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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.

REFERENCES

[1] Soteriou C., Andrews R. and Smith M. "Direct injection

Diesel sprays and the effect of cavitation and hydraulic flip

on atomization" SAE Paper 950080, 1995.

[2] Fath A., Fettes C., Leipertz A. "Investigation of the

Diesel spray break-up close to the nozzle at different

injection conditions", 4th Int. Symp. COMODIA 98, 1998.

[3] Bruneaux G. "Liquid and vapor spray structure in high

pressure common rail Diesel injection", submitted to

Atomization and sprays, 2000.

[4] Lee E., Huh K., Koo J-Y. "Development of a Diesel

spray atomization model considfering nozzle flow

characteristics",

Proc. ICLASS '97, Seoul, August 18-22, 1997.

[5] Chaves H., Obermeier F. "Modelling the effect of

modulations of the injection velocity on the structure of

Diesel sprays", SAE 950426, 1995.

[6] Dumont N., Simonin O. and Habchi C. "Cavitating flow

in Diesel injectors and atomization : a bibliographical

review", 8th Int. Conf. on liquid atomization and spray

systems (ICLASS), Pasadena CA, July 2000.

[7] Chaves H., Knapp M., Kubitzek A., Obermeier F. and

Schneider T. "Experimental study of cavitation in the

nozzle hole of Diesel injectors using transparent nozzles"

SAE Paper 950290, 1995.

[8] Meingast U., Staudt M., Hohmann S., Renz U.

"Untersuchung eines Common-Rail Einspritzstrahls im

Düsennahbereich", Spray '98, Essen, 13-14/10/9

[9] Winklhofer E. "Velocities and structure of atomizing

Diesel sprays", ILASS Florence, july 1997.

19

[10] Bergwerk W. "Flow pattern in diesel nozzle spray

holes", Proc. Instn. Mech. Engrs., Vol. 173, n° 25, pp 655-

674, 1959.

[11] Ohrn T., Senser D. and Lefebvre A. "Geometrical

effects on discharge coefficients for plain-orifice atomizers"

Atomization and Sprays, vol. 1, n° 2, pp 137-153, 1991.

[12] Genge O. "Untersuchung und Quantifizierung

strömungsdynamischer Effekte bei der Kavitation in einer

planaren Düse", Diss. Aachen, 1998.

[13] Karasawa T., Tanaka M., Abe K., Shiga S. and

Kurabayashi T. "Effect of nozzle configuration on

atomization of a steady spray", Atomization and Sprays,

vol. 2, pp 411-426, 1992.

[14] Afzal H., Arcoumanis C., Gavaises M. and Kampanis

N. "Internal flow in Diesel injector nozzles: modelling and

experiments", IMechE Int. Conf. on Fuel injection Systems,

London, 1-2 december, 1999.

[15] Arcoumanis C., Badami M., Flora H. and Gavaises M.

"Cavitation in real-size multi-hole Diesel injector nozzles",

SAE Paper 2000-01-1249, 2000.

[16] He L. and Ruiz F. "Effect of cavitation on flow and

turbulence in plain orifices for high-speed atomization",

Atomization and Sprays, vol. 5, pp. 569-584, 1995.

[17] Lush P.A "Dynamics of cavities in a duct flow", Int. J.

Heat and Fluid Flow, vol. 9, n° 1, march 1988.

[18] Schmidt D., Rutland C., Corradini M., Roosen P. and

Genge O. "Cavitation in two-dimensional asymmetric

nozzles", SAE Paper 1999-01-0518, 1999.

[19] Alajbegovic A., "AVL's Two phase flow simulation

activities – Status and Outlook", Fire user's meeting, Graz

1999.

[20] Henry M., Collicot S., Heister S. "Internal structure of

cavitating slot flow", ILASS America, Ottawa, Ontario

1997.

[21] Kim J-H., Nishida K., Yoshizaki T. and Hiroyasu H.

"Characterization of flows in the sac chamber and the

discharge hole of a D.I Diesel injection nozzle by using a

transparent model nozzle", SAE Paper 972942, 1997.

[22] Arcoumanis C., Gavaises M., Nouri J. M., Abdul-

Wahab E. and Horrocks R. "Analysis of the flow in the

nozzle of a vertical multi-hole Diesel engine injector", SAE

Paper 980811, 1998.

[23] http//leo.iff.tu-freiberg.de/Forschung/Projekte

/Injektion/diesel_cav.html

[24] Tamaki N., Shimizu M., Nishida K., Hiroyasu H.

"Effects of cavitation and internal flow on atomization of a

liquid jet", Atomization and Sprays, vol. 8, pp. 179-197,

1998.

[25] Hiroyasu H., Arai M., and Shimizu M. "Break-up

length of a liquid jet and internal flow in a nozzle". Proc. of

the 5th Int. Conf. on liquid atomization and spray system,

(ICLASS-91), pp 275-282, 1991

[26] Prescher K. and Schaffitz W. "Verschleiss von

Kraftstoff-Einspritzdüsen für Dieselmotoren infolge

Kraftstoff- Kavitation", MTZ, Motortechnische Zeitschrift

40 (1979] 4

[27] Eifler W. "Untersuchungen zur Struktur des

instationären Dieselöleinspritzstrahles im Düsennahbereich

mit der Methode der Hochfrequenz-Kinematografie", Diss.

Kaiserslautern, 1990.

[28] Chaves H. and Obermeier F. "Correlation between

light absorption signals of cavitating nozzle flow within and

outside of the hole of a transparent diesel injection nozzle",

ILASS-Europe '98, Manchester 6-8 july 1998.

