Content uploaded by Francesco Duronio
Author content
All content in this area was uploaded by Francesco Duronio on Jun 29, 2023
Content may be subject to copyright.
Phase transition in cavitation-driven chemical processes
Eulerian-Lagrangian modelling of phase transition for application to
cavitation-driven chemical processes
Francesco Duronio,1Andrea Di Mascio,1Angelo De Vita,1, a) Valentina Innocenzi,1and Marina Prisciandaro1
Università degli studi dell’Aquila - Piazzale Ernesto Pontieri, Monteluco di Roio, 67100 L’Aquila (AQ),
Italy
(*Corresponding author: valentina.innocenzi1@univaq.it)
(Dated: 29 June 2023)
Hydrodynamic cavitation (HC) is a promising technology for several applications, like disinfection, sludge treatment,
biodiesel production, degradation of organic emerging pollutants as pharmaceutical, and dye degradation.
Due to local saturation conditions, cavitating liquid exhibit generation, growth, and subsequent collapse of vapour-
filled cavities. The cavities’ collapse brings very high pressure and temperature; this last aspect is essential in some
chemical processes because it induces the decomposition of water molecules into species with a high oxidation poten-
tial, which can react with organic substances.
Properly exploiting this process requires a highly accurate prediction of pressure peak values. To this purpose, we
implemented a multi-phase Eulerian-Lagrangian code to solve the fluid-dynamic problem, coupled with the Rayleigh-
Plesset equation, to capture the evolution of bubbles with the required accuracy. The algorithm was validated against
experimental data acquired with optical techniques for different cavitation-shedding mechanisms.
Then, we used the developed tool to investigate the decolouration of organic substances from a cavitation Venturi
tube operating at different pressure. We compared the obtained results with the experimental observation to assess the
reliability of the developed code as a predictive tool for cavitation and the possibility of using the code itself to assess
scale-up criteria for possible industrial applications.
I. INTRODUCTION
Cavitation occurs in flow regions where the pressures drop
below the vapour pressure. Vapour cavities develop as large
voids or as clouds of bubbles and can be attached to a wall
or travel in the mainstream flow. Experimental observations
show that, depending on the flow conditions, two mechanisms
lead to cavitation with bulk vapour shedding1. The first is
a reentrant jet, typically when the attached or partial vapour
cavity is short. The reentrant jet has the topology of a thin liq-
uid film wedged between the solid boundary and the vapour
cavity. It is periodically generated at the cavity closure re-
gion when the vapour cavity has reached its maximum length
during a shedding cycle2. It then travels upstream until it
causes cavity detachment. This detached cloud travels down-
stream and collapses where pressure has returned to a value
higher than the vapour pressure. At the same time, a new
vapour cavity begins to form. Figure 1 is a schematic view
of the vapour cavity and the reentrant flow corresponding to
the mid-plane in a cavitating Venturi tube. The second mech-
anism is condensation or bubbly shock and typically occurs
when the operating conditions permit the attached vapour re-
gion to extend further downstream. This phenomenon relates
to pressure waves propagating upstream and originated by the
collapse of previously shed vapour3. Once these waves re-
turn at the attached vapour’s rear edge, they trigger condensa-
tion shock fronts, causing vapour separation from the wall and
cloud collapse. Detailed investigations about this mechanism
a)Also at Consiglio Nazionale delle Ricerche, Istituto di Scienze e Tecnologie
per l’Energia e la Mobilità Sostenibili (STEMS),Via G. Marconi 4, 80125
Napoli – Italy.
on wedges and other geometries are in4–6.
FIG. 1: Schematic of re-entrant cavitation regime.
In both mechanisms, cavitation generates extreme condi-
tions useful in specific chemical processes. In recent years, re-
search activities aimed at the exploitation of the phenomenon
of cavitation for biodiesel production7, hydrolysis of oils8,
polymerization/depolymerization9, wastewater treatment10,11.
Most of this research is in the field of water treatment and the
development of sustainable industrial activities.
The chemical effect resulting from the HC process is the
release of hydroxyl radicals (•OH) generated after the disso-
ciation of the water molecules. These radicals strongly oxi-
dise and can degrade organic pollutant compounds from con-
taminated water. The scientific literature reports many re-
search activities concerning the degradation of various organic
substances12–14, pesticides15–17, pharmaceuticals18–25, sludge
treatment26, bacteria and virus disinfection27,28 . Among the
most studied applications, there is the degradation of dyes
mainly from synthetic solutions such as Reactivethe Red 120
dye29, Brilliant Blue30 , Congo red31 reactive orange 4 dye32
methylene blue33,34.
In previous papers, the authors demonstrated hydrody-
Phase transition in cavitation-driven chemical processes 2
namic cavitation’s efficiency in degrading methyl orange dye
(MO)35–38. MO or Sodium 4-[(4-dimethylamino) phenyl-
diazenyl] benzenesulfonate is an azo dye soluble in water;
it is used in textile production and is a residue in industrial
wastewater. It is not easy to degrade MO with conventional
treatment39; to improve the degradation efficiency, advanced
oxidation processes (AOPs) are applied alone or in combina-
tion with other degradation methods. Electrochemical pro-
cesses, photocatalytic oxidations, Fenton, and ultrasonic tech-
nologies are used for a partial degradation of MO40–42. Hy-
brid techniques of AOPs can be applied to increase the degra-
dation efficiency42–44. However, these methods have a series
of limits as the preparation of catalysts, which often requires
complex operations that do not make them applicable on an
industrial scale45. An alternative to these processes is hydro-
dynamic cavitation, often in combination with other processes
in which chemicals substances such as hydrogen peroxide and
ozone increase the number of free radicals produced that react
and destroy organic pollutant molecules46,47.
The limitation of hydrodynamic cavitation for industrial-
scale application is that there are no unique criteria for scaling
up the cavitation reactor, thus restricting the applications to
laboratory or pilot scale47. It follows that simulation tools
are required to model such a phenomenon and to optimize the
operating parameters.
The Single Bubble Dynamic (SBD) models must be men-
tioned among the simplest approaches. They present the so-
lution of the Rayleigh-Plesset equation for a single bubble by
coupling it with continuity, momentum, and energy equations
for a 1D environment20,48.
Numerous efforts and CFD (Computational Fluid-dynamic)
studies investigate the phenomenon of hydrodynamic cavi-
tation. The attention is focused on many different aspects:
the solution algorithm, the modelling of the gas/liquid in-
terface, the phase transition model, the influence of turbu-
lence, the thermal effects, the presence of incondensable
gases, etc... Concerning fully Eulerian continuum approach49
various authors proposed research featuring the single-fluid
method50–52. In contrast, others choose the two-fluid ap-
proach, where conservation equations are solved separately
for each phase, and interaction between them is consid-
ered by using additional source terms53,54. Several incom-
pressible approaches were applied for simulating cavitating
flows55–57 while compressible cavitation solvers use the so-
called barotropic equation of state to relate the density of
vapour–liquid mixture to the pressure field. The results ob-
tained with such models exhibit excellent agreement with the
experiments56,58–60.
Phase transition and gas/liquid interface modelling is rele-
vant in cavitation CFD codes. Sauer et al. develop a model
that exploits a simplified version of the Rayleigh-Plesset equa-
tion where the bubble growth rate is a function of satura-
tion and local pressure61,62. Homogeneous equilibrium mod-
els (HEM) compute the vapour amount present by exploiting
the barotropic equation of state (EoS) and assuming a perfect
mixing between the liquid and the vapour phases56,60,63. Ho-
mogeneous relaxation models (HRM) are used when thermal
non-equilibrium between the phases is present64,65.
Some other CFD codes adopt a combined Eulerian-
Lagrangian approach wherein the bubbles are treated as dis-
crete parcels. Such a modelling approach is essential when
the investigation aims to resolve the physics down to the sin-
gle bubble scale, maintaining computation cost affordable to
capture the overall large-scale flow behaviour66,67.
