comparison of mixture and multifluid models for in-nozzle cavitation prediction

1 Copyright © 2013 by ASME

Proceedings of the ASME 2013 Internal Combustion Engine Division Fall Technical Conference ICEF2013

October 13-16, 2013, Dearborn, Michigan, USA

ICEF2013-19093

COMPARISON OF MIXTURE AND MULTI-FLUID MODELS FOR IN-NOZZLE CAVITATION PREDICTION

Michele Battistoni1,2, Sibendu Som1 and Douglas E. Longman1 1Energy Systems Division, Argonne National Laboratory, Argonne, IL, 60439, USA

2Department of Industrial Engineering, University of Perugia, 06125, Italy

ABSTRACT Fuel injectors often feature cavitation because of large

pressure gradients which in some regions lead to extremely low

pressure levels. The objective of this paper is to compare the

prediction capabilities of two computational fluid dynamics

(CFD) codes for modeling cavitation in small channel flows, like

those used in diesel or gasoline fuel injectors. Numerical results

are assessed against quantitative high resolution experimental

data collected at Argonne National Laboratory using synchrotron

x-ray radiography of a model nozzle. The first numerical

approach is based on the homogeneous mixture model, phase

change is modeled via the Homogeneous Relaxation Model

(HRM), and it is implemented in CONVERGE code. The second

approach is based on the multi-fluid non-homogeneous model as

implemented in AVL-FIRE code, and it uses the Rayleigh bubble

dynamics model to account for cavitation. One key feature of the

work is that both models take into account the presence of

dissolved gases in the multi-phase flow. This effect has been

accounted for by running compressible three-phase flow

simulations.

Results indicate that both modeling approaches are capable

of capturing the local and global effects of cavitation and void

formation due to either phase change or expansion of the

dissolved air. From a quantitative standpoint, the amount of void

predicted by the multi-fluid model is in good agreement with

measurements, while the mixture model overpredicts the values.

Qualitatively, void regions look similar and compare well with

the experimental measurements, especially in the nozzle

entrance region where most of the vapor is produced. Some

difference has been noted in the centerline of the channel towards

the exit, where void due to dissolved gas expansion has been

observed.

The study highlights the importance of accounting for

dissolved gases in the liquid, especially if the outlet pressure is

low. If dissolved gases are neglected, results underestimate the

centerline void formation. Grid converged results have been

achieved for the prediction of mass flow rate, while grid-

convergence for void fraction is still an open issue. The paper

also includes a discussion about the effect of turbulent pressure

fluctuations on cavitation inception.

INTRODUCTION Prediction and control of sprays emerging from fuel

injectors are considered a key aspect for improving the

performance of modern direct injection engines. The flow

characteristics developed inside an injector determine many

aspects of the resulting jet and of the following combustion

process. The ultimate purpose of the spray is to increase the

liquid surface area exposed to the mass, momentum, and heat

transfer phenomena. Cavitation of the fuel is commonly

encountered even at relatively low injection pressure, but can be

enhanced by either increasing the injection pressure or

decreasing the outlet pressure. Nozzles can experience fully

developed cavitation or even string-type cavitation depending on

the sac flow field [1]. Several studies carried out on Diesel or

gasoline direct injection (GDI) injectors [2,3,4,5,6,7,8,9]

reported that cavitation inside the nozzle orifice generates

increased turbulence that contributes greatly to the disintegration

of the liquid jet, improving the primary breakup and the

subsequent atomization process. Another positive outcome of

cavitation, which is usually observed, is a larger spray cone

angle. However, lower injected mass is generally achieved

because of the reduction of the effective outlet area. In addition,

material erosion and damages to the internal surfaces are

observed. These advantages and disadvantages both provide

significant motivation to study the phenomena.

The size of typical fuel nozzles have made the observation

of the flow field extremely difficult for quite long time, and

researchers have used scaled-up and transparent nozzles to study

NOT PUBLISHED


the internal fluid dynamics. However, over the past decade, the

advances in instrumentation technology have allowed more

information to be obtained in real-size injector nozzles

[10,11,12,13]. These experiments have pointed out the presence

of various types of cavitation structures inside the injectors like

attached cavitation, bubble clouds, and string-type cavitation.

More recently, x-ray radiography has been used to provide

quantitative measurements of local void fractions in cavitating

flows, as reported by Duke et al. [14]. Medical x‐ray tomography

has also been used by Bauer et al. [15] to measure void fraction

in a scaled‐up cavitating pipe flow. Lastly, x-ray phase contrast

imaging experiments of internal injector geometry and needle lift

have also been reported by Kastengren et al. [16,17], very

recently.

At the same time, numerous modeling studies of nozzle flow

cavitation can be found in literature. An extensive coverage of

the topic is given by Schmidt and Corradini [18] or by

Giannadakis [19]. Different approaches can be adopted to treat

the two-, or multi-, phase flow. They can be classified either

according to the model of multi-phase fluid adopted or according

to the mass transfer mechanism assumed for cavitation.

Concerning the multi-phase model criteria, and with specific

reference to cavitation inside Diesel fuel nozzles, at least three

approaches can be identified: the homogeneous Eulerian models

[20,21,22,23], the multi-fluid Eulerian models [24, 25], and the

Lagrangian models [27,28].

The common feature of the homogeneous method is the

assumption that all the phases share the same velocity. In this

case the fluid is a continuous mixture of liquid and vapor. They

can assume different forms depending on how density and

pressure are formulated. The mixture is usually compressible. In

some implementation the effects of turbulence have been

neglected [32]. From the numerical perspective the Volume-of-

fluid (VOF) model, which allows the resolution of sharp

interfaces, is very similar to the homogeneous model. A single

momentum equation is calculated for all phases that interact

using the VOF model. An implementation of this has been

proposed by Marcer et al. [23]. The basic drawback of this

approach is the assumption that cavitation area is delimited by a

large-scale interface, but this rules out the possibility of

dispersed bubbly flows as observed in experiments.

The second aforementioned approach is the Eulerian multi-

fluid method. This method is characterized by different sets of

conservation equations, one for each phase, thus the main feature

is that each phase has its own velocity. One of the first

application of this method to fuel injector flows is reported by

Alajbegovic et al. [24,25].

