Focusing on cavitation phenomena caused by high-speed submerged water jets, this paper presents an improved cavitation model for a compressible fluid mixture based on a concise estimation of fluid compressibility that considers phase change effects. The homogeneous two-phase flow assumption is adopted, and the gas phase is assumed to consist of vapor and non-condensable components. Equations of state for a pure liquid and an ideal gas are employed to evaluate the compressibility of the liquid and non-condensable components, and the compressibility of the vapor is treated semi-empirically as a constant. The model is embedded in an unsteady Reynolds-averaged Navier–Stokes solver, with the realizable k-ε model employed to evaluate the eddy viscosity. The turbulent cavitating flow caused by an impulsively started submerged water jet is treated. The pattern of periodic cavitation cloud shedding is acceptably captured, and the mass flow rate coefficient and its fluctuation frequency evaluated by simulations agree with the experimental results well. The validity of the proposed method is confirmed. The results reveal that cavitation occurs when p i n / P in reaches 0.65 and fluid flow begins to pulsate. In the well-developed stage, the leading cavitation cloud and a subsequent cloud are successively shed downstream, and this process is repeated. The subsequent cloud catches the leading cloud, and they coalesce in the range x / d  2–3. The pressure fluctuations concentrate in the range of x / d 2 5 corresponding to the periodic shedding of cavitation clouds. The mass flow rate coefficient pulsates from 0.59 0.66 under the effect of cavitation.

1.
K. M.
Kalumuck
and
G. L.
Chahine
, “
The use of cavitating jets to oxidize organic compounds in water
,”
J. Fluids Eng.
122
,
465
(
2000
).
2.
S.
Shimizu
and
G.
Peng
,
Water Jetting Technology for LOHAS
(
International Academic Printing Co. Ltd
.,
Tokyo
,
2009
).
3.
H.
Soyama
and
O.
Takakuwa
, “
Enhancing the aggressive strength of a cavitating jet and its practical application
,”
J. Fluid Sci. Technol.
6
(
4
),
510
(
2011
).
4.
J.
Foldyna
,
L.
Sitek
,
J.
Ščučka
et al, “
Effects of pulsating water jet impact on aluminium surface
,”
J. Mater. Process. Technol.
209
(
20
),
6174
(
2009
).
5.
M. M.
Wright
,
B.
Epps
,
A.
Dropkin
et al, “
Cavitation of a submerged jet
,”
Exp. Fluids
54
,
1541
(
2013
).
6.
A.
Yamaguti
and
S.
Shimizu
, “
Erosion due to impingement of cavitation jet
,”
J. Fluids Eng.
109
(
4
),
442
(
1987
).
7.
K.
Gopalan
,
J.
Katz
, and
O.
Knio
, “
The flow structure in the near field of jets and its effect on cavitation inception
,”
J. Fluid Mech.
398
,
1
(
1999
).
8.
J.
Foldyna
,
L.
Sitek
,
B.
Svehla
, and
S.
Svehla
, “
Utilization of ultrasound to enhance high-speed water jet effects
,”
Ultrason. Sonochem.
11
(
3
),
131
(
2004
).
9.
S.
Singh
,
J. K.
Choi
, and
G. L.
Chahine
, “
Characterization of cavitation fields from measured pressure signals of cavitating jets and ultrasonic horns
,”
J. Fluids Eng.
135
,
091302
(
2013
).
10.
P. K.
Ullas
,
D.
Chatterjee
, and
S.
Vengadesan
, “
Experimental study on the effect of throat length in the dynamics of internal unsteady cavitating flow
,”
Phys. Fluids
35
,
023332
(
2023
).
11.
S. L.
Ceccio
and
C. E.
Brennen
, “
Observations of the dynamics and acoustics of travelling bubble cavitation
,”
J. Fluid Mech.
233
,
633
(
1991
).
12.
S.
Nishimura
,
O.
Takakuwa
, and
H.
Soyama
, “
Similarity law on shedding frequency of cavitation cloud induced by a cavitating jet
,”
J. Fluid Sci. Technol.
7
(
3
),
405
(
2012
).
13.
K.
Peng
,
S.
Tian
,
G.
Li
et al, “
Cavitation in water jet under high ambient pressure conditions
,”
Exp. Therm. Fluid Sci.
89
,
9
(
2017
).
14.
C. E.
Brennen
,
Cavitation and Bubble Dynamics
(
Oxford University Press
,
New York
,
1995
).
15.
D. D.
Joseph
, “
Cavitation in a flowing liquid
,”
Phys. Rev. E
51
,
R1649
(
1995
).
16.
