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Citation: Giannadakis, E., Gavaises, M. and Arcoumanis, C. (2008). Modelling of
cavitation in diesel injector nozzles. Journal of Fluid Mechanics, 616, pp. 153-193. doi:
10.1017/S0022112008003777
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J. Fluid Mech. (2008), vol. 616, pp. 153–193.
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2008 Cambridge University Press
doi:10.1017/S0022112008003777 Printed in the United Kingdom
153
Modelling of cavitation in diesel injector nozzles
E. GIANNADAKIS, M. GAVAISES AND C. ARCOUMANIS
Centre for Energy and the Environment, School of Engineering and Mathematical Sciences,
City University, London, UK
(Received 27 April 2007 and in revised form 30 July 2008)
A computational fluid dynamics cavitation model based on the Eulerian–Lagrangian
approach and suitable for hole-type diesel injector nozzles is presented and discussed.
The model accounts for a number of primary physical processes pertinent to cavitation
bubbles, which are integrated into the stochastic framework of the model. Its
predictive capability has been assessed through comparison of the calculated onset
and development of cavitation inside diesel nozzle holes against experimental data
obtained in real-size and enlarged models of single- and multi-hole nozzles. For
the real-size nozzle geometry, high-speed cavitation images obtained under realistic
injection pressures are compared against model predictions, whereas for the large-
scale nozzle, validation data include images from a charge-coupled device (CCD)
camera, computed tomography (CT) measurements of the liquid volume fraction and
laser Doppler velocimetry (LDV) measurements of the liquid mean and root mean
square (r.m.s.) velocities at different cavitation numbers (CN) and two needle lifts,
corresponding to different cavitation regimes inside the injection hole. Overall, and
on the basis of this validation exercise, it can be argued that cavitation modelling
has reached a stage of maturity, where it can usefully identify many of the cavitation
structures present in internal nozzle flows and their dependence on nozzle design and
flow conditions.
1. Introduction
Current common-rail fuel injection systems for direct injection diesel engines operate
at very high pressures, up to 1800 bar, while the whole injection process lasts for very
short time intervals – of just a few milliseconds. The injection rate is controlled through
the fast opening and closing of the needle valve, whereas the typical diameter of nozzle
holes is 0.1–0.2 mm. As the flow from the injector enters into the nozzle discharge holes,
it has to turn sharply from the needle seat area, which leads to the static pressure of
the liquid at the entrance of the holes falling below its vapour pressure and initiation
of cavitation. The occurrence of cavitation in orifices and its significant effect on
spray formation have been known for quite some time. From the early experiments of
Bergwerk (1959), using simplified large-scale and real-size single-hole acrylic nozzles,
it was found that the discharge coefficient of the nozzle is mainly dependent on the
cavitation number, which is a non-dimensional parameter indicating the expected
cavitation intensity (see (1)), and is independent of the Reynolds number, i.e.
CN =
p
inj
−p
back
p
back
−p
vapour
(1)
More recent experimental studies of the flow inside real-size and large-scale model
nozzles have revealed the complexity of the two-phase flow structures formed over a
154 E. Giannadakis, M. Gavaises and C. Arcoumanis
number of cavitation regimes; for example see Soteriou, Andrews & Smith (1995);
Chaves et al. (1995); Arcoumanis et al. (1999). Other studies have revealed that
cavitation significantly influences the atomization process of the emerging fuel, which
represents one of the key factors affecting the performance and exhaust emissions
of direct-injection diesel engines; e.g. see He & Ruiz (1995) and Tamaki, Shimizu &
Hiroyasu (2001). Due to the difficulty in obtaining real-time measurements during the
injection process, most of the reported experimental studies have employed devices
operating under simulated conditions approaching those of diesel engines. Although
very useful results have been obtained from large-scale nozzle experiments, e.g. see
Soteriou, Smith & Andrews (1998); Afzal et al. (1999); Henry & Collicott (2000) and
Roth, Gavaises & Arcoumanis (2002), the advances in instrumentation technology
have allowed more information to be obtained in real-size injector nozzles. Such
studies have been recently reported by Arcoumanis et al. (2000), Badock et al. (1999),
Badock, Wirth &, Tropea (1999), Goney & Corradini (2000), Henry & Collicott
(2000), Walther et al. (2000) and Blessing et al. (2003).