The third approach is the Lagrangian method. Here, only the

liquid is a continuum, while the vapor is the dispersed phase and

it is represented by parcels of bubbles. Vapor bubble trajectories

are tracked integrating the Newton equation of motion for each

parcel. Examples of this method applied to nozzle flows are

reported by Giannadakis et al. [27, 28].

Concerning the classification criteria based on the type of

mass transfer model, two macro approaches can be identified.

The first one uses thermodynamic considerations in which the

pressure is treated only as a thermodynamic variable, and

thermal equilibrium (Homogeneous Equilibrium Model - HEM)

between liquid and vapor is assumed [29,30,31]. Also a thermal

non-equilibrium version has been proposed for flash boiling

phenomena (Homogeneous Relaxation Model - HRM) [32,33].

This type of approach is generally implemented in the

homogeneous mixture fluid model.

The second group of cavitation models are based on the

assumption that the pressure difference between the inner bubble

and the surrounding liquid acts as a mechanical force which

determines whether a vapor bubble is expanding or collapsing.

The Rayleigh-Plesset (R-P) equation is generally used for

tracking the bubble dynamics and from its time integration the

mass transfer rate can be derived. Applications of this approach

to nozzle flow simulations are given by Giannadakis et al. [27,

28] and Alajbegovic at al. [24,25,34].

In the context of such a wide panorama it is difficult to find

studies that have addressed the quantitative comparison of

cavitation models with comprehensive experimental data. In the

present paper an attempt to validate numerical models has been

made by comparing the results with void fraction measurements

performed at Argonne National Laboratory using synchrotron x-

rays [14]. The paper focuses on the analysis of cavitation, using

a gasoline type fuel in a 500 m diameter nozzle.

The primary objective of the paper is to present a

quantitative comparison of two CFD codes against the

experimental data. One code is CONVERGE and it is based on

the homogeneous flow assumption. The cavitation model is a

non-equilibrium thermal model (HRM) and it uses an empirical

time-scale correlation to allow for finite-rate evaporation

process. The second code is FIRE and it uses an Eulerian multi-

fluid description. The cavitation model is based on the R-P

equation, and bubble number and size are predicted by a poly-

dispersed model.

The second objective of the paper is to analyze numerically

the effect of the amount of non-condensable gases in the multi-

phase flow, by means of three-phase flow simulations. In

addition, some insight on the effect of turbulent pressure

fluctuations on the cavitation inception is presented.

The paper is organized as follows. First, a description of the

available experimental data is given. Second, the CFD models

concerning the mixture and the multi-fluid approaches are

described. Lastly, numerical results are assessed against

experimental data, and in addition some sensitivity and

parametric studies are discussed.

NOMENCLATURE '''A interfacial area density [m-1]

c speed of sound

CN cavitation number satoutoutin pppp

d orifice diameter

D bubble diameter

f

body forces per unit mass

h enthalpy

k turbulence kinetic energy


L orifice length

m mass

ijM

momentum interfacial exchange between phases i, j

n number of phases involved in the multi-fluid model '''N bubble number density [m-3]

p pressure

R bubble radius

Re Reynolds number

Rj source/sink of bubble number due coalescence/breakup

Rph source/sink of bubble number due to phase change

T temperature

v velocity

Y mass fraction

Greek symbols

volume fraction

ij mass interfacial exchange term between phases i, j

turbulence dissipation rate

time scale

kinematic viscosity

density

stress tensor

j source/sink of interfacial area due to coales./breakup

ph source/sink of interfacial area due to phase change

Subscripts

1 or l liquid phase

2 or v vapor phase

3 or a air phase

b bubble

crit critical (used for p or T of the fluid at the critical point)

D drag

eff effective

g gaseous (air + vapor)

in inlet

out outlet

ph phase change

sat saturation

TD turbulent dispersion

Abbreviations

AMR Adaptive Mesh Refinement

CFD Computational Fluid Dynamics

GDI Gasoline Direct Injection

HEM Homogeneous Equilibrium Model

HRM Homogeneous Relaxation Model

LES Large Eddy Simulation

RANS Reynolds Averaged Navier Stokes

R-P Rayleigh-Plesset

VOF Volume of Fluid

PROBLEM DESCRIPTION AND EXPERIMENTAL DATA All the numerical tests reported in this paper refer to nozzle

flow simulations with a gasoline surrogate. The purpose of this

section is to describe the available data and the conditions in

which they have been collected. Experiments have been carried

out recently at Argonne National Laboratory by Duke et al. [14]

using synchrotron x-ray radiography of a polycarbonate nozzle.

The model nozzle has sharp inlet edges and it is cylindrical with

a diameter (d) = 500 m and a length (L) = 2.5 mm. Geometry is

shown in Figure 1; test conditions and fluid properties are shown

in Table 1. As a result of the low outlet pressure of 0.87 bar (abs),

the cavitation number (CN) = 11.2 is very high, even with a

modest pressure drop of about 10 bar and a relatively low Re

number of 1.5·104. The gasoline surrogate used is called Viscor

which has been extensively used in the past in x-ray

measurements [14,17].

Figure 1. Nozzle geometry tested by Duke et al. [14].

Table 1. Experimental test conditions and fluid

properties [14].

Inlet p, T Outlet p,

T

Mass flow rate Re CN

10.6 bar

25 °C

0.87 bar

25 °C

5.82 g/s 1.58x104 11.2

Fluid type Density

@ 20 °C

Dynamic

Viscosity

@ 20 °C

Saturation

Pressure

@ 25 °C

Viscor 16br

(gasoline

surrogate)

781.8 kg/m3 9.35x10-4 Pa.s 640 Pa

Typical experimental images and data are reproduced in

Figure 2. Figure 2.a provides a high resolution (1.67 m)

qualitative image of cavitation at the entrance to the nozzle.

Figure 2.b provides quantitative void fraction measurements

over a line-of-sight collected by means of an x-ray microprobe

radiography technique. Probe resolution was 5x6 m and it was

moved over a raster grid with 100 transverse points and 19 axial

locations. Images have been then reconstructed by streamwise

interpolation. Both these measurements represent time-averaged


results. It is also important to mention that the experiments are

performed under steady flow conditions (there is no opening or

closing of needle), hence comparing time averaged experimental

data against steady state CFD results is justified.