Turbulence-induced pressure fluctuations strongly influ-
ence cavitating flows68. Many researchers studied the mu-
tual interaction of turbulence and cavitation with differ-
ent approaches59,69?,70. RANS (Reynolds Averaged Navier
Stokes) models, in certain conditions, overestimate turbulent
viscosity in cavitation zones preventing a correct represen-
tation of re-entrant jet motion71,72. At the same time, they
predict cavitation with reasonable accuracy in high-pressure
difference conditions73. Many other scientific works demon-
strate how LES (Large Eddy Simulation) models can ad-
equately describe cavitation inception and evolution5,74,75.
DES approaches were also successfully used in studying cav-
itating flows55,76,77.
In specific cavitating flows, such as those issued in fuel
injectors, incondensable gases and thermal effects are rele-
vant, so various researchers focused their attention on this
topic49,64,69,78.
In the present paper, we implemented a CFD code with an
Eulerian-Lagrangian approach that solves the governing equa-
tions for a multi-phase flow in 3D domains. In the code, a
finite volume approach solves the continuous field of a multi-
phase mixture. We fill the continuous phase with Lagrangian
particles representing the cavitation bubbles to capture the de-
tails of local pressure evolution. A specific injection model,
accounting for the conditions for cavitation bubbles nucle-
ation, ensures a proper parcel initialization. We adopted the
classical fourth-order Runge-Kutta scheme to integrate the
Rayleigh-Plesset equation to accurately capture the cavitation
bubbles’ evolution.
We first validated the CFD approach against experiments of
a cavitating Venturi reactor acquired by optical imaging tech-
niques available in the literature. In particular, we compared
the numerical data with the vapour volume fraction maps ob-
tained with computed tomography (CT). The comparison con-
sidered grid uncertainty obtained by grid refinement in the
preliminary verification procedure.
Next, the decolouration of organic substances from a cavi-
tation Venturi tube, operating at different pressure, was inves-
tigated experimentally at the Laboratory of Chemical Plants
of the University of L’Aquila (Italy). Comparing the mea-
surements with the CFD results further verifies the proposed
approach’s predictive capabilities and benefits in defining the
parameters that mainly affect chemical decolouration, which
helps solve the scale-up problem.
II. NUMERICAL METHODOLOGY
An Eulerian-Lagrangian solver was developed in the Open-
FOAM library to properly model the hydrodynamic cavitation
process. The algorithm solves the continuity equation and mo-
Phase transition in cavitation-driven chemical processes 3
mentum equations for the mixture:
∂ ρm
∂t+∇·(ρmU) = 0 (1)
∂ ρmU
∂t+∇·(ρmUU) = −∇P+∇·[(µeff(∇U+ (∇U)T](2)
where:
•ρmis the mixture density;
• P is pressure;
•Uthe mixture velocity;
•µe f f is the effective viscosity
A pressure correction equation is obtained from the continu-
ity and momentum equations, also exploiting the equation of
state for the mixture79,80. An iterative PIMPLE algorithm is
employed within which the density and the vapour volume
fraction are computed accordingly in the following equations:
ρm= (1−γ)ρ0
l+ΨmP(3)
γ=ρm−ρl,sat
ρv,sat −ρl,sat
(4)
where:
•γis the vapour volume fraction ( γ=0 means no vapour;
when γ=1, a cell is filled with vapour).
•ρ0
lis a liquid reference density equal to:
ρ0
l=ρl,sat −ΨlP
sat (5)
•ρl,sat and ρv,sat are liquid and vapour at saturation pres-
sure densities, respectively.
•Ψmthe mixture compressibility can be modelled in sev-
eral ways. In the present work, a linear model based on
vapour volume fraction was adopted:
Ψm=γΨv+ (1−γ)Ψl(6)
The mixture compressibility is related to the speed of
sound (a):
Ψm=1
a2(7)
Finally, the effective viscosity is given by
µe f f =µm+µt(8)
where
•µtrepresents the turbulent viscosity modelled by a
proper turbulence model.
100 1000 10000 100000 1000000
0.001
0 . 0 1
0 . 1
1
1 0
100
1000
10000
D e n s i t y [ k g / m 3]
P r e s s u r e [ P a ]
C o o l P r o p D e n s i t y
B a r o t r o p i c E o S
FIG. 2: comparison of the density values computed with the
adopted barotropic EoS for water against the real values
obtained from the CoolProp database.
•µmis the mixture viscosity given by:
µm=γµv+ (1−γ)µl(9)
Figure 2 compares the density values computed with the
barotropic EoS for water against the real values obtained from
the CoolProp database.
The Eulerian solution is coupled through the pressure field
to a Lagrangian Particle Tracker (LPT) framework, which de-
scribes the cavitation bubbles’ evolution within the domain.
Each parcel represents a multitude of bubbles that share sim-
ilar characteristics. Position and velocity are updated at each
time step solving differential equations for trajectory and mo-
mentum:
dxp
dt =up(10)
mp
dup
dt =∑Fi(11)
with:
•upparcel velocity.
•mpmass.
•xpposition.
•Figeneric force acting on the parcels (i.e. drag, gravity,
etc ...).
In addition to the fundamental dynamic equations, the LPT
accounts for the solution of the complete Rayleigh-Plesset
equation to describe the expansion and the collapse of the
vapour cavities:
R(t)¨
R(t) + 3
2˙
R2(t) = P
B−P
ρl
−4νl
˙
R(t)
R(t)−
2σst
ρlR(t)(12)
where:
Phase transition in cavitation-driven chemical processes 4
•R(t),˙
R(t),¨
R(t): are instantaneous radius, first and sec-
ond derivative, respectively.
•P: mixture pressure, obtained from the Eulerian frame-
work.
•P
B: bubble pressure.
•µl: liquid kinematic viscosity.
•σl: liquid surface tension
•ρl: liquid density
The coupling with the Eulerian framework is obtained in a
one-way fashion through the pressure field P, which is inter-
polated from the cell value for the single particle. In addition
to the Rayleigh-Plesset equation, it is assumed that the bubble
contains some quantity of non-condensable gas whose partial
pressure is P
Go for some reference size Ro. Virtually all liquids
have some dissolved gas. Indeed, it is impossible to eliminate
this gas from any substantial liquid volume. These dissolved
gases act as nucleation sites for cavitation onset. With no ap-
preciable mass transfer between gas and liquid and thermal
equilibrium, it can be demonstrated that the bubble pressure
is related to the bubble radius by48:
pB(t) = P
V+P
Go Ro
R3k
(13)
where:
•P
Go : is incondensable gas partial pressure at reference
conditions.
•R0: is the initial bubble radius.
•P
V: is the saturation pressure.
•k: is the specific heat ratio.
The Runge-Kutta 4th order scheme integrates the Rayleigh-
Plesset equation with adaptive time-step. The Lagrangian
time step is smaller than the Eulerian time step and dynam-
ically changes according to the tolerance value chosen. The
pressure field Pused on each Runge-Kutta step is linearly in-
terpolated from the continuous phase, using its value at the
current and the previous time step. A specific injection model
was developed within the solver. The model has three main
input parameters:
•fb: temporal injection frequency [s].
•nb: bubble density, number of particle injected per vol-
ume unit [ 1
m3].
•R0: initial radius of the injected bubbles [m]
The Lagrangian particles, representative of the vapour bub-
bles, are injected only in those cells where there are condi-
tions for cavitation (i.e. local pressure lower than the liquid
saturation pressure). The insemination takes place with the
temporal frequency fb; for each injection event, the number
of particles added is such to respect the input bubble density
parameter nb. The injection positions are randomly chosen
within the eligible cells following the method implemented in
OpenFOAM. For further details, see81.