J. P.
Franc
and
J. M.
Michel
,
Fundamentals of Cavitation
(
Kluwer Academic Publishers
,
Netherlands
,
2004
).
17.
G.
Wang
,
I.
Senocak
,
W.
Shyy
,
T.
Ikohagi
, and
S.
Cao
, “
Dynamics of attached turbulent cavitating flows
,”
Prog. Aerosp. Sci.
37
,
551
(
2001
).
18.
G. E.
Reisman
,
Y. C.
Wang
, and
C. E.
Brennen
, “
Observations of shock waves in cloud cavitation
,”
J. Fluid Mech.
355
,
255
(
1998
).
19.
B.
Budich
,
S. J.
Schmidt
, and
N. A.
Adams
, “
Numerical simulation and analysis of condensation shocks in cavitating flow
,”
J. Fluid Mech.
838
,
759
(
2018
).
20.
G.
Wallis
,
One-Dimensional Two-Phase Flow
(
McGraw-Hill
,
New York
,
1967
).
21.
H.
Shamsborhan
,
O.
Coutier-Delgosha
,
G.
Caignaert
et al, “
Experimental determination of the speed of sound in cavitating flows
,”
Exp. Fluids
49
,
1359
(
2010
).
22.
Y.
Kawanami
,
H.
Kato
,
H.
Yamaguchi
et al, “
Inner structure of cloud cavity on a foil section
,”
JSME Intl. J. (B)
45
(
3
),
655
(
2002
).
23.
H.
Ganesh
,
S. A.
Mäkiharju
, and
S. L.
Ceccio
, “
Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities
,”
J. Fluid Mech.
802
,
37
(
2016
).
24.
C.
Wang
,
B.
Huang
,
G.
Wang
et al, “
Numerical simulation of transient turbulent cavitating flows with special emphasis on shock wave dynamics considering the water/vapor compressibility
,”
J. Hydrodyn.
30
(
4
),
573
(
2018
).
25.
P.
Yu
,
S. L.
Ceccio
, and
G.
Tryggvason
, “
The collapse of a cavitation bubble in shear flows—A numerical study
,”
Phys. Fluids
7
(
11
),
2608
(
1995
).
26.
G.
Tryggvasona
,
B.
Bunner
,
A.
Esmaeeli
et al, “
A front-tracking method for the computations of multiphase flow
,”
J. Comput. Phys.
169
(
2
),
708
(
2001
).
27.
M.
Sussman
, “
A second order coupled level set and volume-of-fluid method for computing growth and collapse of vapor bubbles
,”
J. Comput. Phys.
187
(
1
),
110
(
2003
).
28.
B.
Boyd
and
B.
Becker
, “
Numerical modeling of the acoustically driven growth and collapse of a cavitation bubble near a wall
,”
Phys. Fluids
31
,
032102
(
2019
).
29.
R.
Saurel
and
O.
Lemetayer
, “
A multiphase model for compressible flows with interfaces, shocks, detonation waves and cavitation
,”
J. Fluid Mech.
431
,
239
(
2001
).
30.
H.
Takahira
,
T.
Matsuno
, and
K.
Shuto
, “
Numerical investigations of shock-bubble interactions in mercury
,”
Fluid Dyn. Res.
40
(
7
),
510
(
2008
).
31.
I.
Senocak
and
W.
Shyy
, “
A pressure-based method for turbulent cavitating flow computations
,”
J. Comput. Phys.
176
,
363
(
2002
).
32.
O.
Coutier-Delgosha
,
R.
Fortes-Patella
, and
J. L.
Reboud
, “
Evaluation of the turbulence model influence on the numerical simulations of unsteady cavitation
,”
J. Fluids Eng.
125
(
1
),
38
(
2003
).
33.
M.
Gavaises
,
F.
Villa
,
P.
Koukouvinis
,
M.
Marengo
, and
J.-P.
Franc
, “
Visualisation and LES simulation of cavitation cloud formation and collapse in an axisymmetric geometry
,”
Int. J. Multiphase Flow
68
,
14
(
2015
).
34.
A.
Kumar
,
A.
Ghobadian
, and
J. M.
Nouri
, “
Assessment of cavitation models for compressible flows inside a nozzle
,”
Fluids
5
(
3
),
134
(
2020
).
35.
X. L.
Zhang
,
M. M.
Ge
,
G. J.
Zhang
, and
O.
Coutier-Delgosha
, “
Compressible effects modeling for turbulent cavitating flow in a small venturi channel: An empirical turbulent eddy viscosity correction
,”
Phys. Fluids
33
,
035148
(
2021
).
36.
F.