These data show that cavitation is generated near the sharp

inlet edge. At halfway of the channel length the vapor layer

attached to the wall collapses, while a significant cloud of void

starts being visible in the core of the channel and then it extends

up to the channel outlet. Duke et al. [14] observe that the void

accumulation along the centerline is different from the canonical

distribution found in channels of similar size, such as those

investigated by Winklhofer et al. [35] using diesel fuel. The

presence of a void in the center, according to [14] is more likely

due to vapor detached from the walls and transported in the core

rather than due to a new nucleation process, since the pressure in

that region is not supposed to be low enough for cavitation

inception/triggering. At the same time it is also recognized that

dissolved gases in the fuel could also cause the centerline void.

The small amount of air eventually present is estimated to be of

the order of 10-3 mole fraction in the experiments but actual

measurements are not available.

Bauer et al. [15], in a cavitating pipe flow using water,

observed presence of void in the centerline in a similar manner.

They claim that isolated nucleation events occur due to the low

pressure in the core region. It should be noted from all the

experimental studies cited here that the techniques cannot

capture the differences between dissolved gases and fuel vapor.

The experiments can detect only a void cloud in different regions

of the channels.

Figure 2. X‐ray measurements of cavitation in a 500 m

polycarbonate nozzle, obtained via (a) radiographic imaging and

(b) raster‐scan microprobe radiography using a monochromatic

synchrotron source at 8‐10keV. Images reproduced with

permission from Duke et al. [14].

MIXTURE MODEL In this section, the Navier-Stokes equations for a

homogeneous multi-phase mixture will be introduced along with

a description of the HRM non-equilibrium cavitation model.

These models are implemented in the CFD code CONVERGE

[36,37,38].

Basic Equations

In a single-fluid approach, the homogeneous multi-phase

mixture model is governed by one set of conservation equations

for mass, momentum, and energy, with the addition of a

turbulence closure model for Reynolds Averaged Navier-Stokes

(RANS) equations. The homogeneous mixture model is based on

the assumption that the velocity, temperature, and pressure

between the phases are equal. Mass and momentum equations

are given below

0

v

t

(1)

fpvvt

v

(2)

where and v are the mixture density and velocity; p is the

pressure; is the mixture stress-strain tensor due to molecular

and turbulent viscosity. Analogous formalisms apply for energy

conservation and turbulence closure models [36,37,38].

Equations are omitted here for brevity.

In the present study, the multi-phase system is comprised of

a liquid phase (1), a vapor phase (2) and non-condensable gases

(3). The sum of vapor and non-condensable gases will be

referred to with the subscript g. The concept of pseudo-density

is used and the mixture density is computed with the following

equation:

lggg 1332211 (3)

The volume and mass fractions are related through

iii Y (4)

In the present implementation, the void fraction is not

transported directly, but the species are first solved using the

species transport equation and then the void fraction g is

calculated:

iiiii SYDvY

t

Y

(5)

ii

gg

gY

Y

(6)

The liquid phase is treated as incompressible and the gas phases

are treated as compressible.

Cavitation Model

In this code the mass exchange between the liquid and vapor

is based on a Homogenous Relaxation Model (HRM) as

developed by Bilicki and Kestin [32], and recently implemented

by Schmidt and co-workers [20,21,22,32], proposed for

describing non-equilibrium flash boiling processes. Both

cavitation and flash-boiling are evaporation phenomena driven

by a pressure drop. Cavitation is pressure-driven vaporization

(a) (b)


occurring at low temperatures in which the vapor density is so

small that the latent heat flow does not affect the phenomenon.

As a result, the time scale of heat transfer is much faster than the

time scale of bulk motion, therefore the latter is basically

controlled by the inertia of the liquid. On the contrary, a flash-

boiling process occurs at elevated temperatures and in this case

vapor density is much higher, therefore the liquid must provide

more energy per unit volume of vapor. The process takes a non-

negligible time scale and the dynamics is controlled by finite-

rate heat transfer rather than by inertia [39]. In this framework

the HRM approach assumes a first order rate equation for the

evolution of the instantaneous mass fraction of vapor Yv towards

its equilibrium value vY over a given time-scale . The model

as proposed by Bilicki and Kestin [32] is given below

vvv YY

dt

dY (7)

The equilibrium vapor quality vY is a function of the

thermodynamic properties at the local pressure, i.e.

lvlv hhhhY . The time-scale is evaluated using an

empirical fit proposed by Downar- Zapolski [40] 76.154.071084.3 [s] (8)

where

satcrit

sat

pp

pp

(9)

and it was determined in experiments of flashing flows of water

in pipes with upstream pressures of in excess of 10 bar.

EULERIAN MULTI-FLUID MODEL In this section, the governing equations for the multi-phase

flow modeled using a non-homogeneous multi-fluid approach

will be introduced along with a description of the Rayleigh-

Plesset cavitation model. These models are implemented in the

CFD code FIRE.

Basic Equations

The Eulerian multi-fluid approach is based on the

assumption of co-existence of n different phases, treated as

interpenetrating continua. In the present study, for the purpose of

cavitation modeling, the multi-phase system is comprised of a

liquid phase (1), a vapor phase (2) and incondensable gases (3).

Each phase is considered as a continuous medium and an

ensemble averaging procedure is applied to remove the

microscopic interfaces, at sub-grid scale level [44,45]. For each

phase a set of conservation equations is applied which is

analogous to the RANS equations, with the addition of a scalar

quantity, i.e. the volume fraction i of the phase i. Furthermore,

relevant source/sink terms for inter-phase exchange of the

conserved quantities are introduced [46,47]. For n = 1, equations

actually reduce to the RANS equations for a single phase flow.

Mass and momentum averaged equations for the multi-fluid

model are given below, according to [45, 49]

n

ijj ijiiiii v

t ,1

(10)

n

ijj iji

n

ijj ijiiiiii

iiiiiii

vMfp

vvt

v

,1,1

(11)

where , and v are the averaged volume fraction, density, and

velocity for each phase; p is the pressure and it is shared by all

phases, i.e. p = pi; i is the phase stress-strain tensor due to

molecular and turbulent viscosity; subscripts i and j are the phase

indicators; Γij is the mass change rate due to cavitation and ijM

is the inter-phase momentum transfer term. Analogous

formalisms apply for energy conservation and turbulence closure

models [45,49]. Equations are here omitted for brevity.

Each fluid can be treated as incompressible or compressible.

