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Vortex ring-like structures in gasoline fuel
sprays under cold-start conditions
S Begg
1
*, F Kaplanski
2
, S Sazhin
1
, M Hindle
1
,and M Heikal
1
1
Centre for Automotive Engineering, University of Brighton, Brighton, UK
2
Laboratory of Multiphase Physics, Tallinn University of Technology, Tallinn, Estonia
The manuscript was accepted after revision for publication on 26 March 2009.
DOI: 10.1243/14680874JER02809
Abstract: A phenomenological study of vortex ring-like structures in gasoline fuel sprays is
presented for two types of production fuel injectors: a low-pressure, port fuel injector (PFI) and
a high-pressure atomizer that injects fuel directly into an engine combustion chamber (G-DI).
High-speed photography and phase Doppler anemometry (PDA) were used to study the fuel
sprays. In general, each spray was seen to comprise three distinct periods: an initial, unsteady
phase; a quasi-steady injection phase; and an exponential trailing phase. For both injectors,
vortex ring-like structures could be clearly traced in the tail of the sprays. The location of the
region of maximal vorticity of the droplet and gas mixture was used to calculate the temporal
evolution of the radial and axial components of the translational velocity of the vortex ring-like
structures. The radial components of this velocity remained close to zero in both cases. The
experimental results were used to evaluate the robustness of previously developed models of
laminar and turbulent vortex rings. The normalized time, t
¯5t/t
init
, and normalized axial
velocity, V
¯
vx
(t
¯)5V
vx
(t
¯t
init
)/V
vx
(t
init
), were introduced, where t
init
is the time of initial
observation of vortex ring-like structures. The time dependence of V
¯
vx
on t
¯was approximated
as V
¯
vx
(t
¯)5t
¯
22.97
and V
¯
vx
(t
¯)5t
¯
21.14
for the PFI and G-DI sprays respectively. The G-DI spray
compared favourably with the analytical vortex ring model, predicting V
¯
vx
5t
¯
2a
, in the limit of
long times, where a53/2 in the laminar case and a53/4 when the effects of turbulence are
taken into account. The results for the PFI spray do not seem to be compatible with the
predictions of the available theoretical models.
Keywords: gasoline engine, vortex ring, spray, phase Doppler
1 INTRODUCTION
It is foreseeable that port fuel injector (PFI) gasoline
engines will remain as the primary power source for
small-capacity, lower-class production vehicles ow-
ing to their relatively low cost and fuel economy.
They also have the potential to meet future emission
targets with minimal modification and optimization.
This can be achieved through valvetrain improve-
ments as well as through fuel injection technologies.
In mid-class vehicles, direct injection (DI) combus-
tion systems have shifted focus from air- and wall-
guided strategies to a widely adopted spray-guided
model. Central to these concepts is a repeatable,
well-controlled fuel spray vortex ring that produces a
combustible mixture in the vicinity of the spark plug
gap at the optimum time.
The stability of the combustion process in gasoline
engines is principally governed by the mixture pre-
paration process, whereby air and fuel are homo-
geneously combined at the instant of spark ignition.
Modern engines achieve this condition by the
injection of liquid fuel into the intake port or directly
into the combustion chamber. In both of these cases,
the liquid fuel droplets rapidly disperse throughout
the gaseous phase to ensure that evaporation
occurs within the shortest time-scales available.
In the fuel injection process, liquid fuel exits the
nozzle initially as a continuous liquid jet with only its
*Corresponding author: Sir Harry Ricardo Laboratories, Centre
for Automotive Engineering, Faculty of Science and Engineering,
University of Brighton, Brighton BN2 4GJ, UK.
email: s.m.begg@brighton.ac.uk
195
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
periphery finely atomized. Distortion of the jet
region, through the effects of gas shear flows and
internal radial forces within the liquid, initiate
instabilities on the surface of the liquid that have
been observed experimentally [1,2]. Along the
envelope of the spray plume, liquid accelerates the
surrounding gas in the boundary layer from a
laminar to a turbulent state. The rapid transition in
boundary layer thickness creates local regions of gas
recirculation. Mushroom-like patterns of different
scales are formed. The opposing gas velocity gradi-
ents lead to the entrainment of liquid droplets and
enhance gas flow back into the droplet cloud [3–6].
Several key parameters are commonly used to
describe the dynamics of sprays in automotive
applications. These are the spray tip penetration,
break-up length, spread or cone angle, and the
atomization quality (the spatial and temporal dis-
tribution of droplet sizes and velocities). These
parameters can be used to predict the rate at which
the mixing proceeds, and have been well documen-
ted in the literature for PFI [2,7–12] and DI [4,5,13–
19]. However, the effects of spray-induced vortex
ring-like structures have been generally overlooked,
although these play an important role in the rate at
which the liquid evaporates. It is therefore important
to develop predictive models for engine fuel sprays
that take into account the complex formation of
vortex ring-like structures. The analyses are particu-
larly suited to spray-guided, DI combustion con-
cepts that require fundamental knowledge of vortex
ring formation and translation.
These structures can be either laminar or turbu-
lent depending upon the ambient conditions and the
nozzle geometry. The conventional models of classi-
cal laminar rings have been widely discussed in the
literature [20–27]. Automotive fuel sprays, however,
are characterized by a turbulent periphery affected
by gas–wall interactions. In a previous study [28],
vortex ring behaviour in gasoline engine-like condi-
tions could not be adequately described in terms of
the laminar vortex ring models. Instead, it was
suggested that the features of the vortex rings were
more suited to the turbulent ring model developed
by Lugovtsov [29,30]. A generalized vortex ring
model, incorporating both laminar and turbulent
models, has been developed in reference [31]. In this
paper the properties of the vortex ring-like structures
in gasoline engine-like conditions will be studied
experimentally in more depth. The axial and radial
translational velocities in the vortex ring regions of
maximal vorticity will be compared with velocities
predicted by the generalized vortex ring model,
developed in reference [31], which incorporates
both laminar and turbulent vortex ring models. The
focus of this study will be on the fluid dynamics
characteristics of these sprays. A review of recently
developed models for spray heating and evaporation
is given in reference [32].