Giussani
,
F.
Piscaglia
, and
J.
Hèlie
, “
A three-phase VOF solver for the simulation of in-nozzle cavitation effects on liquid atomization
,”
J. Comput. Phys.
406
,
109068
(
2020
).
37.
C. F.
Delale
,
K.
Okita
, and
Y.
Matsumoto
, “
Steady-state cavitating nozzle flows with nucleation
,”
J. Fluids Eng.
127
,
770
(
2005
).
38.
T.
Yano
,
R.
Egashira
, and
S.
Fujikawa
, “
Linear analysis of dispersive waves in bubbly flows based on averaged equations
,”
J. Phys. Soc. Jpn.
75
(
10
),
104401
(
2006
).
39.
R. F.
Kunz
,
H. J.
Gibeling
,
M. R.
Maxey
et al, “
Validation of two-fluid Eulerian CFD modeling for microbubble drag reduction across a wide range of Reynolds numbers
,”
J. Fluids Eng.
129
(
1
),
66
(
2007
).
40.
Y.
Tamura
and
Y.
Matsumoto
, “
Improvement of bubble model for cavitating flow simulations
,”
J. Hydrodyn.
21
(
1
),
41
(
2009
).
41.
Z.
Wang
,
H.
Cheng
, and
B.
Ji
, “
Numerical prediction of cavitation erosion risk in an axisymmetric nozzle using a multi-scale approach
,”
Phys. Fluids
34
,
062112
(
2022
).
42.
S.
Yang
and
C.
Habchi
, “
Real-fluid phase transition in cavitation modeling considering dissolved non-condensable gas
,”
Phys. Fluids
32
,
032102
(
2020
).
43.
F. L.
Brandao
,
M.
Bhatt
, and
K.
Mahesh
, “
Numerical study of cavitation regimes in flow over a circular cylinder
,”
J. Fluid Mech.
885
,
A19
(
2020
).
44.
J.
Decaix
and
E.
Goncalvès
, “
Compressible effects modeling in turbulent cavitating flows
,”
Eur. J. Mech. B
39
,
11
(
2013
).
45.
S.
Park
and
S. H.
Rhee
, “
Comparative study of incompressible and isothermal compressible flow solvers for cavitating flow dynamics
,”
J. Mech. Sci. Technol.
29
(
8
),
3287
(
2015
).
46.
C.
Wang
,
G.
Wang
, and
B.
Huang
, “
Characteristics and dynamics of compressible cavitating flows with special emphasis on compressibility effects
,”
Int. J. Multiphase Flow
130
,
103357
(
2020
).
47.
A.
Kubota
,
H.
Kato
, and
H.
Yamaguti
, “
A new modeling of cavitating flow: A numerical study of unsteady cavitation on a hydrofoil section
,”
J. Fluid Mech.
240
,
59
(
1992
).
48.
S.
Gopalan
and
J.
Katz
, “
Flow structure and modeling issues in the closure region of attached cavitation
,”
Phys. Fluids
12
(
4
),
895
(
2000
).
49.
R. F.
Kunz
,
D. A.
Boger
,
D. R.
Stinebring
et al, “
A preconditioned Navier–Stokes method for two phase flows with application to cavitation prediction
,”
Comput. Fluids
29
,
849
(
2000
).
50.
A. K.
Singhal
,
M. M.
Athavale
,
H.
Li
, and
Y.
Jiang
, “
Mathematical basis and validation of the full cavitation model
,”
J. Fluids Eng.
124
,
617
(
2002
).
51.
P.
McGinn
,
G.
Tretola
, and
K.
Vogiatzaki
, “
Unified modeling of cavitating sprays using a three-component volume of fluid method accounting for phase change and phase miscibility
,”
Phys. Fluids
34
,
082108
(
2022
).
52.
S.
Venkateswaran
,
J. W.
Lindau
,
R. F.
Kunz
, and
C. L.
Merkle
, “
Computation of multiphase mixture flows with compressibility effects
,”
J. Comp. Phys.
180
(
1
),
54
(
2002
).
53.
Y.
Iga
,
M.
Nohmi
,
A.
Goto
,
B. R.
Shin
, and
T.
Ikohagi
, “
Numerical study of sheet cavitation breakoff phenomenon on a cascade hydrofoil
,”
J. Fluids Eng.
125
,
643
(
2003
).
54.
G. H.
Schnerr
,
I. H.
Sezal
, and
S. J.
Schmidt
, “
Numerical investigation of three-dimensional cloud cavitation with special emphasis on collapse induced shock wave dynamics
,”
Phys. Fluids
20
,
040703
(
2008
).