In the latter case, for a gas phase the density is determined based

on the equation of state for an ideal gas:

iii TRp (12)

For the liquid phase the density is determined using the

barotropic relation:

2

, irefrefii cpp (13)

where ci is the liquid speed of sound.

Cavitation Model: Mass Transfer between Liquid and Vapor

In order to describe the mass transfer due to cavitation, a

vaporization/condensation rate model is required. In this code

the Rayleigh equation for the dynamics of a spherical bubble

growth constitutes the starting point [41]

RR

ppR

l

effsat 3

2 (14)

Plesset [39,42] also added two terms accounting for surface

tension and liquid viscosity effects. Although these terms could

be easily incorporated into the model their contribution is

marginal, therefore they have been neglected in this study.

The concept of effective pressure peff has been introduced to

reflect the fact that the dynamics of a single bubble can be

affected by other flow parameters as well. In particular, due to

the local liquid turbulence the pressure around a bubble is

subjected to fluctuations, which leads to the instantaneous

pressure experienced by the bubble becoming lower than the

mean pressure, at least for certain time intervals. In view of this

the following expression is used:

llEeff kCpp 3

2 (15)

where CE is the Egler coefficient with a value of 1.2 as suggested

by Hinze [43] and Giannadakis et al. [27], and kl is the local

turbulent kinetic energy in the surrounding liquid. In this model

therefore the effective surrounding pressure peff is assumed lower

than the RANS averaged pressure p. This is a way for

representing the likelihood of cavitation inception at average

pressure levels higher than the saturation pressure because of

turbulent pressure fluctuations. At the same time it is implicitly

assumed that bubble dynamics and turbulent eddies dynamics

are different, so that once bubble growth is started, a rapid re-


increase of the pressure will not necessarily cause a

corresponding quick collapse.

The mass change rate of a single vapor bubble due to

isothermal evaporation or condensation is

RRtm vb24 (16)

In general, assuming a poly-dispersed flow, a probability

distribution function for bubble radius Rf can be defined.

Therefore, the integral mass exchange of the population can be

calculated. Statistically, two average quantities are required to

close the problem, e.g. the bubble number density '''N and the

interfacial area density '''A . The related transport equations are

given below

phj RRvN

t

N )0('''''' (17)

phjvA

t

A )2('''''' (18)

where )0(v

and )2(v

are the 0th and 2nd moment average veloci-

ties, respectively, of the bubble size distribution, defined as:

max

min

max

min

)(R

R

kR

R

kk dRRfRdRRfRvv . (19)

The terms on the right hand side of the equations (17) and (18),

i.e. jR and

j , represents the net rate of change in the

corresponding transported quantities due to fluid particle

interactions such as coalescence and disintegration, while phR

and ph are the fluid particle source/sink rates due to phase

change. Closure models for these terms along with further

detailed descriptions of the model are provided in [47,48]. By

means of '''N and '''A the mass exchange term 12 in eq. (10) and

(11) can be calculated.

Momentum Exchanges

The momentum exchange term 12M

takes into account at the

grid scale level the microscopic effects exerted at the interface

between liquid and vapor.

2112121

'''

21128

1 kCvvvvACM TDD

(20)

Drag and turbulent dispersion effects are considered, whereas

inertia and lift effects are neglected. The drag coefficient CD is

based on the drag law of a single sphere. The turbulent dispersion

force, which accounts for the bubble dispersion due to the

turbulent mixing process, has been modeled setting CTD = 0.1. In

addition, since a bubbly flow induces turbulence at a

microscopic level and basically this is caused by momentum

interactions at the interface, a bubble induced turbulent viscosity

term is added to the liquid phase turbulent viscosity term,

according to Sato [50].

221

1

2

1,1 2

vvRC

kC Satot

(21)

The value here used is SatoC = 0.6.

The momentum exchange between liquid and air 13M

was

accounted for using formally a model identical to that already

described in eqns. (20) and (21) for the vapor. Air was considered

the dispersed phase and liquid the continuous phase. The only

difference is the assumption of a constant diameter for air

bubbles, set to 1 m. Therefore in eq. (20) the interfacial area

density simplifies to

33

3/2

3

'''3/1'''2

3

'''

3 636 DNNDA (22)

The coefficients for turbulence dispersion CTD and for bubble-

induced turbulent viscosity CSato associated with air have been

set to the same values adopted for vapor. If locally 3 becomes

greater than 0.5, the continuous and dispersed phases are

switched. In addition, no momentum exchange between air and

vapor was assumed, i.e. we set 023 M

.

NUMERICAL SETUP Acknowledging that the two codes are based on different

models, it is understandable that the implementations are also

different. In this study, extensive effort has been made in order

to present a fair comparison, hence, same settings have been

applied wherever possible. Most of the numerical parameters are

summarized in the following Table 2 and are discussed in the

following paragraphs. Some intrinsic differences exist, which in

essence characterize each code and each approach, and they will

be discussed below as well.

Table 2. Baseline numerical set-up.

CONVERGE FIRE

Fluid model Mixture model Multi-fluid model

Cavitation model HRM Rayleigh

Effective pressure

which triggers

cavitation

ppeff llEeff kCpp 3

2

setting CE = 0

Species / phases liquid, vapor, air liquid, vapor, air

Dissolved gases at

the inlet boundary

2·10-5 by mass 2·10-5 by mass

Compressibility compressible compressible

Turbulence model k- k-

Grid type hexahedral, with

fixed embedding

mainly hexahedral

Base grid size [m] 150 150

Min grid size [m] 9 9

Geometry 90 deg sector with

periodic boundar.

90 deg sector with

periodic boundar.

Pressure-Velocity

coupling

PISO SIMPLEC

Time integration Euler 1st order Euler 1st order

Spatial discretiz. 2nd order 2nd order

Time step [s] lower than 5·10-9 lower than 5·10-8

The main difference as outlined in the previous section lies

in the homogeneous vs. non-homogeneous approach used for

representing the multi-phase system. CONVERGE uses a

mixture model which is intrinsically homogeneous, for example

velocity is common to all the phases, therefore no slip exists.


This is reasonably true in a cell where the volume fraction of one

phase largely dominates over all the others; in this case the latter

can be thought as transported without velocity slip.