In section 2, the experimental methods used to
determine the location and time evolution of the
axial and radial velocities of the regions correspond-
ing to maximal vorticity are described. The experi-
mental results are presented in section 3. In section
4, some predictions of the generalized vortex ring
model are summarized. A comparison between the
experimental results and the model predictions, and
a summary of the main results, are presented in
sections 5 and 6 respectively.
2 EXPERIMENTAL METHODS
2.1 Experimental set-up
Experimental investigations were performed on two
modern production gasoline injectors: a low-pres-
sure PFI (injector A) and a high-pressure, direct fuel
injector G-DI (injector B). The contrasting choice of
fuel injection systems was adopted to highlight the
difference in the spray dynamics and to assess the
robustness of the vortex ring models. Each injector
was controlled by a production injector driver unit.
Fuel was delivered using a production fuel pump
and pressure-regulated fuel rail in both cases. The
high-pressure fuel pump was driven independently
using a direct current (d.c.) motor with variable
speed control. The low-pressure fuel return circuit to
the tank was passed through a shell-in-tube heat
exchanger to maintain the fuel temperature. The
most important criterion for the comparison was
that both injectors were operated under ‘steady
state’ flow conditions (following initial mass flowrate
fluctuations) when the vortex rings were tracked.
Previous studies [28,33–35] have shown that the
identification of specific structures in gasoline fuel
sprays is often obscured by the turbulent gas
boundary layer around sprays or through the
interaction of sprays with the incoming airflows. A
typical example of complex vortex ring-like struc-
tures in a high-pressure, DI gasoline fuel spray (G-
DI), in a motored, optically accessed, single-cylinder
research engine at 1000 r/min, is shown in Fig. 1. In
this study, the formation of vortex ring-like struc-
tures was observed in a large, atmospheric pressure
and temperature, quiescent chamber. For both
injectors, homogeneous engine operating conditions
196 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
occurred when the gas pressure at the start of fuel
injection is close to atmospheric at wide-open
throttle. These conditions were used to determine a
typical fuel injection duration for stoichiometric
engine operation. The specifications of the fuel
injection systems are given in Table 1. The chamber
and fuel conditions were considered representative
of cold-start cranking or early engine warm-up
conditions.
The experimental set-up is illustrated in Fig. 2.
Each injector was mounted vertically in the quies-
cent chamber of square cross-section. The large
windows permitted a full view of the spray without
impingement or boundary interference with the
walls. The injector orientation was carefully selected
to ensure that the measurement plane bisected the
injection axis. A programmable traverse was used to
move the instruments through the spray. The
positional error in the traverse was ¡0.1 mm. The
location of the protruding tip of the injector was
taken as the origin for all measurements. The axes
are defined vertically downwards along the spray
axis and radially, perpendicular to the spray axis as
shown in Fig. 2. Two sets of experiments were
performed to determine the characteristics of vortex
ring-like structures.
2.2 High-speed cine and still photography
A series of high-speed cine films and short-exposure
digital still photographs were used to observe the
formation of vortex ring-like structures. A combin-
ation of a pulsed laser light sheet and backlit
shadowgraphy were applied. A Phantom V7.1 high-
speed camera was used for the cine photography
with framing rates in the range of 14.4–100 kHz. In
addition, a high-resolution charge-coupled device
(CCD) camera (128061024 pixels) was used to
reconstruct the injection event from a series of
sequential, time-stepped, still images captured over
consecutive injections. At each step, 20 images were
acquired to produce a single average image. The
optimum camera exposure time used was 1 ms, and
the laser sheet thickness was limited to approxi-
mately 1 mm. Both cameras were synchronized to
the start of injection (SOI) trigger pulse provided to
the engine management system.
2.3 Phase Doppler anemometry
A Dantec Dynamics Classical two-component phase
Doppler anemometer (PDA) was used to measure
the droplet size and velocity distributions over a
1mm65 mm measurement grid close to the nozzle
and a 2 mm610 mm grid further downstream. The
extents of the grid were determined from a series of
preliminary measurements performed across the
vertical injector axis of symmetry to the periphery
of the spray. Constraints in the experimental layout
Fig. 1 A sequence of high-speed photographs showing
the formation of vortex ring-like structures in a
G-DI spray in a motored optical engine at
1000 r/min. The start of injection timing was
60ucrank angle (CA) after top dead centre
(ATDC) non-firing. The photographs were
recorded at 1.5uCA intervals starting with 60uCA
ATDC. Fuel injection starts from the top left-
hand corner
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 197
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
limited the PDA configuration to side-scatter only. In
addition, forward scattering through the dense spray
was subject to a poorer signal to noise ratio. The
ideal scattering angle for first-order refraction and
parallel polarization with iso-octane was close to 78u.
The PDA processor internal clock was synchro-
nized to the SOI trigger pulse. The time taken from
the start of injection was recorded as t.Non-
coincident data collection was used to maximize
the validated data rate in the axial direction, as this
was of predominant interest in the modelling
approach. The data rates were generally far higher
in this direction. Non-coincident data was collected
for either 10 000 measurements or 60 s duration for
the axial velocity component (V
x
), and 5000 mea-
surements or 60 s duration for the radial velocity
component (V
r
). The trade-off between injection
frequency and data validation rate (data quality) and
total data rate was optimized in the chamber. The
data rate varied with spatial location in the spray and
with spent fuel vapour concentration in the cham-
ber. A frequency of 1 Hz was considered a good
balance between acceptable data validation rate and
experiment duration. However, it was not always
possible to obtain a full data set in sparsely
populated regions on the periphery of the spray or
further from the nozzle, even though the axial
velocity data validation rate was almost 100 per
cent. In the central region, data validation of the
axial velocity component was lower (approximately
70 per cent at worst) but the data rate attained a
maximum of 400 Hz close to the nozzle exit.
Within these constraints, the velocity and droplet
size measurement range was optimized to ensure
that all of the acquired, validated data were within
the range of ¡3 standard deviations of the respective
mean values. The total data validation rates were
greater than 90 per cent in both sprays except for
locations close to the nozzle exit and either side
of the vertical injector axis of symmetry (for
x(10 mm). In these regions, the data validation
rates were of the order of 70 per cent. The op-
Table 1 Fuel injection equipment specifications
Injector A Injector B
Fuel injector type Port (PFI) Direct (G-DI)
Nominal fuel pressure (bar) 3.5 100
Fuel temperature (uC) 22 22
Fuel type Iso-octane (2,2,4 TMP) Iso-octane (2,2,4 TMP)
Injection frequency (Hz) 1 1
Injection duration (ms) 5 2
Injected mass of fuel (mg) 11 17
Air pressure (bar) 1 1
Air temperature (uC) 20 20
Orifice size (mm) 200 250
TMP, trimethylpentane.
