continuity of a particular phase (equation (1)). For the jth phase, the equation is ...
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... e. at low injection pressures and ambient densities). The VOF method has been applied since 2000 for internal and external spray flow computations [4,5,6,7,8,9,10], due to the accurate interface-tracking capability across all the phases involved (up to three in cavitating injectors). However, the extremely high computational cost required for the resolution of the spray breakup process does not allow the VOF integration into complete engine cycle modeling. ...
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