Unfortunately, the approach is questionable when comparable

amounts of different fluids are present in the same cell. FIRE

instead solves different sets of equations for each fluid, therefore

slip phenomena can be taken into account.

The numerical solution is based on a finite volume

discretization of the governing equations, as available in both the

commercial codes. Simulations are transient, so all calculations

were run until both the inflow and outflow had stabilized. This

usually required at least 1.0 ms of physical time, but in some

cases simulations were run beyond 1.5 ms. The first-order

implicit Euler scheme has been employed for the time

integration. For the baseline case comparison, shown in Table 2,

calculations were run taking into account gas-phase

compressibility, therefore time step size was extremely small for

stability reasons. FIRE required a maximum time step of the

order of 5·10-8 s and CONVERGE an even smaller time-step of

approximately 5·10-9 s.

Even though large gradients exist in nozzle flows and

stability has been a main concern, in the present study convection

terms have been discretized using second-order spatial schemes

for all quantities. To this end, calculations have been started with

first order upwind schemes when needed and after reaching a

steady state solution they have been extended with second order

schemes. The overall iterative solution procedure has been based

on the PISO algorithm for CONVERGE and on the SIMPLEC

algorithm for FIRE. Turbulence has been accounted for using the

standard k- model with standard wall treatment in both codes.

Grids are slightly different in terms of cell type and hence in

terms of total number of cells. This is a consequence of different

meshing approaches used by the two codes. CONVERGE uses a

modified cut-cell Cartesian method for grid generation and the

grid is generated internally at runtime. Both fixed cell

embedding and AMR cell embedding can be employed.

Nevertheless, in this study only fixed embedding has been used

for fairer comparison of the results. FIRE can handle structured

and unstructured grids and it uses polyhedral cells in the

transition zone between two neighboring refinement levels.

Examples of the meshes are given in Figure 3. In both cases the

base grid size is 150 m and 4 levels of refinements are used to

reach the minimum value of approximately 9 m. The whole

channel region has been uniformly discretized using the smallest

cells, while coarsening has been applied only outside, in the inlet

and outlet chambers. Lastly, only one quarter of the channel has

been modeled and periodic boundaries have been applied on the

side surfaces.

A noteworthy aspect of this work is that the effects of

dissolved gases in the multi-phase calculations were accounted

for in the simulations. As outlined in a previous section, the main

reason is that the experimental tests present low outlet pressures

therefore expansion of small fractions of air inside the channel

could be emphasized at these low pressures. In both codes the

amount of air introduced as a third species in the fluid was set to

2.10-5 mass fraction. This value corresponds approximately to the

standard value for gasoline (or water) exposed to the ambient

pressure [51]. It is worth noting that in CONVERGE this input

is straightforward, since it solves for the species mass transport,

while in FIRE it has been converted to the corresponding amount

of volume fraction at the inlet pressure (approximately 1.31.10-3

by vol. at 10.6 bar).

Figure 3. Examples of grids generated in CONVERGE (a)

and FIRE (b) with 150 m base grid size and 9 m min. cell size.

The cavitation models are also different, as described in the

previous section. CONVERGE uses the HRM model, while

FIRE uses the Rayleigh equation based model. Both codes

assume that cavitation occurs if the fluid pressure drops below

the saturation pressure of the fluid at the local temperature. Since

the flow is turbulent and pressure fluctuations can exists, the

problem of what would be the effective triggering pressure

arises, especially in a RANS approach. In this context, FIRE

allows to include this contribution by means of the Egler

coefficient CE shown in eq. (15). Nevertheless, the value chosen

is arbitrary since the only theoretical basis [43] is valid for

isotropic turbulent flows. Hence the value can be changed to

enhance or reduce cavitation by influencing the local pressure

distribution. Furthermore, the present implementation of

CONVERGE neglects this aspect; therefore in this study for the

baseline set-up, in order to present a fair comparison, the value

of CE in FIRE has been set to 0, as shown in Table 2.

RESULTS AND DISCUSSION In this section CONVERGE and FIRE results concerning

the cavitating nozzle under the conditions described in Table 1

will be compared against x-ray data [14].

Model assessment

Figure 4 shows the results of the simulations for the

cavitating nozzle compared to the experimental data from Duke

et al. [14]. Simulation settings and boundary conditions for both

codes are reported in Table 2 and Table 1, respectively. Results

shown in this sub-section refers to the minimum grid size of 9

m for both codes. CFD data have been post-processed to

evaluate the projected amount of void, computed as the line

integral of g along the transverse direction, throughout all the

channel width. The spatial resolution of this evaluation

corresponds to the cell size and shape existing on the mid cut-

(a)

(b)


plane, i.e., the integration is performed starting from each cell

belonging to this plane and moving normally to the plane itself

in both directions. This type of result will also be referred to as

radiographic image.

In terms of global mass flow rate the experimental value is

5.82 g/s, and the computed values are 5.25 g/s (-10%) and 5.36

g/s (-8%) with CONVERGE and FIRE respectively.

Figure 4. Predicted void fraction integrated along the

transverse direction using CONVERGE and FIRE codes,

compared to experimental data [14]. Dimensions are in mm.

Flow is from bottom to top.

Observing radiographic void contours in Figure 4, a

reasonable qualitative agreement can be seen for both codes.

Each model is capable of reproducing the presence of cavitation

just after the inlet corners. In this region the predicted integrated

void along the line-of-sight has a peak which occurs slightly off

from the wall, and 0.25 mm downstream of the inlet edge. The

CONVERGE mixture model predicts a peak value around

220 m. The FIRE multi-fluid model, instead, predicts a local

peak value of 150 m. In this region, measurements show two

different local maxima (they are not symmetrical due to

machining imperfections of the nozzle), whose average can be

estimated close to 150 m. In view of that, the mixture model

overestimates the local void peak, while the multi-fluid model is

quite close to the experimental data.

As far as the void cloud in the middle of the channel is

concerned, both models capture this effect and this is

noteworthy, since in the literature it is fairly common to find

simulation results with cavitation clouds developing just along

the wall, without penetrating significantly into the core

[20,26,31,33]. Anyway, for both codes the predicted void cloud

is a bit shorter, and it does not extend up to the outlet section as

in the experiment. FIRE predicts a shorter central cloud

compared to the experimental data. CONVERGE predicts

slightly better the axial location of this central void, but the local

maximum still occurs early. Another interesting point is that in

the experiments the centerline void is rather disconnected from

the inlet edge void. This aspect is better captured by the multi-

fluid model.