Fig. 2 Schematic of the quiescent spray chamber;
locations of the phase Doppler anemometer
(PDA) and traverse
198 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
timization of the PDA parameters and the statistical
accuracy of the measurements, specific to each
spray, formed an important part of the study (see
reference [34] for the details). The optimized PDA
operating parameters are summarized in Table 2.
The temporal spray data at each measurement
node was averaged within 60 equal time bins of
0.25 ms duration from the SOI trigger signal. A
spatial droplet velocity and diameter data field,
ensemble-averaged within each time step of refer-
ence, was then constructed. A weighted, inverse
distance to a power method was then applied to
interpolate the measurement data on to a regularly
spaced, destination grid. It was assumed that the
average value of the vorticity at the destination data
point (j
d
) was given by j
d
5S(j
s
D
2E
)/S(D
2E
), where
Dis the distance from the measured points of
vorticity (j
s
)to the destination points of vorticity
(j
d
). The weighting exponent (E) was taken to be
equal to 3.5 to ensure that closer data points were
given a higher fraction of the overall weighting. The
summation was performed between each point on
the destination grid and the eight closest adjacent
measurement points to smooth out oscillations in
the data field. The accuracy of the optimized
interpolation method was considered good when
the resolution of the destination grid was small
compared with the diameter of the vortex ring-like
structure.
3 RESULTS
3.1 Temporal evolution of droplet velocities and
diameters
In general, the fuel sprays produced by both
injectors were observed to comprise three widely
recognized phases. Initially, a developing phase was
observed where a poorly atomized, unsteady, high-
velocity jet was formed, combined with small
dispersed droplets. This was followed by the main,
quasi-steady period, where the mass flow of liquid
fuel was approximately constant and the position of
the leading edge of spray penetration was observed
to move almost linearly with time. In the final phase,
the spray momentum decayed and vortex ring-like
structures were seen to be formed, translated and
destroyed as the jet collapsed inwards towards the
injector axis.
The main difference between the sprays produced
by both injectors was observed in the relative
phasing and duration of these three periods, for a
given injection pulse width, and the mean droplet
velocities and diameters associated with each phase.
The interpretation of these phases is complicated by
the position of the measurement probe in the spray.
In Figs 3(a) and (b) the development of the droplet
velocities and diameters are shown from the start of
injection for two locations in the PFI spray: along the
spray axis (r50 mm and x515 mm; Fig. 3(a)) and at
the spray periphery (r56 mm and x555 mm;
Fig. 3(b)). In the first case, fuel droplets arrived at
the measurement volume at approximately 2.5 ms
after SOI. There was a large spread in the size
and velocity distribution but generally the higher-
momentum droplets arrived first. Following the
initial phase, there was a small hesitation in the
mean velocity followed by a recovery that initiates
the quasi-steady phase which was dominated by a
clustering of high-velocity droplets, with V
x
approxi-
mately equal to 25 ms
21
and diameters in the range
of 80–100 mm. The trailing edge of the spray shows an
exponential decay in velocities and a reduction of
the mean droplet diameters.
The hesitation in the spray delivery was observed
in the magnified high-speed photography image
sequences in Fig. 4 for t52.45 ms after SOI. Irregu-
lar, axial pulsations were observed in the early
needle-lift phase. Close to the nozzle, partial detach-
ment of the liquid jet was recorded and a void
developed that was transported downstream with
Table 2 Optimized PDA operating parameters for Dantec FibreFlow Classical PDA
Laser Spectra-Physics Stabilite 2017 5W argon-ion
Signal processor Dantec Dynamics BSA P70
Sphericity validation (%) 10
Diameter range (mm) 1.0–114
Refractive index 1.389
Forward scattering angle (degrees) 77.5
Wavelength (nm) 514.5 488
Focal length of objective (mm) 310 310
Measuring volume diameter (mm) 47 44
Measuring volume length (mm) 1.044 0.990
Beam half-angle (degrees) 2.56 2.56
Fringe spacing (mm) 5.76 5.46
Number of fringes 8 8
Velocity range (ms
21
)248 to 82 25to16
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 199
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
Fig. 3 Distribution of droplet diameters and velocities in the PFI injector spray plotted against
time from SOI when (a) r50 mm and x515 mm and (b) when r56 mm and x555 mm
Fig. 4 High-speed cine image sequence (measuring 9618 mm) of liquid fuel injection close to
the nozzle at 100 kHz frame rate in a PFI injector at t52.45, 2.56, 4.07, and 5.62 ms after
SOI
200 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
the mean flow. Comparable characteristics were
observed in the jets of three other identical produc-
tion fuel injectors over a broad range of injection
periods using both the magnified high-speed cine
technique and PDA. The hesitation characteristics
were present to some degree in the same region in
each jet of each injector and occurred independently
of injection duration. In one case it was less evident
in a single jet that showed particularly poor
atomization characteristics. The period of the injec-
tion was chosen to ensure that the measured,
instantaneous, mass flowrate of fuel was approxi-
mately constant. Further from the nozzle, the
developed flow from the main injection period
merged with the liquid issued in the initial stages
of injection and spray hesitation was not detected by
the PDA or observed in the high-speed cine
sequences. This is shown in Fig. 3(b) for r56mm
and x555 mm, where the three phases were not
defined as clearly as in Fig. 3(a). The first droplets
arrived at approximately 4 ms after start of injection,
with a mean axial droplet velocity of V
x
520 ms
21
.
The mean droplet diameter of the higher momen-
tum droplets remained unchanged although the
spread in droplet diameters during the main injec-
tion phase was reduced, the measured population
being skewed towards the larger diameters. This
would suggest that the smaller droplets became
stripped from the periphery of the spray plume and
entrained into the gas phase. This is confirmed by
the observation that a small proportion of fine
droplets were observed between approximately 4
and 5 ms after SOI that had a negative (or upward)
axial velocity component but were, however, co-
incident with the timing of the onset of the main
injection phase in Fig. 3(a). It should be noted also
that at this location the effects of collision, coales-
cence, and dispersal further complicate the inter-
pretation of the measurements.