55.
A.
Gnanaskandan
and
K.
Mahesh
, “
Numerical investigation of near-wake characteristics of cavitating flow over a circular cylinder
,”
J. Fluid Mech.
790
,
453
(
2016
).
56.
T.
Yabe
, “
A universal cubic interpolation solver for compressible and incompressible fluids
,”
Shock Waves
1
,
187
(
1991
).
57.
S.
Acharya
,
B. R.
Baliga
,
K.
Karki
et al, “
Pressure-based finite-volume methods in computational fluid dynamics
,”
J. Heat Transfer
129
(
4
),
407
(
2007
).
58.
R. I.
Issa
, “
Solution of the implicitly discretised fluid flow equations by operator-splitting
,”
J. Comput. Phys.
62
,
40
(
1986
).
59.
T.
Yabe
and
P. Y.
Wang
, “
Unified numerical procedure for compressible and incompressible fluid
,”
J. Phys. Soc. Jpn.
60
(
7
),
2105
(
1991
).
60.
G.
Peng
,
M.
Ishizuka
, and
S.
Hayama
, “
An improved CIP-CUP method for submerged water jet flow simulation
,”
JSME Int. J., Ser. B
44
(
4
),
497
(
2001
).
61.
H.
Soyama
, “
Cavitating jet: A review
,”
Appl. Sci.
10
(
20
),
7280
(
2020
).
62.
G.
Peng
,
A.
Wakui
,
Y.
Oguma
,
S.
Shimizu
, and
H.
Ji
, “
Periodic behavior of cavitation cloud shedding in submerged water jets issuing from a sheathed pipe nozzle
,”
J. Flow Control, Meas. Vis.
6
(
1
),
15
(
2018
).
63.
D.
Beattie
and
P.
Whally
, “
A simple two-phase frictional pressure drop calculation method
,”
Int. J. Multiphase Flow
8
,
83
(
1982
).
64.
B. E.
Launder
and
D.
Spalding
, “
The numerical computation of turbulent flows
,”
Comput. Methods Appl. Mech. Eng.
3
(
2
),
269
(
1974
).
65.
D. C.
Wilcox
,
Turbulence Modeling for CFD
(
DCW Industries
,
California
,
2002
).
66.
G.
Peng
,
C.
Yang
,
Y.
Oguma
, and
S.
Shimizu
, “
Numerical analysis of cavitation cloud shedding in a submerged water
,”
J. Hydrodyn.
28
(
6
),
986
(
2016
).
67.
T.-H.
Shih
,
W. W.
Liou
,
A.
Shabbir
et al, “
A new k-ε eddy-viscosity model for high Reynolds number turbulent flows—model development and validation
,”
Comput. Fluids
24
(
3
),
227
(
1995
).
68.
L.
Zeng
,
G.
Liu
,
J.
Mao
et al, “
A novel numerical simulation method to verify turbulence models for predicting flow patterns in control valves
,”
J. Fluid Sci. Technol.
10
(
1
),
00484
(
2015
).
69.
G.
Peng
,
S.
Shimizu
, and
S.
Fujikawa
, “
Numerical simulation of cavitating water jet by a compressible mixture flow method
,”
J. Fluid Sci. Technol.
6
(
4
),
499
(
2011
).
70.
T.
Sriveerakul
,
S.
Aphornratana
, and
K.
Chunnanond
, “
Performance prediction of steam ejector using computational fluid dynamics. I. Validation of the CFD results
,”
Int. J. Therm. Sci.
46
(
8
),
812
(
2007
).
71.
R.
Shaheed
,
A.
Mohammadian
, and
H. K.
Gildeh
, “
A comparison of standard k–ε and realizable k–ε turbulence models in curved and confluent channels
,”
Environ. Fluid Mech.
19
,
543
(
2019
).
72.
V. S.
Yakhot
,
S. A.
Orszag
,
S.
Thangam
,
T. B.
Gatski
, and
C.
Speziale
, “
Development of turbulence models for shear flows by a double expansion technique
,”
Phys. Fluids
4
(
7
),
1510
(
1992
).
73.
A.
Prosperetti
, “
The speed of sound in a gas–vapour bubbly liquid
,” Interface Focus
5
,
20150024
(
2015
).
74.
R. H.
Cole
,
Underwater Explosions
(
Dover Publications
,
New York
,
1965
).
75.
V. M.
Viitanen
,
T.
Sipilä
,
A.
Sánchez-Caja
, and
T.
Siikonen
, “
Compressible two-phase viscous flow investigations of cavitation dynamics for the ITTC standard cavitator
,”
Appl. Sci.