Lastly, FIRE is able to capture the pressure recovery and

hence the reattachment of the liquid phase to the wall which

occurs approximately at one third of the channel length.

CONVERGE is not able to capture this effect and it predicts

presence of void along the wall all the way up to the exit.

Figure 5 shows a second quantitative comparison in terms

of total void fraction along the channel axis in ten specific axial

locations as available from the experiments [14]. Numerical

values are calculated as the volume integral of the void fraction,

normalized to the volume itself, over small channel slices with a

thickness of one cell layer. Experimental data are in fact

evaluated on 6 m width cross cuts. Right after the inlet edge (at

x/L = 0.1-0.2) we can observe the maximum value of void

formation. Thereafter, both codes predict a fairly constant

amount of total void up to x/L = 0.6. Then a certain decrease as

a function of the axial position is predicted, while the

experimental value is still constant around 21%. Results obtained

with CONVERGE over-predict by 30% the experimental data in

the first half of the channel, while downstream results tend to

approach the measured values. Void fractions computed by FIRE

agree very well for the most part of the channel, and only at the

exit the value is underestimated. It should be noted that the

experimental uncertainties associated with such measurements

are not available.

Figure 5. Computed total volume fraction g (vapor + air)

along the channel axis evaluated in one cell layer thick slices,

compared to the experimental data [14].

It is worth noting that results shown so far (Figure 4 and

Figure 5) refer to the total void g as the sum of vapor and air,

without addressing the separate contribution of cavitation and

expansion of dissolved gases. While it is understood and

accepted (and it will be shown in the next sub-section) that the

void after the inlet edge is generated by cavitation due to the

extremely low pressures, the most difficult region to be predicted

is the second part of the channel, along the axis. In this region

either new cavitation or vapor transport or air expansion or a

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tota

l vo

id f

ract

ion

g

x/L

Experimental dataCONVERGEFIRE


combination of all these effects can play a significant role in

contributing to the void. A specific sub-section, following a grid

sensitivity study, will address this issue in more detail.

Grid sensitivity

Since both codes have similar mesh embedding capabilities,

the cell sizes shown are the minimum grid sizes. For both codes

two sizes have been tested, i.e., 18 m and 9 m. Results in terms

of mass flow rates are given in Table 3. As mentioned previously,

predicted values are underestimated within a range of 7% to

11%, but the effect of changing the grid size is limited to 1.5%

for each code, therefore the calculations can be considered grid-

converged with respect to this parameter. Future studies will

consider finer cell sizes to comprehensively address the issue of

grid-convergence.

Figure 6 shows the overall void fraction predicted along the

channel axis. This quantity is affected by higher dispersion, but

taking into account inherent modeling difficulties, the results are

still encouraging. A good trend is shown by both codes, since

refining the mesh numerical results tend to get closer to the

experiments. In fact, it can be noted that mesh refinement has

opposite effects on the two codes, i.e., with CONVERGE mesh

refinement results in increase of the total void fraction while in

FIRE the vice versa happens.

Table 3. Cell size effect on mass flow rate.

Min. cell size[m] Mass flow rate [g/s]

CONVERGE FIRE

18 5.17 5.46

9 5.25 5.36

Experimental value 5.82

Figure 6. Cell size effects on total void fraction g along the

channel. Experimental data [14] are also plotted for comparison

purposes.

Further Discussion

In order to better understand the underlying physics and to

further assess the modeling capabilities and limitations, the

computational results can be used to analyze other features of

interest, such as velocity, pressure, and split of vapor fraction and

air fraction. Figure 7 plots all the aforementioned quantities

along a cut-plane through the nozzle axis; the two columns show

the comparison between the two codes.

The total void g on the cut-plane is shown in Figure 7.c.

Despite the fact that these patterns are quantitatively different

from the radiographic images described in Figure 4, they

substantially present common features, therefore this allow us to

analyze the void formation process also on this basis.

The pressure field is mainly responsible for the void

production and can be seen in Figure 7.b (please note that the

scale is narrowed to 0-1 bar) and pressure profiles along the

channel wall and axis can be seen in Figure 8 (scale is zoomed

as well).

The flow separates off of the inlet edge and an extremely

low pressure region is generated just behind it. In this region both

codes predict local values well below the saturation pressure

(Figure 8.b) where local values close to the wall are of the order

of a few tens of Pa. As a result vapor is produced and expands

along the wall (Figure 7.c,d,f). It is worth to point out that

because of the compressible model, there is no unphysical

undershooting of the pressure below 0 bar, as would occur using

an incompressible approach. In the region downstream of the

inlet edge, there is significant amount of cavitation predicted by

both codes. Here, vapor mass fraction is distributed along the

channel wall and sticks to it without penetrating significantly

towards the axis (Figure 7.d,f). The total void g reaches 100%

locally and the cavitation contribution to the total void is very

important. However, it is worth noting that the predictions about

the amount of vapor mass generated after the inlet corner (cf.

Figure 7.f,g) are very different between the two codes. The Y2

peak value predicted by CONVERGE is around one tenth of the

air mass fraction Y3, while the corresponding value predicted by

FIRE is close to ten times the air mass fraction. Such relatively

big differences are due to the fact that FIRE using a multi-fluid

model, predicts separation of air from vapor in this region as a

result of momentum exchange (cf. Figure 7.g at the inlet edge),

while CONVERGE using a homogeneous model, keeps the air

mass fraction constant everywhere.

Along the wall at one third of the nozzle length, FIRE

predicts a pressure recovery (marked by the arrow in Figure 8.a

at x/L = 0.3) which causes reattachment of the liquid (Figure

7.a,c). This feature has been already pointed out in Figure 4 and

matches with experiments. CONVERGE does not capture this

effect.

Along the nozzle axis pressure rapidly decreases and

reaches a new local minimum at x/L = 0.8-0.9 (see second arrow

in Figure 8.b). In this region the predicted pressure is not low

enough to generate additional vapor, but it is low enough to allow

a significant air expansion (Figure 7.c,d,e). Here the local void

fraction g reaches 50% and is entirely due to air expansion.