In Figs 5(a), (b), and (c), the temporal evolution of
the high-pressure G-DI spray is shown for several
locations. In all cases, the mean droplet velocities
were always greater than those observed in the PFI
spray and the mean droplet diameters generally
smaller with a reduction in spread. In Fig. 5(a), the
results are presented for r50 mm and x515 mm.
These results are consistent with those reported
previously in reference [34]. In the initial phase of
needle lift, a high-speed jet, with velocities in excess
of 100 ms
21
, penetrated directly along the spray axis
and was first observed at approximately t51ms
after SOI. This jet accelerated the low-pressure liquid
fuel that was resident in the nozzle tip from the
previous injection. This is manifested in poor
atomization and a spread of droplet diameters and
velocities. As the needle reached full lift, the liquid
exited the orifice radially to form the hollow cone
shape. In doing so, the spray moved away from the
probe volume location and an apparent void was
recorded. In the closing phase, the spray returned to
its axis and an exponential trailing edge was
recorded that comprised small droplets with mean
diameters in the range of 5–10 mm.
These observations are corroborated by Fig. 5(b),
which shows the results recorded for r510 mm and
x515 mm, i.e. close to the nozzle but away from the
spray axis. The initial phase is similar to Fig. 5(a),
except that the maximum velocities were of the
order of 30–40 ms
21
(although the radial component
was higher) and a large number of small droplets
were present with a negative (entrainment) axial
velocity component. In the second phase, the void
was replaced by a broad range of droplet sizes and
velocities. The majority of droplets were small with
negative or near-zero (highly fluctuating) velocity
components. This behaviour suggested the presence
of small vortex ring-like structures near the spray
boundary that readily dispersed the lower momen-
tum droplets into the gas phase. The trailing phase
was marked by an absence of droplets, after about
3 ms, as they moved closer to the axis. A broad
spread of larger diameter droplets were ejected after
about 2 ms as the needle closed.
In Fig. 5(c), the same results as in Figs 5(a) and (b)
are plotted for the region far from the nozzle and
further from the spray axis (r514 mm and
x570 mm). At this location, the spray did not arrive
at the measurement point until about t53 ms after
SOI, and the peak velocities were reduced to
approximately 25 ms
21
. The largest droplets of mean
diameters of 20–30 mm arrived first followed by a
cloud of finer droplets. As mentioned previously, the
data recorded furthest from the nozzle were the
most difficult to interpret. However, within the
trailing phase, a significant proportion of smaller
droplets was recorded with V
x
in the range of 210 to
220 ms
21
. The smallest droplets had the highest
negative axial velocity component, in contrast to that
generally observed in the other locations and spray
phases. These features were attributed to the inter-
action of the spray with the surrounding gas
boundary layer through entrainment and the for-
mation of vortex ring-like structures. In the next
section, the results of investigations of the timescales
of these structures using high-speed photography
and digital still images are presented.
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 201
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
3.2 Identification of vortex ring-like structures
The vortex ring-like structures were observed in both
sprays over a wide range of timescales. In the PFI
case, these structures were seen to occur shortly
after injection, close to the nozzle, but were not
easily identified for experimental investigation.
These early vortex ring-like structures are shown in
Fig. 4 in a sequence of consecutive images recorded
at 100 kHz, close to the nozzle of the PFI injector. In
the G-DI case, reverse flow structures of differing
scales were observed over the entire injection
duration. In both cases, however, the vortex ring-
like structures observed in the decaying phase were
used in this study as these could be identified with
sufficient precision for quantitative analysis.
In the first instance, the high-speed photographic
results were used to identify the regions of investi-
Fig. 5 (a) Distribution of droplet diameters and velocities in the G-DI injector spray at r50mm
and x515 mm plotted against time from SOI. Droplets were observed during the initial
and trailing phases but not during the quasi-steady injection phase. (b) Distribution of
droplet diameters and velocities in the G-DI injector spray at r510 mm and x515 mm
plotted against time from SOI. Droplets were observed during all three phases up to about
3 ms after SOI. (c) Distribution of droplet diameters and velocities in a G-DI injector spray
at r514 mm and x570 mm plotted against time from SOI. Droplets were observed
during the trailing phase after about 3 ms from SOI. The vortex ring-like structures were
clearly identifiable between about 3 ms and 4.5 ms after SOI
202 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
gation for the PDA study. Second, the centres of the
vortex ring-like structures were located by identify-
ing the regions corresponding to maximal vorticity
magnitude computed from the interpolated PDA
measurement grid. These occurred from approxi-
mately 1.75 to 5 ms after SOI in the high-pressure G-
DI sprays, and from 12 to 15 ms after SOI in the low-
pressure PFI spray.
An example of the vortex ring-like structures and
typical scales of interest are shown in Fig. 6 for the
G-DI spray photographed within the optical engine,
using a pulsed laser-light sheet of 40 mm in height,
towards the end of the fuel injection period. Several
vortex ring-like structures can be clearly traced
despite the complex nature of the spray. This single
image can be used to estimate the radii of these
structures, and their distance from the spray axis.
The translational and radial displacements and the
velocities of these structures can be estimated from a
sequence of images. In reference [28], the points
where the fluid velocities are close to zero were
manually traced between frames (see points V
x1
,V
x2
,
and V
x3
in Fig. 6). In this paper, computer tracing of
the regions of maximal vorticity, calculated from a
velocity field measured by the PDA, is considered
instead. A schematic of the key scales used in the
vortex ring models is included in Fig. 6.
In the PFI case, vortex ring-like structures were
clearly observed to form close to the nozzle exit. In
the latter stages of fuel injection and in regions far
from the nozzle, the low concentration of droplets
did not provide sufficient illumination for photo-
graphic evidence of vortex ring-like structures.