Turbulence is modelled using Spalart-Allmaras DES (De-
tached Eddy Simulation)82. This hybrid model switches be-
tween a pure LES approach in the core turbulent region where
large unsteady turbulence scales can be resolved by grid size,
and a RANS Spalart-Allmaras model near solid walls, where
the typical length scale of the turbulent eddies is significantly
smaller than the grid dimensions.
III. PRELIMINARY VALIDATION
The proposed numerical approach was validated in compar-
ison with the experimental recordings of a hydrodynamic cav-
itation device acquired by Hogendoorn et al.83 and Jahangir et
al.84,85. The shadowgraph optical technique acquired images
of the flow within a Venturi tube.
CFD simulations replicate the exact geometry used in the
experiments. Figure 3 reports all the characteristics of the
considered Venturi tube.
FIG. 3: Geometrical Characteristics of the Venturi tube used
for the code validation.
OpenFOAM’s utility cartesianMesh was used to discretize
the geometry. Three grids were adopted: a fine grid of 0.5 mm
base dimension with the throat region refined up to a dimen-
sion of 0.25 mm; an intermediate grid with dimensions of 1
mm for the base grid and 0.5 mm for the throat and a coarse
grid of 2mm in the base grid and 1 mm in the contraction.
Figure 4 shows the fine grid where the total cell count for the
fine grid is 3 M.
A variable time step was chosen to keep the Courant num-
ber equal to 0.25. The data for investigated conditions are re-
ported in Table I (inlet flow rate and outflow pressure). They
are representative of the two cavitation regimes previously ex-
posed and experimentally investigated.
This combination of boundary conditions guarantees com-
putational stability86,87.
Non-dimensional quantities were used to compare the ex-
perimental and CFD results. In particular, the Strouhal num-
ber Stdis used to measure the shedding frequency and is de-
Phase transition in cavitation-driven chemical processes 5
FIG. 4: Computational domain: particular view of the
convergent-divergent section.
TABLE I: Simulated conditions parameters.
Regime Inlet Flow-Rate [dm3/s] Outflow Pressure [bar]
Re-entrant jet 2.94 0.8
Condensation shock 2.99 0.4
fined as:
Std=f D
U(14)
where:
• f is the shedding frequency.
• D is a characteristic length considered equal to the
throat diameter
•Uis the average velocity at the throat section.
The cavitation number can express the presence of vapour and
the possibility of cavitation:
Cv=P
2−P
v
1
2ρU2(15)
where:
•P
2is the downstream pressure, measured at a distance
of 400 mm and sufficiently far downstream the critical
section
•P
vis the vapour pressure,
•ρis the reference liquid density.
In Table II, the values of these parameters obtained from the
CFD simulations are compared with the one experimentally
measured.
The experimental and numerical value of the Strouhal num-
ber in table II shows that the shedding frequency is reasonably
predicted for both flow conditions. Figure 5 reports the run-
ning average of pressure and velocity sampled along the axis,
together with the computed uncertainty.
TABLE II: Cavitation Parameters and GCI.
Regime Strouhal
Number
GCI Cavitation
Number
GCI
Re-entrant jet Experiments 0.24 - 0.86 -
CFD Sim. 0.21 3.5% 0.88 3.2%
Condensation Experiments 0.056 - 0.25 -
shock CFD Sim. 0.051 3.3% 0.28 3.6%
0 50 100 150 200 250 300
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1 . 0
1 . 2
1 . 4
1 . 6
1 . 8
2 . 0
P r e s s u r e [ b a r ]
A x i a l D i s t a n c e [ m m ]
P r e s s u r e r e - e n t r a n t
P r e s s u r e c o n d e n s a t i o n s h o c k
0
2
4
6
8
1 0
1 2
1 4
1 6
1 8
2 0
A x i a l V e l o c i t y r e - e n t r a n t
A x i a l V e l o c i t y c o n d e n s a t i o n s h o c k
A x i a l V e l o c i t y [ m / s ]
FIG. 5: Temporal averages of pressure and velocity.
The vertical grey line represents the throat section. The un-
certainty of the numerical results was assessed as in the clas-
sical paper by Roache88,89, in which the uncertainty assess-
ment procedures, now adopted by AIAA, ITTC and IEEE are
described and discussed in detail. The Grid Convergence In-
dex for the Strouhal number in the two test cases is reported
in table II. The figure shows that the minimum pressure is
reached downstream of the throat section and, as expected, is
lower in the condensation shock mechanism. The axial veloc-
ity reaches its maximum value in the same axial position in
both cases.
To verify the adequacy of the adopted grid far from the
Venturi tube surface, where the turbulence model reduces to a
large-eddy simulation, the modelled kinetic energy was eval-
uated and compared with the total energy. To do that, we fol-
lowed Di Mascio et al.90. Figure 6 shows the ratio between the
modelled kinetic energy and the total kinetic energy for the re-
entrant flow (top) and condensation shock regime (bottom) on
the longitudinal mid-plane for a representative temporal in-
stant.
For both conditions, the modelled kinetic energy is reason-
ably small in the bulk flow compared to the resolved one; the
ratio between the modelled and total kinetic energy is almost
everywhere smaller than the value of 0.3, and therefore, the
grid can be considered adequate for LES resolution, accord-
ing to the Pope criterion91 (there are only a few spots where
the latter exceeds 0.3, reachings at most 0.5). Of course, the
ratio becomes larger than 0.3 in the boundary layer on the tube
Phase transition in cavitation-driven chemical processes 6
FIG. 6: Kmod /(Kmod +Kres)for re-entrant jet and
condensation shock regimes; longitudinal section.
surfaces, where we have a Reynolds-averaged Navier–Stokes
equation simulation.
Figure 7 reports a comparison of frames extracted from
the high-speed images acquired with the shadowgraph
technique84,85 and the CFD simulation results for the re-
entrant jet shedding regime (fine grid). The high-speed shad-
owgraphy experiment is shown on the left while the numerical
simulation is on the right. The vapour volume fraction is com-
puted accordingly with Eq. 4.
A complete shedding cycle is reported. The numerical re-
sults for the vapour structures reasonably match the experi-
mental observations. In particular, the numerical simulation
captures the re-entrant jet shedding mechanism, as seen in the
velocity field reported in Figure 8 for a shedding cycle.
The white iso-lines represent a vapour volume fraction
equal to 0.1. The attached cavity grows at the contraction
while the previously shed vapour flows downstream. After 2-
3 ms, a cavitating vortex ring appears, with a clear re-entrant
jet (white zone) close to the solid boundary. As it arrives at
the throat section, it splits the attached cavity from the wall
shedding another vortex ring.
Figure 9 compares the numerical results with the experi-
mental observation for the condensation shock regime (fine
grid). Images comparison demonstrates good agreement in
terms of vapour distribution. As shown, the lower downstream
pressure brings more extended cavitation clouds.
Computed Tomography by Jahangir et al.92 was used to re-
construct 2D experimental time-averaged x-ray images of the
cavitating Venturi nozzle to generate a 3D quantitative rep-
resentation of the averaged vapour content. The x-ray im-
ages show the average vapour distribution at different planes
across the divergent section starting from the throat section
FIG. 7: Comparison of vapour volume fraction for re-entrant
jet conditions. Left: experimental data (reproduced from
Brunhart et al85 - Physics of Fluids 32, 083306 (2020), with
permission of AIP Publishing); right: CFD vapour volume
fraction.
Phase transition in cavitation-driven chemical processes 7
FIG. 8: Cross section of the velocity field. The volume
fraction’s contour line γ=0.1 is overlapped to identify the
vapour.
and equally spaced as described in Figure 10 (further exam-
ples of CT reconstruction can be found in Bauer et al.93). The
CFD results for the vapour volume fraction were averaged
over several shedding cycles (for the whole simulated period
of 0.2 s ) to compare them with the averaged CT slices.