Further downstream, a steep pressure rise occurs (the flow goes

through local shocks) and thereafter pressure oscillates before

finally recovering the outlet chamber level. Both codes predict a

similar behavior, the only difference lies in the location of the

pressure rise.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tota

l vo

id f

ract

ion

g

x/L

Experimental dataCONVERGE - 18 micronsCONVERGE - 9 micronsFIRE - 18 micronsFIRE - 9 microns


Figure 7. Predicted velocity (a), pressure (b), volume fractions (c,d,e) and mass fractions (f,g). Left column shows CONVERGE

results, right column shows FIRE results. Color scale range is shown above each row.

0.0 0.2 0.4 0.6 0.8 1.0-20

0

20

40

60

80

100

120

0.0 0.2 0.4 0.6 0.8 1.0-20

0

20

40

60

80

100

120

Pre

ssure

[kP

a]

x/L

nozzle axis - FIRE

nozzle axis - CONVERGE

Pre

ssure

[kP

a]

x/L

nozzle wall - FIRE

nozzle wall - CONVERGE

Figure 8. Pressure profiles along the nozzle wall (a) and

along the axis (b).

As far as the comparison between the two codes is

concerned, velocity and total void fractions are similarly

predicted. Two main differences can be observed. The first one

lies on the prediction of the amount of vapor generated at the

inlet edge, since FIRE predicts more vapor production than

CONVERGE. The second one lies on the reattachment of the

liquid predicted by FIRE at x/L = 0.3, which CONVERGE is not

able to reproduce.

From this discussion it is evident that although the total

amount of void matches the experimental data quite well (Figure

4 and Figure 5), the relative amount of vapor and air is still not

fully assessed. This is mainly due to the fact that the experiments

cannot distinguish between vapor and dissolved gases. A

stronger validation would be possible if pressure data was

available, since the vapor formation and the air expansion in

simulations are extremely sensitive to these low pressure levels.

In addition, the exact knowledge of the dissolved air fraction

would be essential. With respect to the latter, the following

section will give more details.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(a)

(b)

Mixture Model

(CONVERGE)

Multi-Fluid Model

(FIRE)


PARAMETRIC STUDY Following the validation of the two codes, some parametric

studies have been carried out using parameters of interest as

outlined below in Table 4. First, a study about changing the inlet

and outlet pressure, at equal cavitation number CN is shown.

Afterwards, the influence of the amount of dissolved gases on

cavitation is presented, since some uncertainties on the

knowledge of this quantity exists. Both these assessments are

performed with CONVERGE code. Lastly cavitation-turbulence

interaction is discussed with FIRE code. All the other parameters

not mentioned in Table 4 have been kept unchanged, as already

shown in Table 2.

Table 4. Parametric studies with CONVERGE and FIRE

codes.

CONVERGE FIRE

Inlet – outlet pressure [bar]

106.0 – 8.7,

10.6 – 0.87

(same CN=11.2)

Amount of dissolved air,

mass fractions [-]

0, 2E-7, 2E-6,

2E-5, 2E-4

Turbulent pressure

fluctuations on effective

pressure llEeff kCpp

3

2

CE = 0,

CE = 1.2

Influence of outlet pressure on cavitation

In order to quantify the importance of using a three-phase

model which includes dissolved air in the fuel, instead of a

simpler two-phase approach, a simple test is presented. The

baseline case (pin = 10.6 bar and pout = 0.87 bar) already

discussed in the previous sections, is compared to a higher

injection and back-pressure case, i.e., pin = 106 bar and pout = 8.7

bar. This allows a comparison at the same CN number.

The comparison is shown in Figure 9 and Figure 10.

Concerning the void production at the inlet corner no significant

differences can be noticed, since the local pressure is always

extremely low and sufficient to trigger cavitation (Figure 9.a).

On the contrary, along the channel axis the pressure minimum,

occurring at approximately x/L = 0.8, presents very different

values (Figure 9.b). Pressure is above 5 kPa for the high outlet

pressure case, while it is as low as 1 kPa for the low outlet

pressure case. This means that in this region, the appearance of

void can be substantially offset by high back-pressure

conditions, regardless of the amount of dissolved gases or the

value of the saturation pressure of the fuel (at least for the typical

values encountered).

The difference visible in Figure 9.a up to x/L = 0.6 occurs

because the 8.7 bar case has higher velocities that bring the

pressure down to very low levels for a greater length.

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

0.0 0.2 0.4 0.6 0.8 1.00

10

20

30

40

50

Pre

ssure

[kP

a]

x/L

Axis profiles:

outlet pressure = 8.7 bar


Pre

ssure

[kP

a]

x/L

Wall profiles:



Figure 9. Outlet pressure effects under equal CN value.

Pressure profiles along the nozzle wall (a) and along the axis (b)

are shown using CONVERGE code.

Figure 10. Outlet pressure effects under equal CN value.

Predicted total volume fractions g using CONVERGE code.

Dissolved air set to the baseline value of 2.10-5 mass fraction

(Table 2).

Dissolved air effect

The influence of the amount of non-condensable gases on

void formation is discussed here. The experimental uncertainty

concerning the exact knowledge of this data motivated us to

quantify the effects of different concentrations. All the

parameters have been kept unchanged as shown in Table 4, apart

from the air fraction Y3 which has been varied from 0 to 2·10-4

by mass.

Results are shown in Figure 11 and Figure 12. The former

plots the value of g integrated and normalized over channel

slices at different axial locations, the latter shows the contours of

vapor mass fraction Y2 and of total void fraction g. From Figure

11 it can be inferred that the volume fraction of vapor and non-

condensable gases remains fairly constant as the amount of air is

increased, except for the highest level of Y3 = 2·10-4. This

indicates that adding air to the liquid inhibits the inception of

cavitation at very low pressures. This occurs because while the

pressure decreases, the expansion of non-condensable gases

opposes a further pressure drop and in turn hinders cavitation by

keeping the local pressure at high values.

(b)

(a)


Figure 11. Effect of dissolved air mass fraction Y3 on total

void fraction g predicted along the channel using CONVERGE

code.

At extreme conditions, for Y3 = 0 the vapor and void

formation is confined along the wall; while for Y3 = 2·10-4 there

is a practical absence of cavitation but a huge production of void

at the wall and in the core due to the air expansion. It is also

important to point out that only introducing the highest level of

non-condensable gases (Y3 = 2·10-4) the total void produced

starts to get increased, with respect to all the other cases (Figure

11), as it would be intuitively expected.