However, the presence of rotating structures could
be indirectly identified using the PDA droplet
velocity data. The normalized turbulence intensity
components, defined as the ratio of the amplitude of
the velocity component oscillation to the absolute
value of the average velocity component, were used
to visualize the results. The axial and radial compo-
nents of normalized turbulence intensity, calculated
from the PDA data over consecutive injections,
plotted against the radial distance from the spray
axis, are shown in Fig. 7 for x555 mm. The results
show that this intensity in the axial direction was
almost always less than that recorded in the radial
direction and did not exceed 100 per cent at this
location. The greatest fluctuation (.360 per cent)
occurred in the radial component of normalized
turbulence intensity. Close to the injector axis, the
normalized velocity fluctuations in the axial direc-
tion were at their smallest while the radial compo-
nents were an order of magnitude greater. This is
related to the fact that the mean axial velocity of the
jet in this location is maximal, while its radial
velocity is close to zero. As the radial distance, r,
was increased, the magnitude of the axial compo-
nent increased until a maximum was observed at
approximately r512 mm. This was coincident with a
local maximum in the radial component. As rwas
increased further, both components of normalized
turbulence intensity rose and fell in a similar pattern
Fig. 6 A typical high-speed photograph of a G-DI spray (left); typical parameters of vortex ring-
like structures in this spray (right) including the location of the points with zero velocities
(V
x1
,V
x2
,V
x3
), average distance of these points from the spray axis (R
0
), thickness of these
structures (,)(‘~ffiffiffiffiffiffiffi
2nt
p, where nis the gas kinematic viscosity and tis the time, in the case
of a laminar vortex ring), spray width, and the penetration depth
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 203
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
of coincident peaks and troughs. The maxima in the
measured fluctuation intensities were identified with
the regions corresponding to the centres of vortex
ring-like structures where the mean velocities were
the smallest. Normalized turbulence intensity max-
ima were observed in the axial and radial velocity
components over the approximate ranges of r511–
13 mm, r517–20 mm, and r525–28 mm. The ap-
proximate location of the first peak observed in the
turbulence intensity (r512 mm) was used in the
subsequent analyses.
3.3 Translational velocities in the regions
corresponding to maximal vorticity
In each time interval, the mean vorticity magnitude
of the observed spatial velocity distribution was
calculated where clearly defined vortex centres were
identified. An example of the time evolution of the
spatial field of computed vorticity magnitude is
shown in Figs 8(a) to (c) and Figs 9(a) to (c) for the
PFI spray between 13.75 and 14.25 ms, and for the G-
DI spray between 4.25 and 4.75 ms respectively. The
regions corresponding to maximal vorticity magni-
tude are denoted by crosses. The time evolution of
the axial (V
vx
) and radial (V
vr
) velocity components
at these regions were obtained by direct measure-
ment.
In addition, the velocity components of the
translation of these regions at each location were
computed from the displacement of the vortex
centres divided by the time interval between frames.
Implicit within this method was the assumption that
the velocities were constant within a given time step
of 0.25 ms. This method showed a greater fluctuation
in the prediction of the vortex centre precession for
the G-DI injector than in the case of direct
measurements. It was concluded that this method
proved less accurate than the direct method owing
to the greatest experimental error being present in
the determination of the vortex centre. The effect
was further accentuated in the low-pressure PFI
spray. In what follows the focus will be on direct
measurements of translational velocities.
The maximum vorticity magnitude recorded for
the PFI injector for 13.75 (t(14.25 ms was
0.72 s
21
. In the G-DI injector case for 4.25 (t(
4.75 ms it was 2.78 s
21
. It is interesting to note
that the regions corresponding to maximal vorticity
consistently showed the smallest mean droplet
diameters within each incremental time step for
the G-DI injector spray.
The evolution of the values of the axial and radial
velocities, obtained by direct measurement in the
mixture of air and fuel droplets in the region
corresponding to maximal vorticity, is shown in
Fig. 10 for the PFI injector from 12 to 15 ms and in
Fig. 11 for the G-DI injector from 1.75 to 5 ms after
SOI. The uncertainty in the determination of the
regions of maximal vorticity was estimated to be
between ¡0.2 and ¡0.5 mm for the PFI and G-DI
injectors respectively. The greatest uncertainty in the
velocity as a result of this positional error in the
interpolated field was ¡0.5 ms
21
for the PFI case
and ¡1ms
21
for the G-DI spray. These are shown as
error bars in the plots. There are several common
features that can be seen in both Figs 10 and 11.
First, V
vr
shows the most scatter in data. In most
cases, V
vr
is close to zero, being either positive or
negative. This is particularly clearly visible in the
case of the G-DI injector for tgreater than approxi-
mately 3.5 ms after SOI. Second, in both cases, V
vx
is
positive for all values of tand tends to decrease with
time over the range identified. There is more scatter
in the data for V
vx
for the port injector than for the
direct injection case but a trend in each data set can
be traced. The experimental results were approxi-
mated as V
vx
(t)5At
B
, where tis the time after SOI.
Constants Aand Bwere found using a linear least-
squares curve-fitting technique by minimizing the
sum of the squares of the vertical distance of the
Fig. 7 Normalized turbulence intensity (defined as the
ratio of the amplitude of the velocity compo-
nent oscillation to the absolute value of the
average velocity component) of the axial and
radial velocity components plotted against the
radial distance rfrom the spray axis for
x555 mm for the PFI spray. The turbulence
intensity maxima were identified with the
centres of vortex ring-like structures. They were
observed in the axial and radial velocity
components over the approximate ranges of
r511–13 mm, r517–20 mm, and r525–28 mm
204 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
points from the curve. In the PFI case, A51239 ms
21
and B522.21. In the G-DI case, A574.1 ms
21
and
B521.57. The corresponding curves, At
B
, are also
plotted in Figs 10 and 11. Comparing Figs 10 and 11,
it can be seen that the G-DI data show the best
approximation to this curve fit. Third, oscillations in
both the axial and radial velocities, with a frequency
of approximately 1 kHz, can be seen in both sprays,
especially far from the nozzle during the decaying
phase of injection. The analysis of these oscillations,
Fig. 8 The distribution of the vorticity magnitude for the PFI spray for (a) t513.75 ms, (b)
t514.0 ms, and (c) t514.25 ms. The crosses show the locations of the regions of maximal
vorticity
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 205
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
which are likely to be related to spray instabilities, is
beyond the scope of this paper.
Although vortex ring-like structures observed in
gasoline engines (see Figs 4, 6, 8, and 9) look rather
different from the structures of the classical vortex
rings (see reference [21]), it can be expected that the
integral characteristics of these structures, such as
translational axial and radial velocities, are similar.