[29] Favennec A.G and Fruman D. "Effect of the needle

position on the cavitation of Diesel injectors", 3rd

ASME/JSME Joint Fluids engineering conference, July 18-

23, San Francisco, California.

[30] Chaves H., Mulhem B. and Obermeier F. "Comparison

of high speed stroboscopic pictures of spray structures of a

Diesel spray with numerical calculations based on a

convective instability theory", ILASS-Europe '99, Toulouse

5-7 july 1999.

[31] Nakahira T., Komori M., Nishida M. and Tsujimura K.

"The shock wave generation around the Diesel fuel spray

with high pressure injection", SAE Paper 920460, 1992.

[32] Huh K.Y. and Gosman D. "A phenomenological model

of Diesel spray atomization", Proc. Int. Conf. on Multiphase

flow, Tsukuba, Japan, 24-27 September 1991.

[33] Arcoumanis C., Gavaises M. and French V. "Effect of

fuel injection processes on the structure of sprays", SAE

Paper 970799,1997.

[34] Kampmann S., Dittus B., Mattes P. and Kirner M.

"The influence of hydro grinding at VCO nozzles on the

mixture preparation in a DI Diesel engine", SAE 960867,

1996.

[35] Ventikos Y., Tzabiras G. "A numerical study of the

steady and unsteady cavitation phenomenon around

hydrofoils", International Symposium on Cavitation,

CAV’95, 2-5 May Deauville, France, 1995.

[36] Delannoy B., Kueny H. "Two phase flow approach in

unsteady cavitating modeling". ASME Cavitation and

Multiphase flow forum, 98:153-158, 1990.

[37] Schmidt D., Rutland J., Corradini M. "A numerical

study of cavitating flow through various nozzle shapes".

SAE 971597, 1997.

[38] Kubota A. Kato H., Yamaguchi H. "A new modeling

of cavitated flow : a numerical study of unsteady cavitation

on a hydrofoil section", J. Fluid Mech., Vol. 240, pp.59-96,

1992.

[39] Chen Y., Heister S.D. "Modeling cavitating flows in

diesel injectors", Atomisation and Sprays, vol. 6, pp. 709-

726, 1996.

[40] Grogger H.A., Alajbegovic A. "Calculation of the

cavitating Flow in venturi geometries using two fluid

model", Proceed. Of FEDSM’98, ASME Fluids Eng.

Division Summer Meeting, June 21-25, Washington,

D.C.,USA, 1998.

[41] Alajbegovic A., Grogger H.A. "Calculation of transient

cavitation in nozzle using the two-fluid model", 12th

Annual Conference on Liquid Atomization and Spray

Systems, May 16-19, Indianapolis, Indiana, USA, 1999.

20

[42] Von Dirke M., Krautter A., Ostertag J., Mennicken M.

and Badock C. "Simulation of cavitating flows in Diesel

injectors", Multidimensional simulation of engine internal

flows, Rueil-Malmaison, France (1998).

[43] Guignard S., Marcer R., Rey V., Fraunié P. "Solitary

wave breaking on sloping beaches : two phase flow

numerical simulation by SL-VOF method", accepted to Eur.

J. of Mec., 1999.

[44] Diéval L., Marcer R., Arnaud M. "Modélisation de

poches de cavitation par une méthode de suivi d’interface

de type VOF", La Houille Blanche, n°4/5, 1997.

[45] Brackbill J.U., Kothe D.B., Zemach C. "A continuum

method for modeling surface tension", J. Comp. Phys., vol.

100, pp. 335-354, 1992.

[46] Kolcio K. and Helmicki A.J. "Development of

Equations of State for Compressible Liquids, J.Propulsion.

Vol.12. No. 1 , Technical Notes, 1995.

[47] Viviand H. "Analysis of pseudo-compressibility

systems for compressible and incompressible flows",

Computational Fluid Dynamics, Review 1995, Wiley

publishers, Editors Halez Oshima.

[48] De Jouette C., Viviand H., Wornom S., Le Gouez J.M.

"Pseudo compressibility method for incompressible flow

calculation", 4th International Symposium on Comp. Fluid

Dyn., of California at Davis, sept. 9-12, 1991.

[49] Jameson A., Schmidt A., Turkel E. "Numerical

Solutions for the Euler Equations by finite volumes methods

using Runge-Kutta Time-Marching Schemes", AIAA 14 Th.

Fluid and Plasma Dyn., Conf. Palo Alto California, June

1981.

[50] Hirt C.W., Nichols B.D "Volume Of Fluid (VOF)

Method for the Dynamics of Free Boundaries", J. of Comp.

Physics, Volumes 39, pages 201-255, 1981.

[51] Li J. "Piecewise Linear Interface Calculation", C. R.

Acad. Sci. Paris, t. 320, Serie II b,pp 391-396., 1995.

[52] Badock C., Wirth R. and Tropea C. "The influence of

hydro grinding on cavitation inside a Diesel injection nozzle

and primary break-up under unsteady pressure conditions",

ILASS-Europe '99, Toulouse, July 5-7 1999.

[53] Heimgärtner C. and Leipertz A. "Investigation of the

primary spray breakup close to the nozzle of a common-rail

high pressure Diesel injection system", SAE 2000-01-1799

[54] Stojanoff C., Schaller J., Bilstein W. "Holographische

Untersuchungen der Dieselmotorischen Einspritzung",

Kollokium des Sondersforschungsbereichs 244 "Motorische

Verbrennung", Aachen, 19-20/3/1996

[55] Cossali G., Brunello G. and Coghe A. "LDV

characterization of air entrainment in transient Diesel

sprays¨", SAE paper 910178, 1991.