Figure 11 compares experimental CT results to CFD im-
ages.
The agreement is satisfactory; the only differences can be
observed in the last slice (the farthest from the throat section),
where the CFD code predicts a greater vapour cloud. It is
also interesting to investigate the flow field in the condensa-
tion shock regime.
Figure 12 reports the axial velocity field with the 10 % con-
tour of volume fraction overlapped. At first, vapour shed from
the previous cycle is present in the last part of the divergent
duct. Vapour grows in the throat section and expands down-
stream until 12 ms from the beginning of the shedding cycle.
Then, the condensation shock wave and the negative axial ve-
locity detach the vapour from the walls. The negative velocity
(white areas) extends even more and brings an almost com-
plete collapse of the vapour volume attached to the walls at
the end of the cycle.
IV. EXPERIMENTAL ACTIVITIES FOR DYE
DECOLOURATION
Once validated, the CFD code developed was applied to
study dye decolouration. This paragraph reports the exper-
imental decolouration of methyl orange carried out at the
FIG. 9: Comparison of numerical data and experimental
observations for condensation shock regime. Left:
experimental data (reproduced from Brunhart et al85 -
Physics of Fluids 32, 083306 (2020), with permission of AIP
Publishing); right: CFD vapour volume fraction.
Phase transition in cavitation-driven chemical processes 8
FIG. 10: Locations where computed tomography (CT) slices
were taken. Letters from A to F covers 93 mm across the
divergent section.
Chemical Plants the Laboratory of the University of L’Aquila
(Italy). The availability of a reliable simulation tool allows
the investigation of parameters that are not available from the
experiments and that are relevant for scaling up the process.
A. Chemical Substances
Methyl orange dye (chemical formula: C14H14N3NaO3S)
and sulfuric acid (98%) were provided by Carlo Erba94. MO
was diluted in distilled water (0.1% in concentration) to pre-
pare the synthetic solutions for experimental tests, while sul-
furic acid was used to adjust the solution pH to 2 to enhance
oh radical production.
B. Experimental set-up and procedure
The hydrodynamic cavitation apparatus reactor is shown in
Figure 13.
The system is composed of: a jacket holding tank (maxi-
mum useful volume 1 L), a centrifugal pump of power rating
375 W and rotating speed in the range 1100-3500 rpm, man-
ual valves, a flow meter, two electrical manometers (Barks-
dale Control Products, UPA2 KF16809D) and a Venturi tube.
Pipelines and fittings are in plastic materials (Rilsan and
polypropylene, respectively); pumps and valves are in stain-
less steel. The cavitation device ( Venturi tube) is made of
Plexiglas. Table III reports the geometrical characteristics.
The tank is connected to the suction tube of the pump, and
the dye solution is fluxed into the main and bypass line. A
thermostatic bath connected to the tank keeps the temperature
constant at 20 ◦C. The manometers are installed upstream and
downstream of the Venturi tube and placed in the main line to
measure the inlet and outlet pressure. The manual valves con-
trol the flow, take samples for analysis, and empty the entire
plant at the end of the test.
Decolouration of MO was carried out at different inlet pres-
sure conditions (from 3 bar min to 6 bar). A 60 minutes treat-
ment time was considered, and samples for analysis were col-
FIG. 11: Comparison of experimental observation and
numerical data for the condensation shock regime. Vapour
Volume Fraction averaged over several shedding cycles. The
experimental results are obtained with computed tomography
(CT). Left: experiment (reproduced with permission from
Jahangir et al.92, Int. J. Multiphase Flow 120, 103085 (2019).
Copyright 2019 Elsevier.); right: simulation
lected every ten minutes. The scope of the experiments was
to study the effect of the inlet pressure on the HC process and,
hence, on MO decolouration. The samples were analyzed
on UV-spectrometer (Cary 1E, UV Visible spectrophotome-
ter Varian). Dye decolouration is determined as a function of
absorbance reduction at the maximum wavelength of 507nm.
A calibration curve was used to measure the concentration of
dye. The decolouration efficiency ηwas calculated by the
following equation:
η=Co−Ct
Co
·100 (16)
where Co (mg/L) and Ct (mg/L) are the initial concentration
Phase transition in cavitation-driven chemical processes 9
FIG. 12: Cross section of the velocity field with represented
volume fraction contour γ=0.1.
TABLE III: Geometrical specifications of the Venturi tube
used for hydrodynamic cavitation tests.
Venturi tube for dye decolouration e
Length of the converging 12 mm
Length of the diverging 30 mm
Maximum diameter 32 mm
Orifice diameter 2 mm
and the concentration at the specific time (t), respectively. Ex-
perimental procedures for the MO degradation are also de-
scribed in35–37,47.
C. Experimental Results
Figure 14 shows the behaviour of dye decolouration effi-
ciency and the cavitation number versus different Venturi inlet
FIG. 13: Experimental apparatus adopted for the HC
experiments.
pressures (pin ).
3 4 5 6
0
5
1 0
1 5
2 0
2 5
3 0
3 5
M O D e c o l o r a t i o n E f f i c i e n c y [ - ]
pi n [ b a r ]
M O D e c o l o r a t i o n
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
C a v i t a t i o n N u m b e r
C a v i t a t i o n N u m b e r [ - ]
FIG. 14: MO decolouration (%) and Cavitation number as a
function of inlet pressure to the Venturi tube.
The MO decolouration efficiency is approximately equal to
20 % at 3, 5 and 6 bar while, at 4 bar is greater reaching its
maximum value of 32.5 %. The experimental uncertainty was
1.8%, obtained by replicating the tests three times.
The cavitation number (Cv, Eqn. 15) slightly decreases with
an increasing inlet pressure.
V. DYE DECOLOURATION CFD INVESTIGATION
A. Setup of Simulations
The computational domain consists of a convergent-
divergent geometry that reproduces the experimental device.
Phase transition in cavitation-driven chemical processes 10
Experimentally measured static pressure values were imposed
as boundary conditions at the upstream and downstream sec-
tions of the Venturi tube, respectively. Six cases with inlet
pressure ranging from 3 to 6 bar were investigated. The outlet
pressure was always 1 bar. The vapour volume fraction was
set to zero at the inlet section. Figure 15 shows the grid com-
puted with the OpenFOAM’s utility cartesianMesh. Three
grids were adopted: a fine grid of 0.5 mm with the throat
section zone refined up to a dimension of 0.25 mm; an inter-
mediate grid with dimensions of 1 mm, 0.5 mm for the throat
section, and a coarse grid of 2-1 mm.
(a)
(b)
FIG. 15: Computational domain: (a) general overview and
(b) particular view of the convergent-divergent section..
A variable time step was chosen to keep the Courant num-
ber below 0.25. The simulations last 0.12 s and reproduce
several vapour-shedding cycles.
B. Numerical results
The different operating conditions simulated can be briefly
synthesized with Strhoual and cavitation numbers as reported
in Figure 16.
Increasing the inflow pressure, both the Strouhal and cavita-
tion number decrease. The cavitation number experimentally
measured and the one computed from the CFD results are in
reasonably good agreement; some differences are present only
with pin = 6 bar.
Figure 17 shows two frames experimentally acquired by
Capocelli et al.20, with an ultra-fast camera using the same
3 4 5 6
0 . 0 0
0 . 0 2
0 . 0 4
0 . 0 6
0 . 0 8
S t r o u h a l N u m b e r [ - ]
pi n [ b a r ]
S t r o u h a l N u m b e r
0 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
C a v i t a t i o n N u m b e r E x p e r i m e n t a l
C a v i t a t i o n N u m b e r C F D
C a v i t a t i o n N u m b e r [ - ]
FIG. 16: Strhoual and cavitation numbers plotted against
inlet pressure.
device investigated in this work. In the same figure 17, CFD
simulations, at the same pin conditions and at three different
time steps, are reported.