This can be supported by the fact that, although the
finite values of the Reynolds number lead to notice-
able changes in vorticity distribution, especially far
from the ring core, the effects of this number on
Fig. 9 The distribution of the vorticity magnitude for the G-DI spray for (a) t53.75 ms, (b)
t54.50 ms, and (c) t54.75 ms. The crosses show the locations of the regions of maximal
vorticity
206 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
these velocities is weak. This can justify the authors’
attempt to explain the properties of these velocities
based on classical vortex ring models. In the
following section, these models are briefly discussed.
4 VORTEX RING MODELS
Since the pioneering papers [20,36–38], the theory
of vortex rings has been extensively developed, and
the results have been reported in several review
papers and monographs (for example, references
[21] and [39]). Among more recent publications are
references [22]to[24], [26], [27], [31], and [40]to
[42]. The analysis of the various modelling ap-
proaches is beyond the scope of this paper. In what
follows, formulae describing the axial translational
velocity of the vortex ring under various approxi-
mations will be summarized.
Although some approaches to the modelling of
turbulent vortex rings have been suggested – for
example, references [29], [30], and [43]; see also
Fig. 10 The time evolutions of the values of axial (V
vx
) and radial (V
vr
) velocities inferred from
the analysis of experimental data for the PFI injector (filled and unfilled diamonds), and
the approximation of these evolutions (solid curve refers to V
vx
; dashed line refers to
V
vr
). Time is measured from SOI
Fig. 11 The time evolutions of the values of axial (V
vx
) and radial (V
vr
) velocities inferred from
the analysis of experimental data for the G-DI injector (filled and unfilled diamonds),
and the approximation of these evolutions (solid curve refers to V
vx
; dashed line refers to
V
vr
). Time is measured from SOI
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 207
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
references [44] and [45], the quantitative models
have been developed mainly for laminar rings. An
attempt to develop a model incorporating both
laminar and turbulent vortex ring features was
reported in reference [31]. The parameter commonly
used in modelling of vortex rings is h5R
0
/,, where
R
0
is a free parameter of the model which is usually
identified with the initial value of rat which the
absolute velocity reaches its local minimum, and ,is
the characteristic vortex ring thickness (see Fig. 6).
In the case of laminar vortex rings, ‘~ffiffiffiffiffiffiffi
2nt
p, where t
is time and nis the kinematic viscosity of the fluid
(mixture of air and droplets in the current case). In
the generalized vortex ring model described in
reference [31], ,was defined as ,5at
b
, where ais
a constant (a~ffiffiffiffiffi
2n
pin the case of laminar vortex
rings) and 1/4 (b(1/2 (the case of b51/4 refers to
turbulent rings, while the case of b51/2 refers to
laminar rings).
Using these assumptions, the following general
equation for the normalized vortex ring axial
translational velocity has been obtained [26,31]
Ux~Vx
vn
~ffiffiffiffiffiffi
ph
p3exp {h2
2
I1
h2
2
zh2
12
2F2
3
2,
3
2;5
2,3;{h2
{3h2
52F2
3
2,
5
2;2,
7
2;{h2
ð1Þ
where the generalized hypergeometric function
2
F
2
[a
1
,a
2
;b
1
,b
2
;x] is defined as
2F2a1,a2;b1,b2;x½~X
‘
k~0
a1
ðÞ
ka2
ðÞ
kxk
b1
ðÞ
kb2
ðÞ
kk!ð2Þ
with the coefficients defined as
aðÞ
0~1,aðÞ
1~a,aðÞ
k~aaz1ðÞ... azk{1ðÞ
k¢2ðÞ
vn~M
4p2R3
0
~C0
4pR0
C0~MpR2
0
is the initial circulation of the vortex
ring; Mis the vortex ring momentum divided by the
density of the ambient fluid. Equation (1) was
obtained under the assumption that the Reynolds
number Re 5j
0
,
2
/nis small (f
0
is the characteristic
vorticity). The applicability of equation (1) follows
from the direct comparison of predictions of this
equation with the results of numerical simulations
[46] and experimental data [47] (see references [48]
to [50]). If, following reference [31], ,is defined as
,5at
b
, then equation (1) describes both laminar and
turbulent vortex rings depending on the value of b.
In the limit of long times (small h), equation (1) is
simplified to
Ux~7ffiffiffi
p
ph3
30 ð3Þ
Equation (3) is identical to the equation obtained in
reference [22].
In the limit of short times (large h), equation (1) is
simplified to
Ux~ln hz3{c
2{y3=2ðÞ ð4Þ
where c<0.577 215 66 is the Euler constant and y(x)
is the di-gamma function defined as
yxðÞ~d log CxðÞ
dxð5Þ
where C(x) is the gamma function. Note that y(1) 5
c. Equation (4) is identical to the equation obtained
earlier in reference [20], as shown in reference
[50].
Note that the velocities, U
x
, predicted by equa-
tions (1), (3), and (4) are different from the velocities
of the fluid in the region of maximal vorticity,
U
vx
5V
vx
/v
n
. In the case of long times (small h),
they are linked by
Uvx~Uxz2ph2ð
‘
0
merfc m
ffiffiffi
2
p
J1hmðÞJ0mðÞdmð6Þ
where
erfc xðÞ~2
ffiffiffi
p
pð
‘
x
exp {t2
dt
and J
1
and J
0
are Bessel functions of the first and zero
order respectively. U
x
is defined by equation (1) in
the general case and by equation (3) in the case of
long times. In the latter case, for sufficiently large m
0
,
the contribution of m.m
0
in the integral in equation
(6) can be ignored. On the other hand, for h%m
0
21
J1hmðÞ~1
2hm ð7Þ
Combining equations (3), (6), and (7)
208 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
Uvx~7ffiffiffi
p
p
30 zpð
‘
0
m2erfc m
ffiffiffi
2
p
J0mðÞdm
2
43
5h3ð8Þ
Comparing equations (3) and (8), it can be seen that
both U
x
and U
vx
are proportional to h
3
,t
23/2
.
An alternative approach to the derivation of U
x
was suggested by Saffman [20], who based his
analysis on simple dimensional considerations
rather than on rigorous solution of the underlying
equations. This analysis led him to the following
expressions for the vortex ring radius and trans-
lational velocity
R2~R2
0zk’ntð9Þ
Ux~4p2
k1zk’2h2
3=2ð10Þ
where kand k9are fitting constants of the model.