In contrast to the insight obtained from experimental studies such as the
aforementioned ones, theoretical and modelling studies of nozzle flow cavitation
were somewhat less forthcoming; an extensive coverage of the topic can be found in
the review of Schmidt & Corradini (2001). One of the earliest efforts was reported by
Delannoy & Kueny (1990), who employed a single-fluid mixture approach, combined
with an empirical barotropic law model for the calculation of the mixture density
variation as a function of pressure and the speed of sound. Two-dimensional
simulations of cavitation in a Venturi nozzle have shown only qualitative rather
than quantitative agreement with experiments. A similar two-dimensional model
based on enthalpy considerations (Avva, Singhal & Gibson 1995) assumes thermal
equilibrium between liquid and vapour and has shown acceptable agreement for the
discharge coefficient dependency on cavitation number for nozzle holes with sharp
inlet edges. In the barotropic law model of Schmidt, Rutland & Corradini (1997, 1999),
cavitation was simulated by a continuous compressible liquid–vapour mixture, with
the speed of sound, based on the homogeneous equilibrium model (HEM) of Wallis
(1969). Although it was argued that compressibility is beneficial both physically and
numerically, the authors have acknowledged that a drawback of barotropic models
is the allowance of gradual density changes when gradual pressure gradients exist,
which means that such models would not be easily applicable to large-scale lower-
speed cavitation calculations, where there are very small pressure differences but quite
steep density gradients. Another point about this approach is that it does not take
into account turbulence effects. A similar model was also considered by Dumont,
Simonin & Habchi (2001), who extended it to three dimensions. Another cavitation
model developed and applied to simulations of diesel nozzles is that proposed by
Marcer et al. (2000) and Marcer & LeGouez (2001), in which a volume-of-fluid
(VOF) method was modified and combined with an energy-derived mass transfer
model. The basic model assumption was that cavitation could be approximated by a
larger-scale interface, which rules out the possibility of dispersed bubbles as observed
in experimental studies. Although liquid compressibility was taken into account,
turbulence modelling was not considered in their investigations.
In contrast to the cavitation models that are based on thermodynamic consi-
derations, in which the pressure is considered only as a thermodynamic variable, there
is a group of cavitation models that are based on the assumption that the pressure
difference between the inner bubble and the surrounding liquid, acting as a mechanical
force, is responsible for the appearance of cavitation. Furthermore, models whose basic
Modelling of cavitation in diesel injector nozzles 155
assumption is that cavitation occurs due to the growth of bubble nuclei belong to
the same group. One of the early models in this category is that of Kubota, Kato &
Yamaguchi (1992), although it has not been applied to nozzle flow simulations; in it,
cavitation was treated as a viscous fluid whose density changed significantly due to the
presence of vapour, with pure liquid and pure vapour treated as incompressible media.
A bubble number density, which was assumed to be constant, and a local bubble
radius were used to determine at each location the vapour fraction. The Rayleigh–
Plesset (R-P) bubble dynamics equation expressed on a Eulerian frame of reference
was used for calculating bubble growth and collapse; nevertheless, viscous and
surface tension effects were not taken into account. Moreover, the behaviour inside the
bubble was assumed to be isothermal, but possible existence of contaminant gas was
not considered. In a similar effort by Chen & Heister (1996a, b) the number of bubble
nuclei was assumed to be constant per unit mass of mixture. The flow inside single-
hole diesel-like sharp-edged and rounded nozzles was simulated in two dimensions
(Chen & Heister 1996a), and the above model was further validated (Bunnell et al.
1999) and extended to three dimensions in a single-hole geometry reminiscent of a
diesel nozzle (Bunnell & Heister 2000). A similar model was developed by Grogger &
Alajbegovic (1998) in which, although cavitation was considered as a mixture, the
two-fluid method was employed; two sets of conservation equations were solved,
one for the liquid phase and one for the vapour phase, allowing for a slip velocity.
The mass transfer rate from one phase to the other was calculated by a simplified
version of the classical R-P bubble dynamics equation, which is known as the
Rayleigh or asymptotic equation; it assumes that the growth/collapse of bubbles
depends solely on the difference between the liquid and the vapour pressures. Two-
and three-dimensional simulations of venturi cavitating flows were performed. The
model was able to reproduce the observed cavitation regimes, while the predicted
pressure distribution agreed rather well with the experiment, although in terms of
the vapour volume fraction there were significant differences. Subsequently, a revised
version of the model was presented (Alajbegovic 1999; Alajbegovic, Grogger &
Philipp 1999a, b), in which an empirical equation was used to account for the reduction
of bubble number density with increasing vapour fraction. Variants of this bubble
model were more recently presented (Sauer & Schnerr 2000; Schnerr & Sauer 2001),
in which the classical interface-capturing VOF method was converted into a mixture
model. In subsequent improvements to the initial approach the k-ω turbulence model
was implemented (Yuan, Sauer & Schnerr 2000; Yuan & Schnerr 2001); predictions
of cavitating flow in a single-hole sharp-edged nozzle showed that the inclusion of
a turbulence model led to a steady-state solution, with no evidence of transient
behaviour.