The proposed numerical approach seems to properly re-
produce the cavitation phenomenon experimentally observed.
The cavitation process is initiated in the throat section, where
bubbles nucleate on the channel’s lateral surfaces. It devel-
ops in the subsequent divergent section where bubbles grow
because of local low-pressure values22. This behaviour was
experimentally observed in various studies and also theoreti-
cally predicted22,48,95. It can also be observed that the higher
the pin is, the greater the vapour volume becomes. This is
another aspect clearly visible in experimental observations of
hydrodynamic cavitation by optical techniques,48,95.
Figure 18 reports the velocity field for the case with pin=3
and 6 bar to understand the shedding regime that is taking
place in the device.
As can be clearly observed, the vapour cavities are quite
short, and the white-coloured areas highlight negative axial
velocities; as highlighted by the volume fraction contours
γ=0.1, when increasing pin the shedding mechanism remains
almost unchanged.
Figure 19 reports the mean values of the number of parcels
generated and the collapse peak pressure as a function of the
inlet pressure. The overall averages were performed consider-
ing several shedding cycles.
As can be observed, the number of parcels increases as the
inlet pressure becomes greater while the peak pressure at the
bubble collapse reaches values of the order of 350 bar with
minimum oscillation.
Figure 20 shows the evolution of the bubbles coloured and
scaled accordingly with their radius for all the inlet pressures
investigated, i.e. from 3 to 6 bar.
The images start from the detachment of the bubble cloud
from the throat section and show that the higher the inlet pres-
sure, the greater the bubble cloud.
To relate this behaviour with the decolouration experiments
Phase transition in cavitation-driven chemical processes 11
FIG. 17: Vapour volume fraction. Experimental visualization
(reproduced with permission from Capocelli et al.20,
Chemical Engineering Journal 254, 1-8 (2014). Copyright
2014 Elsevier) and CFD simulations for inlet pressures of 3
bar and 6 bar. 3 bar (a - top) and 6 bar (bottom).
previously illustrated, it must be considered that two oppo-
site effects co-exist when increasing the inlet pressure: greater
vapour clouds appear, but, at the same time, the coalescence
phenomenon is enhanced too. At first, with relatively low in-
let pressure, the bubbles, where hydrolysis can occur, are very
few and, as a consequence, the OH radicals production is very
low (figure 20- pin =3). Increasing the pin the presence of
more bubbles has a positive effect enhancing the decoloura-
tion efficiency, while coalescence effects are still secondary
(figure 20- pin =4). Finally, if an excessively high inlet
pressure is adopted, the bubble density becomes so high that
these cavities coalesce to form much larger cavitation bubbles
where recombination of oxidizing radicals takes place (figure
20- pin =5−6). This behaviour was already discussed by var-
ious researchers in the field, demonstrating the presence of an
optimal operating condition for the hydrodynamic cavitation
as a method for chemical substances treatment96–99. Hence
this condition should be a trade-off between these two oppo-
site tendencies previously discussed.
FIG. 18: Mid-plane cross-section with represented vapour
volume fraction iso-lines at 10% over-imposed to an axial
velocity plot for pin=3 and 6 bar.
3 4 5 6
0 E + 0
1 E + 4
2 E + 4
3 E + 4
4 E + 4
5 E + 4
6 E + 4
7 E + 4
8 E + 4
9 E + 4
1 E + 5
N u m b e r o f P a r c e l s G e n e r a t e d [ - ]
pi n [ b a r ]
N u m b e r o f B u b b l e s G e n e r a t e d
0
5 0
100
150
200
250
300
350
400
450
500
P e a k P r e s s u r e @ B u b b l e C o l l a p s e
P e a k P r e s s u r e @ B u b b l e C o l l a p s e [ b a r ]
FIG. 19: Peak pressure at bubble collapse and number of
parcels present in ther domain for pin=3 to 6 bar.
VI. CONCLUSIONS
A combined numerical-experimental analysis of the hy-
drodynamic cavitation process was carried out. An Eule-
rian–Lagrangian code purposely developed for hydrodynamic
cavitation processes was successfully implemented within the
OpenFOAM library. It features an LPT framework to follow
the bubble’s development within the integration domain and,
in particular, the complete solution of the Rayleigh-Plesset
equation to calculate the vapour’s bubble diameter evolution.
Phase transition in cavitation-driven chemical processes 12
FIG. 20: Bubble evolution for pin=3 to 6 bar.
This CFD code was validated against comprehensive exper-
imental data acquired with the shadowgraph optical technique
of a cavitating Venturi tube. Both re-entrant jet and condensa-
tion shock-shedding mechanisms were considered. Computed
tomography data of vapour volume fraction in the divergent
section were also considered for a quantitative comparison
with numerical results.
The decolouration of organic substances with hydrody-
namic cavitation was also investigated. Initially, an exper-
imental campaign was performed using methyl orange in
synthetic aqueous solutions. Various experiments were per-
formed to change the inlet pressure. The results showed the
effects of this parameter on dye decolouration. In particular, it
was found that the highest HC process efficiency was recorded
at 4 bar inlet pressure with a MO decolouration yield of 32%
while, for lower (3 bar) and higher inlet pressures (5 and 6
bar), the efficiency is about 20 %.
Then, the developed code was tested on the same geometry
experimentally investigated. The main findings are:
• The numerical approach adopted is capable of repro-
ducing the cavitation process experimentally analyzed
as shown by both image and a-dimensional numbers
comparisons.
• When increasing the upstream pressure, both Strouhal
and cavitation number decrease.
• The bubble formation begins on the lateral surfaces of
the throat section. The bubbles flow in the divergent
Phase transition in cavitation-driven chemical processes 13
section collapse and reach an average peak pressure of
300-400 bar, depending on the inlet pressure.
• The bubble cloud reproduced by the CFD code becomes
more extended as the inlet pressure increase. However,
the coalescence phenomenon plays an important role
because it triggers the radicals recombination diminish-
ing the decolouration efficiency.
Finally, the CFD code exhibits a good predictive capability
and can be used to assess scale-up criteria for possible indus-
trial applications of hydrodynamic cavitation. Future works
will regard the investigation of scaled-up applications, differ-
ent geometries and operating conditions as well as the imple-
mentation of a coalescence model to better represent the pro-
cess of the radicals’ recombination.
ACKNOWLEDGMENTS
All the simulations were performed with the developed
solver on the Galileo100 cluster of HPC CINECA facilities
within the agreement between DIIIE - Università degli Studi
dell’Aquila and CINECA.
DATA AVAILABILITY STATEMENT
The data supporting this study’s findings are available in the
paper.
REFERENCES
1U. Gawandalkar and C. Poelma, “The structure of near-wall re-entrant flow
and its influence on cloud cavitation instability,” Experiments in Fluids 63,
77 (2022).
2R. A. Furness and S. P. Hutton, “Experimental and The-
oretical Studies of Two-Dimensional Fixed-Type Cavi-
ties,” Journal of Fluids Engineering 97, 515–521 (1975),
https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/97/4/515/5630720/515_1.pdf.
3J. K. Jakobsen, “On the Mechanism of Head Breakdown in Cav-
itating Inducers,” Journal of Basic Engineering 86, 291–305
(1964), https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/86/2/291/5496981/291_1.pdf.
4C. Wang, B. Huang, G. Wang, M. Zhang, and N. Ding, “Unsteady pres-
sure fluctuation characteristics in the process of breakup and shedding of
sheet/cloud cavitation,” International Journal of Heat and Mass Transfer
114, 769–785 (2017).
5B. Budich, S. J. Schmidt, and N. A. Adams, “Numerical simulation and
analysis of condensation shocks in cavitating flow,” Journal of Fluid Me-
chanics 838, 759–813 (2018).