Equation (10), sometimes known as the second
Saffman formula, has been widely used for inter-
pretation of experimental data (for example, refer-
ence [46]).
Note that in the limit of long times, both equations
(3) and (10) predict that
Ux!t{að11Þ
where a53/2 for laminar vortex rings and a53/4 for
turbulent vortex rings. Remembering the analysis
presented earlier in this section, it can be anticipated
that V
vx
,t
23/2
for laminar rings and V
vx
,t
23/4
for
turbulent rings.
Equation (1) predicts that the radial component of
velocity in the region of maximal vorticity at any
fixed moment of time is equal to zero [26]. A small
non-zero radial component can be expected if a drift
of the region of maximal vorticity in the radial
direction is considered [31]. This prediction of the
theory is consistent with the experimental results
referring to both G-DI and PFI injectors (see Figs 10
and 11).
As follows from the experimental plots shown in
the previous section, the values of translational
velocity of vortex ring-like structures (V
vx
) decrease
with time. This agrees with all models described in
this section. A more careful comparison between
experimental data referring to the translational
velocity and predictions of the model is performed
below.
5 DISCUSSION
A comparison of the experimental data described in
section 3 with the available models of the vortex
rings, as summarized in section 4, identified several
unknown parameters including the values of R
0
and
initial time. In order to eliminate the effect of these
values, the axial velocities presented in Figs 10 and
11 were normalized by the times at which the vortex
rings were first observed, t5t
init
. In addition, the
non-dimensional time, t
¯5t/t
init
was introduced. The
non-dimensional axial velocity component then
becomes V
¯
vx
(t
¯)5V
vx
(t
¯t
init
)/V
vx
(t
init
). The plots of
V
¯
vx
versus t
¯are shown in Fig. 12 for both the G-DI
and PFI cases. In both cases, the normalized axial
velocity component decreased with increasing nor-
malized time. In addition, during the period where
vortex rings were identified in each spray, the rate of
decay in the range 1 (t
¯(1.25 is approximately 1.5
times greater in the PFI spray than that observed in
the high-pressure spray case. It should be noted,
however, that the very high scatter of experimental
results for the PFI sprays indicates that these are less
reliable than those for the G-DI sprays.
As in the case of Figs 10 and 11, the data presented
in Fig. 12 are approximated by a power function. In
contrast to Figs 10 and 11, an additional restriction
V
¯
vx
(t
¯51) 51 is applied in the case of Fig. 12. Hence
this approximation is presented in the form
V
¯
vx
(t
¯)5t
¯
2C
. In a similar manner to the cases shown
in Figs 10 and 11, the value of constant Cwas
calculated by the use of the least-squares best-fit
method. For the PFI case it was found that C5
2.97 and for the G-DI spray, C51.14. The plots
of V
¯
vx
(t
¯)5t
¯
22.97
and V
¯
vx
(t
¯)5t
¯
21.14
are included in
Fig. 12 for both sprays. The best curve fit approx-
imation was achieved for the G-DI spray data.
As follows from the analysis in section 4, V
¯
vx
,t
¯
2a
in the long time limit where a53/2 in the laminar
case and a53/4 in the turbulent case. The plots t
¯
2a
for a53/4 and a53/2 are also shown in Fig. 12. It
can be seen that for the high-pressure sprays,
approximately 60 per cent of the experimental points
occur within the boundaries of the theoretical curves
t
¯
23/2
and t
¯
23/4
. In this case, the experimentally
predicted value of the power is a51.14, which falls
between the predictions for laminar and turbulent
limits. In contrast, only two of the data points for the
PFI spray are located near to the curve correspond-
ing to a53/2. The remainder are far from the
prediction of the models; the measured values of
the normalized axial velocity component are con-
sistently less than values predicted by both the
laminar and turbulent models. The experimentally
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 209
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
determined value of a52.97 suggests a much greater
rate of decaying of vortex ring axial translational
velocity with time. It remains unclear, however,
whether this should be attributed to the different
underlying physics of the process, compared with
that described in section 4, or to excessive scatter of
experimental data.
Note, however, that the only significant error
introduced in the acquisition of PDA data would be
due to the difference in droplet concentration in the
region of the probe volume and the subsequent
obscuration of the transmitted and collected laser
light. In the case of large distances from the nozzle
and during the decaying phase of the injection, these
experimental uncertainties were considered mini-
mal. This was confirmed by the high acquired data
validation rates, especially along the spray periphery
under these conditions.
6 CONCLUSIONS
A port-injected (low-pressure) and a DI (high-
pressure) fuel spray have been studied experimen-
tally in gasoline-engine-like cold-start or warm-up
conditions using production fuel injection equip-
ment. Fuel was injected into an ambient pressure
and temperature, quiescent, optical chamber. High-
speed photography was used to show that both fuel
sprays comprised complex, vortex ring-like struc-
tures that exhibit a range of spatial scales that
occurred over a broad range of time-scales.
An optimized, classical, forward-scatter PDA was
used to measure the spray droplet diameters and
velocities over a fine measurement grid. A spatial
field of the droplet properties was reconstructed
from data that were ensemble-averaged within
0.25 ms time steps from the start of injection trigger.
A weighted, inverse-distance method was used to
interpolate between the measurement points.
Analysis of the temporal evolution of droplet axial
velocities and diameters showed that both sprays are
composed of the classical three phases: initial
unsteady, main quasi-steady, and exponential trail-
ing phases. The results showed that the timing and
main features of these phases are highly dependent
upon the location of the probe volume in the spray;
the relative phasing of these periods differed be-
Fig. 12 Normalized experimental values of the axial velocity component V
¯
vx
5V
vx
(t)/V
vx
(t
init
)
against normalized time t
¯5t/t
init
for the G-DI and PFI injectors (filled and unfilled
diamonds). Approximations of the experimental results are given by V
¯
vx
5t
¯
21.14
(G-DI
injector) and V
¯
vx
5t
¯
22.97
(PFI injector). The values of V
¯
vx
5t
¯
2a
predicted by the model
in the limit of long times (dash–dotted curves) for a53/4 (turbulent ring) and 3/2
(laminar ring) are indicated near the curves
210 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
tween the injectors studied. In general, the mean
droplet velocities measured in the G-DI spray were
always greater than those of the PFI spray, and the
mean droplet diameters were smaller for the G-DI
sprays than for the PFI sprays, with less apparent
spread. The PFI spray showed a hesitation in the
initial period of fuel delivery and the presence of
vortex ring-like structures that propagated from
close to the nozzle. The G-DI spray showed the
characteristic axial velocity distribution of a hollow-
cone spray for measurements recorded close to the
nozzle along the spray axis. The characteristics of the
droplet sizes and velocities in both sprays revealed a
small proportion of the smallest-diameter droplets
that were entrained into vortex ring-like patterns
corroborated by the high-speed photography.