10
(
19
),
6985
(
2020
).
76.
S. T.
Miller
,
H.
Jasak
,
D. A.
Boger
et al, “
A pressure-based compressible two-phase flow finite volume method for underwater explosions
,”
Comput. Fluids
87
,
132
(
2013
).
77.
P. J.
Zwart
,
A. G.
Gerber
, and
T.
Belamri
, “
A two-phase flow model for predicting cavitation dynamics
,” in
Fifth International Conference on Multiphase Flow
,
Yokohama, Japan
,
2004
, Paper No. 152.
78.
L.
Geng
and
X.
Escaler
, “
Assessment of RANS turbulence, models and Zwart cavitation model empirical coefficients for the simulation of unsteady cloud cavitation
,”
Eng. Appl. Comput. Fluid Mech.
14
(
1
),
151
(
2020
).
79.
A.
Wei
,
Y.
Yu
,
R.
Gao
et al, “
Unsteady cloud cavitation mechanisms of liquid nitrogen in convergent–divergent nozzle
,”
Phys. Fluids
33
,
092116
(
2021
).
80.
J.
Jablonská
,
M.
Kozubková
,
M.
Mahdal
et al, “
Spectral analysis of gaseous cavitation in water through multiphase mathematical and acoustic methods
,”
Phys. Fluids
33
,
085128
(
2021
).
81.
W.
Li
,
Z.
Yu
, and
S.
Kadam
, “
An improved cavitation model with thermodynamic effect and multiple cavitation regimes
,”
Int. J. Heat Mass Transfer
205
,
123854
(
2023
).
82.
M. S.
Seif
,
A.
Asnaghi
, and
E.
Jahanbakhsh
, “
Implementation of PISO algorithm for simulating unsteady cavitating flows
,”
Ocean Eng.
37
,
1321
(
2010
).
83.
B.
Ran
and
J.
Katz
, “
Pressure fluctuations and their effect on cavitation inception within water jets
,”
J. Fluid Mech.
262
,
223
(
1994
).
84.
W. H.
Nurick
, “
Orifice cavitation and its effect on spray mixing
,”
J. Fluids Eng.
98
,
681
687
(
1976
).
85.
P. J.
Roache
,
Verification and Validation in Computational Science and Engineering
(
Hermosa Publisher
,
New Mexico
,
1998
).
86.
F.
Elizalde-Blancas
,
E.
Karaismail
, and
I. B.
Celik
, “
Application of error scaling method in conjunction with GCI
,” in
Proceedings of the 3rd Workshop on CFD Uncertainty Analysis
(
2008
).
87.
H. B.
Karplus
, “
Velocity of sound in a liquid containing gas bubbles
,”
J. Acoust. Soc. Am.
29
,
1261
(
1957
).
88.
R. E.
Henry
,
M. A.
Grolmes
, and
H. K.
Fauske
, “
Pressure-pulse propagation in two-phase one and two-component mixture
,”
Report No. ANL-7792
,
1971
.
89.
X.
Wu
and
G. L.
Chahine
, “
Characterization of the content of the cavity behind a high-speed supercavitating body
,”
J. Fluids Eng.
129
(
2
),
136
(
2007
).
90.
M.
Brunhart
,
C.
Soteriou
,
M.
Gavaises
et al, “
Investigation of cavitation and vapor shedding mechanisms in a Venturi nozzle
,”
Phys. Fluids
32
,
083306
(
2020
).
91.
B.
Huang
,
Y.
Zhao
, and
G.
Wang
, “
Large eddy simulation of turbulent vortex-cavitation interactions
,”
Comput. Fluids
92
,
113
124
(
2014
).
92.
Y.
Liu
,
B.
Huang
,
H.
Zhang
,
Q.
Wu
, and
G.
Wang
, “
Experimental investigation into fluid–structure interaction of cavitating flow
,”
Phys. Fluids
33
,
093307
(
2021
).
93.
Z.
Zuo
,
H.
Zhang
,
Z.
Ren
,
H.
Chen
, and
S.
Liu
, “
Thermodynamic effects at Venturi cavitation in different liquids
,”
Phys. Fluids
34
,
083310
(
2022
).
94.
G.
Peng
,
R.
Egashira
,
T.
Yano
, and
S.
Fujikawa
, “
A compressible two-phase flow bubble cavitation model for computation of cavitating flows
,” in
Proceedings of the 1st International Colloquium on Dynamics, Physics and Chemistry of Bubble and Gas-Liquid Boundaries (ICBB2007)
,
Hokkaido, Japan
,
2007
, Paper No. 2512.
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