6K. R. LABERTEAUX and S. L. CECCIO, “Partial cavity flows. part 1.
cavities forming on models without spanwise variation,” Journal of Fluid
Mechanics 431, 1–41 (2001).
7V. Innocenzi and M. Prisciandaro, “Technical feasibility of biodiesel pro-
duction from virgin oil and waste cooking oil: Comparison between tra-
ditional and innovative process based on hydrodynamic cavitation,” Waste
Management 122, 15–25 (2021).
8A. B. Pandit and J. B. Joshi, “hydrolysis of fatty oils: effect of cavitation,”
Chemical Engineering Science 48, 3440–3442 (1993).
9V. S. Moholkar, P. S. Kumar, and A. B. Pandit, “Hydrodynamic cavitation
for sonochemical effects,” Ultrasonics Sonochemistry. 6, 53–65 (1999).
10M. G ˛agol, A. Przyjazny, and G. Boczkaj, “Wastewater treatment by means
of advanced oxidation processes based on cavitation – a review,” Chemical
Engineering Journal 338, 599–627 (2018).
11Y. Tao, J. Cai, X. Huai, B. Liu, and Z. Guo, “Application of hydrodynamic
cavitation to wastewater treatment,” Chemical Engineering and Technology
39, 1363–1376 (2016).
12V. Innocenzi, M. Prisciandaro, and F. Vegliò, “Effect of the hydrodynamic
cavitation for the treatment of industrial wastewater,” Chemical Engineer-
ing Transactions 67, 529–534 (2018).
13P. Thanekar and P. Gogate, “Application of hydrodynamic cavitation reac-
tors for treatment of wastewater containing organic pollutants: Intensifica-
tion using hybrid approaches,” Fluids. 3, 98 (2018).
14P. D. S. S. S. S. Raut-Jadhav, M. P. Badve and A. Pandit, “Treatment of
the pesticide industry effluent using hydrodynamic cavitation and its com-
bination with process intensifying additives (h2 o2 and ozone,” Chemical
Engineering Journal 295, 326–335 (2016).
15S. Raut-Jadhav, D. Saini, S. Sonawane, and A. Pandit, “Effect of process
intensifying parameters on the hydrodynamic cavitation based degradation
of commercial pesticide (methomyl) in the aqueous solution,” Ultrasonics
Sonochemistry. 28, 283–293 (2016).
16R. K. Joshi and P. R. Gogate, “Degradation of dichlorvos using hydrody-
namic cavitation based treatment strategies,” Ultrasonics Sonochemistry.
19, 532–539 (2012).
17B. Li, S. Li, L. Yi, H. Sun, J. Qin, J. Wang, and D. Fang, “Degradation
of organophosphorus pesticide diazinon by hydrodynamic cavitation: Pa-
rameters optimization and mechanism investigation,” Process Safety and
Environmental Protection 153, 257–267 (2021).
18D. Panda and S. Manickam, “Hydrodynamic cavitation assisted degradation
of persistent endocrine-disrupting organochlorine pesticide dicofol: Opti-
mization of operating parameters and investigations on the mechanism of
intensification,” Ultrasonics Sonochemistry. 51, 526–532 (2019).
19M. Zupanc, T. Kosjek, M. Petkovšek, M. Dular, and B. Kompare, “B.
Širok, m,” Strazar, E. Heath, Shear-induced hydrodynamic cavitation as
a tool for pharmaceutical micropollutants removal from urban wastewater,
Ultrasonics Sonochemistry. 21, 1213–1221 (2014).
20M. Capocelli, M. Prisciandaro, A. Lancia, and D. Musmarra, “Hydrody-
namic cavitation of p-nitrophenol: A theoretical and experimental insight,”
Chemical Engineering Journal 254, 1–8 (2014).
21D. Musmarra, M. Prisciandaro, M. Capocelli, D. Karatza, P. Iovino, S. Can-
zano, and A. Lancia, “Degradation of ibuprofen by hydrodynamic cavita-
tion: Reaction pathways and effect of operational parameters,” Ultrasonics
Sonochemistry. 29, 76–83 (2016).
22M. Capocelli, D. Musmarra, M. Prisciandaro, and A. Lancia, “Chemical
effect of hydrodynamic cavitation: Simulation and experimental compari-
son,” AIChE Journal 60, 2566–2572 (2014).
23J. Lalwani, A. Gupta, S. Thatikonda, and C. Subrahmanyam, “Oxidative
treatment of crude pharmaceutical industry effluent by hydrodynamic cavi-
tation,” Journal of Environmental Chemical Engineering 8, 1 (2020).
24P. G. M. v. Bagal, “Degradation of diclofenac sodium using combined pro-
cesses based on hydrodynamic cavitation and heterogeneous photocataly-
sis,” (2014).
25G. Mancuso, M. Langone, G. Andreottola, and L. Bruni, “Effects of
hydrodynamic cavitation,” low-level thermal and low-level alkaline pre-
treatments on sludge solubilisation, Ultrasonics Sonochemistry. 59, 10475
(2019).
26F. P. P. R. M. H. B. M. Š. Zezulka, E. Maršálková, “High-pressure jet-
induced hydrodynamic cavitation as a pre-treatment step for avoiding
cyanobacterial contamination during water purification„” Journal of Envi-
ronmental Management. 255 (2020) 109862.
27K. K. Jyoti and A. B. Pandit, “Water disinfection by acoustic and hydrody-
namic cavitation,” Biochemical Engineering Journal 7, 201–212 (2001).
28V. K. Saharan, M. P. Badve, and A. B. Pandit, “Degradation of reactive
red 120 dye using hydrodynamic cavitation,” Chemical Engineering Journal
178, 100–107 (2011).
29S. Baradaran and M. T. Sadeghi, “Coomassie brilliant blue (cbb) degrada-
tion using hydrodynamic cavitation hydrogen peroxide and activated per-
sulfate (hc-h2o2-kps) combined process,” Chemical Engineering and Pro-
cessing - Process Intensification. 4 145, 10767 (2019).
Phase transition in cavitation-driven chemical processes 14
30Z. Askarniya, M.-T. Sadeghi, and S. Baradaran, “Decolorization of congo
red via hydrodynamic cavitation in combination with fenton’s reagent,”
Chemical Engineering and Processing - Process Intensification 150, 4
(2020).
31M. M. Gore, P. v. C. V. K. Saharan, Pinjari, and A. Pandit, “Degradation
of reactive orange dye using hydrodynamic cavitation based hybrid tech-
niques„” Ultrasonics Sonochemistry. 21 4, 1075–1082 (2014).
32B. Wang, T. Wang, and H. Su, “A dye-methylene blue (mb)-degraded by
hydrodynamic cavitation (hc) and combined with other oxidants,” Journal
of Environmental Chemical Engineering 10, 7 (2022).
33M. S. Kumar, S. H. Sonawane, and A. Pandit, “Degradation of methylene
blue dye in aqueous solution using hydrodynamic cavitation based hybrid
advanced oxidation processes,” Chemical Engineering and Processing: Pro-
cess Intensification 122, 288–295 (2017).
34J. P. Guin, Y. K. Bhardwaj, and L. Varshney, “Mineralization and
biodegradability enhancement of methyl orange dye by an effective ad-
vanced oxidation process,” Applied Radiation and Isotopes 122, 153–157
(2017).
35V. Innocenzi, M. Prisciandaro, and F. Vegliò, “Study of the effect of op-
erative conditions on the decolourization of azo dye solutions by using hy-
drodynamic cavitation at the lab scale,” The Canadian Journal of Chemical
Engineering 98, 1980–1988 (2020).
36V. Innocenzi, M. Prisciandaro, F. Tortora, and F. Vegliò, “Optimization of
hydrodynamic cavitation process of azo dye reduction in the presence of
metal ions,” Journal of Environmental Chemical Engineering 6, 6787–6796
(2018).