Another bubble-based model is that of Singhal et al. (2001, 2002), in which
a mixture formulation was also followed, combined with the Rayleigh equation
for vapour mass production/destruction. It should be pointed out that ad hoc
coefficients and some empirical assumption about the maximum attainable bubble
radius were employed, which effectively led to different equations for the evaporation
and condensation rates. Simulations of steady-state cavitation through a sharp-
edged orifice matched the dependence of the discharge coefficient on cavitation
number, in agreement with the correlation of Nurick (1976). In contrast to the
aforementioned bubble-based models, which were purely Eulerian, Sou, Masaki &
Nanajima (2001) and Sou, Nitta & Nakajima (2002) utilized a Lagrangian frame
of reference for the tracking of cavitation bubbles. However, in this investigation
the researchers did not consider bubble dynamics; rather, in an arbitrary manner,
156 E. Giannadakis, M. Gavaises and C. Arcoumanis
constant-diameter bubbles were introduced in the liquid, and when pressure reached
a pre-selected value they were forced to collapse immediately. As a result, bubble
breakup and coalescence could not be taken into account.
So far, no cavitation model has been presented which assesses the physical processes
such as bubble nucleation, growth/collapse, breakup, coalescence and turbulent
dispersion taking place in hole-type nozzles. Furthermore, past modelling studies
have not considered the inherent stochastic nature of the phenomenon, evident in
all experimental data available. As part of ongoing research on cavitation within
the authors’ group, numerous experimental data relevant to cavitation in diesel
injectors have been presented (Arcoumanis et al. 1998; Arcoumanis et al. 1999;
Arcoumanis et al. 2000; Roth, Gavaises & Arcoumanis 2002; Roth et al. 2005),
together with early modelling efforts (Gavaises & Giannadakis 2004; Giannadakis
et al. 2004). The stochastic flow processes incorporated into the developed model are
thought to provide an improved theoretical framework relative to other models repor-
ted in the literature that adopted a thermodynamic or Eulerian–Eulerian approach for
representing cavitation; this has been demonstrated in a recent study (Giannadakis
et al. 2007) in which a comparison between cavitation models against experimental
data, some of which are also used in the present study, has been performed.
This study aims to present the details of the complete form of this Eulerian–
Lagrangian stochastic computational fluid dynamics (CFD) cavitation model and its
application to diesel injector nozzles. The flow processes considered are described,
and their relative importance is addressed, providing new physical insight into such
flows. Special emphasis is given to detailed model validation against the in-house
experimental data. These include CCD and high-speed images of the cavitation
development in single- and multi-hole nozzles, LDV measurements of the liquid
velocity as well as unpublished CT measurements of the cavitation volume fraction
obtained in single-hole enlarged nozzles. In addition, cavitation images obtained in
real-size nozzles incorporating transparent windows as well as measurements of the
nozzle discharge coefficient have further been used for model validation. Finally, it
is worth mentioning that extensions of the present model, linking cavitation with
erosion and its application to new, emerging fuel systems for gasoline direct injection
engines have been recently presented by Gavaises et al. (2007) and Papoulias et al.
(2007), respectively, adding to the model’s predictive capability over a wide range of
applications and nozzle configurations.
In what follows the developed numerical model is described, highlighting the
key assumptions and its applicability limits. Although it is not possible to validate
independently each of the flow processes taking place in the subgrid time and length
scales relevant to cavitation, the variety of the nozzle geometries and operating
conditions tested have allowed macroscopic validation of the developed model, which
has not been previously possible with existing models. The results from the numerous
validation studies are then described followed by a summary of the most important
conclusions.
2. Mathematical model formulation
In this section the developed model is described. Within the framework of the
Eulerian–Lagrangian approach, cavitation is treated as a two-phase flow comprising
the ‘continuous’ liquid and the ‘dispersed’ gas/vapour bubbles. The origin of cavitation
is attributed to small bubble nuclei which are nucleated in the liquid after certain
criteria are met and which grow once they experience steep depressurization.