The fact that the total void evaluated as integral at different

axial location (cf. Figure 11) is quite independent of the amount

of non-condensable gases suggests that further data are needed

to fully validate the models and to accurately predict the physics.

Both a direct measure of the dissolved air and the measurement

of the pressure at some locations could help in complete

understanding.

Figure 12. Effect of the dissolved air mass fraction Y3 on the vapor production and on the total void fraction patterns using

CONVERGE code. Left column shows vapor mass fraction Y2, right column shows total void fraction g. Color scale range is shown

above each column.

Turbulence effect

In this last sub-section the influence of the turbulent pressure

fluctuations accounted for by means of the Egler factor CE, are

presented using FIRE code. In all the previous results, this

parameter has been set to 0, therefore possible turbulent effects

on the cavitation inception have been ignored.

Locally, the effective liquid pressure peff that is responsible

for cavitation inception (cf. eqns. [12] and [13]) could be lower

than the average value p, because of turbulent fluctuations. To

the best of our knowledge many Authors [32,25,27] recommend

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tota

l vo

id f

ract

ion

g

x/L

Y3 = 0

Y3 = 2E-07

Y3 = 2E-06

Y3 = 2E-05

Y3 = 2E-04


the use of CE = 1.2, but no evident proof of that assumption has

been given.

A comparison between assuming CE = 0 and CE = 1.2 is

shown in Figure 13 and Figure 14. The former plots the value of

g integrated and normalized over channel slices at different

axial locations, the latter shows the contours of various fluid

dynamic variables. Clearly, the value of g is increased when

turbulence is included in the cavitation model, as visible in

Figure 14, but the overall value is still comparable to the

experimental data especially near the exit of the channel.

Observing Figure 14.d,f, when a value of CE = 1.2 is used, a

larger cavitation cloud is generated downstream of the inlet edge

and in addition cavitation is predicted in the middle of the

channel too. The corresponding pressure levels (Figure 14.b) are

slightly increased as a countereffect of more momentum

exchange between the vapor and the liquid phase. These results

suggest that the phenomena are extremely complex and further

validation is required to shed light on the physics. It would also

be advisable to model the effect of turbulence in a more

straightforward manner, i.e., by means of large eddy simulations

(LES) to remove great part of the uncertainties associated to this

aspect.

Figure 13. Effect of turbulent pressure fluctuations on total

void fraction g predicted along the channel using FIRE code.

Figure 14. Effect of turbulent pressure fluctuations on cavitation development using FIRE code.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

tota

l vo

id f

ract

ion

g

x/L

Experimental data

CE=0

CE=1.2

(a)

(b)

(c)

(d)

(e)

(f)

(g)


CONCLUSION In this paper two cavitation models have been studied. We

compared a mixture model in conjunction with the HRM phase

change model, as implemented in CONVERGE, with a

multi-fluid model with Rayleigh bubble dynamics for phase

change, as implemented in FIRE. The numerical results have

been assessed against experimental data collected at Argonne

National Laboratory using x-ray techniques. The study focused

on a cavitating nozzle tested with gasoline fuel at atmospheric

outlet pressure. Numerical results were compared to

measurements of mass flow rate and high-resolution void

fraction pattern, evaluated as radiographic projections.

From an engineering point of view, the two models showed

good predictive capabilities. In more detail, on equal grid size,

mass flow rates are underestimated by 8% using the multi-fluid

model, and by 10% using the mixture model. The multi-fluid

model is able to match reasonably well the total void fraction as

a function of the axial position. The mixture model over-predicts

this quantity by up to 30% for large part of the channel. The

multi-fluid model is also able to predict the liquid stream

reattachment to the wall occurring approximately at one third of

the channel length, while the mixture model does not capture this

effect. The axial location of the centerline void is slightly better

predicted by the mixture model instead, but the multi-fluid

model captures the disconnection between the void region at the

inlet edge and the void in the centerline, as it is observed in the

experiments.

The simulations have been carried out using a compressible

three-phase flow approach, in both codes, and this aspect was

shown to be essential for capturing the formation of the void

cloud in central region of the channel. Grid convergence has

been checked, and with respect to the total void fraction some

work is still needed. Future studies will consider finer cell sizes

to comprehensively address this issue. Parametric study on the

amount of non-condensable gases revealed that this quantity has

a strong effect on the amount of cavitation produced. The higher

the amount of air, the lower the vapor generation. However,

surprisingly, the total amount of the two gaseous phases is quite

independent of the amount of non-condensable gases. Therefore,

further data is needed to fully validate the models and split the

two contributions. If the operating pressures are increased, the

void cloud in the channel core is reduced, while the cavitation at

the inlet edge remains largely unaffected. Future work will

address further validations including the comparison with

experimental pressure measurements inside the channel. Also

LES studies will be performed to better address the effect of

turbulence on cavitation inception.

ACKNOWLEDGMENTS The submitted manuscript has been created by UChicago

Argonne, LLC, Operator of Argonne National Laboratory

(“Argonne”). Argonne, a U.S. Department of Energy Office of

Science laboratory, is operated under Contract No. DE-AC02-

06CH11357. The U.S. Government retains for itself, and others

acting on its behalf, a paid-up nonexclusive, irrevocable

worldwide license in said article to reproduce, prepare derivative

works, distribute copies to the public, and perform publicly and

display publicly, by or on behalf of the Government. This

research was funded by DOE’s Office of Vehicle Technologies,

Office of Energy Efficiency and Renewable Energy under

Contract No. DE-AC02-06CH11357. The authors wish to thank

Gurpreet Singh, program manager at DOE, for his support.

We gratefully acknowledge the computing resources

provided on "Fusion," a 320-node computing cluster operated by

the Laboratory Computing Resource Center at Argonne National

Laboratory.

The authors would also like to acknowledge Dr. Eric

Pomraning and Dr. Hongwu Zhao at Convergent Science Inc. for

helping with the set-up in CONVERGE software.

We also acknowledge AVL-Graz for providing support with

the FIRE code during the research activity.

Lastly, the authors would also like to acknowledge Dr.

Daniel Duke and Dr. Chris Powell at Argonne National

Laboratory for sharing the experimental data and many helpful

discussions.

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