37V. Innocenzi, M. Prisciandaro, M. Centofanti, and F. Vegliò, “Comparison
of performances of hydrodynamic cavitation in combined treatments based
on hybrid induced advanced fenton process for degradation of azo-dyes,”
Journal of Environmental Chemical Engineering 7(2019).
38Z. Abbas-Shiroodi, M.-T. Sadeghi, and S. Baradaran, “Design and opti-
mization of a cavitating device for congo red decolorization: Experimental
investigation and cfd simulation,” Ultrasonics Sonochemistry. 71, 6 (2021).
39Y. L. Pang, A. Z. Abdullah, and S. Bhatia, “Review on sonochemical meth-
ods in the presence of catalysts and chemical additives for treatment of or-
ganic pollutants in wastewater,” Desalination. 277, 1–14 (2011).
40J. Luo, X. Zhou, L. Ma, X. Xu, J. Wu, and H. Liang, “Enhanced pho-
todegradation activity of methyl orange over ag2cro4/sns2 composites un-
der visible light irradiation,” Materials Research Bulletin 77, 291–299
(2016).
41I. M. S. Pillai and A. K. Gupta, “Effect of inorganic anions and oxidiz-
ing agents on electrochemical oxidation of methyl orange, malachite green
and 2,4-dinitrophenol,” Journal of Electroanalytical Chemistry 762, 66–72
(2016).
42R. Mohammadi and M. Mohammadi, “Photocatalytic removal of methyl
orange using ag/zn–tio 2 nanoparticles prepared by different methods,” De-
salination and Water Treatment 57, 11317–11325 (2016).
43M. Hassanpour, H. Safardoust, D. Ghanbari, and M. Salavati-Niasari, “Mi-
crowave synthesis of cuo/nio magnetic nanocomposites and its application
in photo-degradation of methyl orange,” Journal of Materials Science: Ma-
terials in Electronics 27, 2718–2727 (2016).
44S. Yang, R. Jin, Z. He, Y. Qiao, S. Shi, W. Kong, Y. Wang, and X. Liu,
“An experimental study on the degradation of methyl orange by combin-
ing hydrodynamic cavitation and chlorine dioxide treatments,” Chemical
Engineering Transactions 59, 289–294 (2017).
45S. Das, A. P. Bhat, and P. R. Gogate, “Degradation of dyes using hydrody-
namic cavitation: Process overview and cost estimation,” Journal of Water
Process Engineering 42, 6 (2021).
46B. Wang, H. Su, and B. Zhang, “Hydrodynamic cavitation as a promising
route for wastewater treatment – a review,” Chemical Engineering Journal
412, 5 (2021).
47V. Innocenzi, A. Colangeli, and M. Prisciandaro, “Methyl orange decolour-
ization through hydrodynamic cavitation in high salinity solutions,” Chem-
ical Engineering and Processing - Process Intensification 109050 (2022).
48C. E. Brennen, “Cavitation and bubble dynamics, cambridge university
press,” 2013.
49S. Yang and C. Habchi, “Real-fluid phase transition in cavitation modeling
considering dissolved non-condensable gas,” Physics of Fluids 32, 032102
(2020).
50F. Giussani, F. Piscaglia, G. Saez-Mischlich, and J. Hèlie, “A three-phase
vof solver for the simulation of in-nozzle cavitation effects on liquid atom-
ization,” Journal of Computational Physics 406, 109068 (2020).
51Y. Chen, X. Chen, J. Li, Z. Gong, and C. Lu, “Large eddy simulation
and investigation on the flow structure of the cascading cavitation shedding
regime around 3d twisted hydrofoil,” Ocean Engineering 129, 1–19 (2017).
52M. Gavaises, F. Villa, P. Koukouvinis, M. Marengo, and J.-P. Franc, “Vi-
sualisation and les simulation of cavitation cloud formation and collapse in
an axisymmetric geometry,” International Journal of Multiphase Flow 68,
14–26 (2015).
53H. Liu, W. Zhang, M. Jia, Y. Yan, and Y. He, “An improved method for cou-
pling the in-nozzle cavitation with multi-fluid-quasi-vof model for diesel
spray,” Computers Fluids 177, 20–32 (2018).
54W. Yuan and G. H. Schnerr, “Numerical Simulation of Two-Phase
Flow in Injection Nozzles: Interaction of Cavitation and Exter-
nal Jet Formation ,” Journal of Fluids Engineering 125, 963–969
(2004), https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/125/6/963/5833866/963_1.pdf.
55R. E. Bensow and G. Bark, “Implicit LES Predictions
of the Cavitating Flow on a Propeller,” Journal of Flu-
ids Engineering 132 (2010), 10.1115/1.4001342, 041302,
https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/132/4/041302/5517504/041302_1.pdf.
56M. S. Mihatsch, S. J. Schmidt, and N. A. Adams, “Cavitation erosion
prediction based on analysis of flow dynamics and impact load spectra,”
Physics of Fluids 27, 103302 (2015), https://doi.org/10.1063/1.4932175.
57M. Cristofaro, W. Edelbauer, P. Koukouvinis, and M. Gavaises, “A numer-
ical study on the effect of cavitation erosion in a diesel injector,” Applied
Mathematical Modelling 78, 200–216 (2020).
58A. Žnidarˇ
ciˇ
c, R. Mettin, and M. Dular, “Modeling cavitation in a rapidly
changing pressure field – application to a small ultrasonic horn,” Ultrason-
ics Sonochemistry 22, 482–492 (2015).
59P. Koukouvinis, H. Naseri, and M. Gavaises, “Performance of turbu-
lence and cavitation models in prediction of incipient and developed cav-
itation,” International Journal of Engine Research 18, 333–350 (2017),
https://doi.org/10.1177/1468087416658604.
60N. Kyriazis, P. Koukouvinis, and M. Gavaises, “Modelling cavitation dur-
ing drop impact on solid surfaces,” Advances in Colloid and Interface Sci-
ence 260, 46–64 (2018).
61J. Sauer and G. H. Schnerr, “Development of a new cavitation model based
on bubble dynamics,” ZAMM - Journal of Applied Mathematics and Me-
chanics / Zeitschrift F"ur Angewandte Mathematik Und Mechanik. 81,
561–562 (2001).
62W. Yuan, J. Sauer, and G. H. Schnerr, “Modeling and computation of un-
steady cavitation flows in injection nozzles,” Mécanique Industries 2, 383–
394 (2001).
63Modelling Injector Flow Including Cavitation Effects for
Diesel Applications, Fluids Engineering Division Summer
Meeting, Vol. Volume 2: Fora, Parts A and B (2007)
https://asmedigitalcollection.asme.org/FEDSM/proceedings-
pdf/FEDSM2007/42894/465/2669848/465_1.pdf.
64M. Battistoni, D. Duke, A. B. Swantek, F. Z. Tilocco, C. F. Powell, and
S. Som, “Effects of noncondensable gas on cavitating nozzles,” Atomiza-
tion and Sprays 25, 453–483 (2015).
65F. Duronio, A. D. Mascio, C. Villante, M. Anatone, and A. D. Vita, “Ecn
spray g: Coupled eulerian internal nozzle flow and lagrangian spray simu-
lation in flash boiling conditions,” International Journal of Engine Research
0, 14680874221090732 (0).
66J. Ma, C.-T. Hsiao, and G. L. Chahine, “Euler–Lagrange Sim-
ulations of Bubble Cloud Dynamics Near a Wall,” Journal of
Fluids Engineering 137 (2015), 10.1115/1.4028853, 041301,
https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/137/4/041301/6193360/fe_137_04_041301.pdf.