In both cases, the vortex ring-like structures were
observed mainly during the decaying phase of spray
development. The results of the PDA data in the PFI
case showed that both the axial and radial compo-
nents of normalized turbulence intensity varied in a
similar oscillatory manner with increasing radial
distance from the spray axis. The normalized
turbulence intensity maxima were identified with
the regions corresponding to the centres of vortex
ring-like structures that were observed in both
velocity components. The mean vorticity magnitude
was calculated within consecutive time intervals
where vortex ring-like structures could be identified.
The maximum vorticity in the G-DI case was shown
to be approximately four times greater than that
computed in the PFI spray. In the G-DI spray, the
regions corresponding to the maximal value of the
mean vorticity consistently showed the smallest
mean droplet diameters in the range considered.
In both fuel sprays, the radial component showed
the most scatter in data and was close to zero for
both the G-DI and PFI injectors. The axial compo-
nents of velocity were positive for all values of t, and
the PFI injector showed the most scatter in the data.
In the PFI case, the axial data have been approxi-
mated as V
vx
(t)51239t
22.21
. In the G-DI case, the
axial data have shown a better curve fit that has been
approximated as V
vx
(t)574.1t
21.57
. In both cases, t
refers to time elapsed after the start of injection.
Periodic oscillations of about 1 kHz were noted in
the axial and radial vortex velocity components of
both sprays.
The results were presented by normalizing the
time and velocity with respect to the initial condi-
tions at which the vortex ring-like structures were
first observed. For both injectors, the normalized
velocity decreased with increasing time. In the PFI
case, the experimental data have been approximated
as V
¯
vx
(t
¯)5t
¯
22.97
. For the high-pressure fuel spray,
the best curve fit was achieved for V
¯
vx
(t
¯)5t
¯
21.14
.
Although the experimental results showed some
scatter of data, the time evolution of V
¯
vx
for the G-DI
case showed good agreement with the model that
predicts the time evolution of V
¯
vx
between t
¯
23/2
(laminar case) and t
¯
23/4
(turbulent case). In con-
trast, the agreement of the time dependence of V
¯
vx
predicted by the model and that observed experi-
mentally for the PFI injector has been poor. It is not
clear whether this should be attributed to physical
processes that are different to those described by the
available models, or to excessive scatter of experi-
mental data.
ACKNOWLEDGEMENTS
The authors are grateful to the Engineering and
Physical Sciences Research Council (EPSRC) (grant
EP/E047912/1) for the financial support of this
project. The authors would also like to acknowledge
the EPSRC scientific instrument pool for the loan of
the camera, Mr Bob Gilchrist of Ricardo UK Ltd for
providing technical expertise and equipment, and
Professor Phil Bowen, University of Cardiff, for
providing spray data for comparison.
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APPENDIX
Notation
Aconstant in the approximation of
V
vx
(t)5At
B
Bconstant in the approximation of
V
vx
(t)5At
B
Cconstant in the approximation of
V
¯
vx
(t
¯)5t
¯
2C
Ddistance from the measurement
points to the interpolated destination
points (m)
Eweighting exponent in the inverse
distance interpolation
2
F
2
generalized hypergeometric function
k,k9fitting constants in the formula for U
x
(see equation (10))
,characteristic vortex ring thickness
(m)
Mvortex ring momentum (m
4
/s)
rradial distance from the injector axis
(m)
Rvortex ring radius defined by equa-
tion (9) (m)
R
0
the initial value of rat which the
absolute value of the initial velocity
reaches its local minimum (m)
Re vortex ring Reynolds number
(Re 5j
0
,
2
/n)
ttime taken from the start of injection
(s)
t
¯non-dimensional time, t
¯5t/t
init
t
init
time at which vortex ring-like struc-
tures were first observed (s)
U
x
normalized vortex ring axial trans-
lational velocity
U
vx
normalized axial velocity of the fluid
in the region of maximal vorticity
v
n
M4p2R3
0
(ms
21
)
V
r
radial component of the droplet
velocity (ms
21
)
V
x
axial component of the droplet
velocity (ms
21
)
V
x,1,2,3
locations of the points where the
fluid velocities are close to zero as
shown in Fig. 6
V
vr
the radial component of the trans-
lational velocity in the regions
corresponding to maximal mean
vorticity magnitude (ms
21
)
V
vx
the axial component of the trans-
lational velocity in the regions
corresponding to maximal mean
vorticity magnitude (ms
21
)
V
¯
vx
(t
¯) non-dimensional axial component of
the translational velocity in the
regions corresponding to maximal
mean vorticity magnitude
xaxial distance from the injector tip
(m)
aconstant in the approximation of the
axial velocity component and
parameter in the definition of
2
F
2
cEuler constant
Cgamma function
C
0
initial circulation of the vortex ring
(m
2
/s)
hR
0
/,
Vortex ring-like structures in gasoline fuel sprays under cold-start conditions 213
JER02809 FIMechE 2009 Int. J. Engine Res. Vol. 10
m,m
0
dimensionless parameters used in
the definition of U
vx
nkinematic viscosity of the fluid (mix-
ture of air and droplets) (m
2
/s)
j
d
average value of the vorticity at the
interpolated destination point (s
21
)
j
s
average value of the vorticity at the
measurement point (s
21
)
j
0
characteristic vorticity used in the
definition of the vortex ring Reynolds
number (s
21
)
ydi-gamma function
Subscripts
init initial
r radial component
xaxial component
0 initial
vmaximal vorticity
214 S Begg, F Kaplanski, S Sazhin, M Hindle, and M Heikal
Int. J. Engine Res. Vol. 10 JER02809 FIMechE 2009
Reproducedwithpermissionofthecopyrightowner.Furtherreproductionprohibitedwithoutpermission.