67E. GIANNADAKIS, M. GAVAISES, and C. ARCOUMANIS, “Modelling
of cavitation in diesel injector nozzles,” Journal of Fluid Mechanics 616,
153–193 (2008).
68W. Edelbauer, J. Strucl, and A. Morozov, “Large eddy simulation of cavi-
tating throttle flow,” (2016).
69G. Strotos, P. Koukouvinis, A. Theodorakakos, M. Gavaises, and G. Berge-
les, “Transient heating effects in high pressure diesel injector nozzles,” In-
Phase transition in cavitation-driven chemical processes 15
ternational Journal of Heat and Fluid Flow 51, 257–267 (2015), theme spe-
cial issue celebrating the 75th birthdays of Brian Launder and Kemo Han-
jalic.
70C. O. Iyer and S. L. Ceccio, “The influence of developed cavitation on the
flow of a turbulent shear layer,” Physics of Fluids 14, 3414–3431 (2002),
https://doi.org/10.1063/1.1501541.
71M. Dular and O. Coutier-Delgosha, “Numerical modelling of cavitation
erosion,” International Journal for Numerical Methods in Fluids 61, 1388–
1410 (2009), https://onlinelibrary.wiley.com/doi/pdf/10.1002/fld.2003.
72O. Coutier-Delgosha, R. Fortes-Patella, and J. L. Reboud, “Evalu-
ation of the Turbulence Model Influence on the Numerical Simula-
tions of Unsteady Cavitation ,” Journal of Fluids Engineering 125, 38–
45 (2003), https://asmedigitalcollection.asme.org/fluidsengineering/article-
pdf/125/1/38/5663252/38_1.pdf.
73A. Morozov, J. Struci, and W. Edelbauer, “Large eddy simulation of cav-
itating throttle flow,” in SimHydro 2014. New Trends in Simulation. 11-13
June 2014 Ecole Polytech’ Nice (France) 2014., Journées de l’hydraulique
(2014) included in a thematic issue : SimHydro 2014. New Trends in Sim-
ulation. 11-13 June 2014 Ecole Polytech’ Nice (France).
74M. H. Arabnejad, A. Amini, M. Farhat, and R. E. Bensow, “Numerical
and experimental investigation of shedding mechanisms from leading-edge
cavitation,” International Journal of Multiphase Flow 119, 123–143 (2019).
75X. Long, H. Cheng, B. Ji, R. E. Arndt, and X. Peng, “Large eddy simula-
tion and euler–lagrangian coupling investigation of the transient cavitating
turbulent flow around a twisted hydrofoil,” International Journal of Multi-
phase Flow 100, 41–56 (2018).
76P. K. Ullas, D. Chatterjee, and S. Vengadesan, “Prediction of unsteady,
internal turbulent cavitating flow using dynamic cavitation model,” Interna-
tional Journal of Numerical Methods for Heat Fluid Flow 32, 3210–3232
(2022).
77O. Usta and E. Korkut, “A study for cavitating flow analysis using des
model,” Ocean Engineering 160, 397–411 (2018).
78F. Orley, S. Hickel, S. J. Schmidt, and N. A. Adams, “Large-eddy simu-
lation of turbulent, cavitating fuel flow inside a 9-hole diesel injector in-
cluding needle movement,” International Journal of Engine Research 18,
195–211 (2017).
79J. H. Ferziger, M. Peri´
c, and R. L. Street, Computational methods for fluid
dynamics, Vol. 3 (Springer, 2002).
80M. Darwish and F. Moukalled, The finite volume method in computational
fluid dynamics: an advanced introduction with OpenFOAM® and Matlab®
(Springer, 2016).
81C. J. Greenshields et al., “Openfoam user guide version 6,” The Open-
FOAM Foundation 237, 624 (2018).
82P. R. Spalart, S. Deck, M. L. Shur, K. D. Squires, M. K. Strelets, and
A. Travin, “A new version of detached-eddy simulation,” resistant to am-
biguous grid densities, Theoretical and Computational Fluid Dynamics 20
(2006).
83W. Hogendoorn, “Cavitation: Experimental investigation of cavitation
regimes in a coverging-diverging nozzle,” (2017).
84S. Jahangir, W. Hogendoorn, and C. Poelma, “Dynamics of partial cavita-
tion in an axisymmetric converging-diverging nozzle,” International Journal
of Multiphase Flow 106, 34–45 (2018).
85M. Brunhart, C. Soteriou, M. Gavaises, I. Karathanassis, P. Koukouvinis,
S. Jahangir, and C. Poelma, “Investigation of cavitation and vapor shed-
ding mechanisms in a venturi nozzle,” Physics of Fluids 32, 083306 (2020),
https://doi.org/10.1063/5.0015487.
86H. K. Versteeg and W. Malalasekera, An introduction to computational fluid
dynamics: the finite volume method (Pearson education, 2007).
87F. Duronio, A. Montanaro, L. Allocca, S. Ranieri, and A. De Vita, “Ef-
fects of thermodynamic conditions and nozzle geometry on the methane di-
rect injection process in internal combustion engines,” in WCX SAE World
Congress Experience (SAE International, 2022).
88P. J. Roache, Quantification of uncertainty in computational fluid dynamics
(Annual Review of Fluid Mechanics, 1997).
89F. Duronio, S. Ranieri, A. Montanaro, L. Allocca, and A. De Vita, “Ecn
spray g injector: Numerical modelling of flash-boiling breakup and spray
collapse,” International Journal of Multiphase Flow 145, 103817 (2021).
90A. Di Mascio, G. Dubbioso, and R. Muscari, “Vortex structures in the
wake of a marine propeller operating close to a free surface,” Journal of
Fluid Mechanics 949, A33 (2022).
91S. B. Pope and S. B. Pope, Turbulent flows (Cambridge university press,
2000).
92S. Jahangir, E. C. Wagner, R. F. Mudde, and C. Poelma, “Void fraction
measurements in partial cavitation regimes by x-ray computed tomogra-
phy,” International Journal of Multiphase Flow 120, 103085 (2019).
93D. Bauer, F. Barthel, and U. Hampel, “High-speed x-ray ct imaging of
a strongly cavitating nozzle flow,” Journal of Physics Communications 2,
075009 (2018).
94C. E. Reagents, “Sodium hydroxide, pellets, 98%,” https:
//www.carloerbareagents.com/cerstorefront/cer-exp/c/
Sodium-hydroxide\%2C- pellets\%2C-98\%25/p/A16037 (2023),
[Online; accessed 2/4/2023].
95J. Ozonek, “Application of hydrodynamic cavitation in environmental en-
gineering, crc press,” 2012.
96S. K. Pawar, A. V. Mahulkar, A. B. Pandit, K. Roy, and V. S. Moholkar,
“Sonochemical effect induced by hydrodynamic cavitation: Comparison
of venturi/orifice flow geometries,” AIChE Journal 63, 4705–4716 (2017),
https://aiche.onlinelibrary.wiley.com/doi/pdf/10.1002/aic.15812.
97P. H. L. Alves, P. d. S. L. Silva, D. C. Ferreira, and J. C. d. S. I. Gonçalves,
“Cod removal from sucrose solution using hydrodynamic cavitation and
hydrogen peroxide: a comparison between venturi device and orifice plate,”
RBRH 24 (2019), 10.1590/2318-0331.241920180147.
98A. Mukherjee, A. Mullick, R. Teja, P. Vadthya, A. Roy, and S. Moulik,
“Performance and energetic analysis of hydrodynamic cavitation and po-
tential integration with existing advanced oxidation processes: A case study
for real life greywater treatment,” Ultrasonics Sonochemistry 66, 105116
(2020).
99P. Senthil Kumar, M. Siva Kumar, and A. Pandit, “Experimental quantifica-
tion of chemical effects of hydrodynamic cavitation,” Chemical Engineer-
ing Science 55, 1633–1639 (2000).
