Inferring cryogenic cavitation features from the boundary conditions (BCs) remains a challenge due to the nonlinear thermal effects. This paper aims to build a fast model for cryogenic cavitation prediction from the BCs. Different from the traditional numerical solvers and conventional physics-informed neural networks, the approach can realize near real-time inference as the BCs change without a recalculating or retraining process. The model is based on the fusion of simple theories and neural network. It utilizes theories such as the B-factor theory to construct a physical module, quickly inferring hidden physical features from the BCs. These features represent the local and global cavitation intensity and thermal effect, which are treated as functions of location x. Then, a neural operator builds the mapping between these features and target functions (local pressure coefficient or temperature depression). The model is trained and validated based on the experimental measurements by Hord for liquid nitrogen and hydrogen. Effects of the physical module and training dataset size are investigated in terms of prediction errors. It is validated that the model can learn hidden knowledge from a small amount of experimental data and has considerable accuracy for new BCs and locations. In addition, preliminary studies show that it has the potential for cavitation prediction in unseen cryogenic liquids or over new geometries without retraining. The work highlights the potential of merging simple physical models and neural networks together for cryogenic cavitation prediction.

1.
G.
Brändle
,
M.
Schönfisch
, and
S.
Schulte
, “
Estimating long-term global supply costs for low-carbon hydrogen
,”
Appl. Energy
302
,
117481
(
2021
).
2.
A.
Gomez
and
H.
Smith
, “
Liquid hydrogen fuel tanks for commercial aviation: Structural sizing and stress analysis
,”
Aerosp. Sci. Technol.
95
,
105438
(
2019
).
3.
C. M.
Chen
,
C. F.
Chen
,
J. Y.
Wang
,
R.
Madda
,
S. W.
Tsai
,
P. K.
Wu
, and
W. M.
Chen
, “
Bone morphogenetic protein activity preservation with extracorporeal irradiation- and liquid nitrogen freezing-treated recycled autografts for biological reconstruction in malignant bone tumor
,”
Cryobiology
89
,
82
89
(
2019
).
4.
T.
Kachaamy
,
R.
Prakash
,
M.
Kundranda
,
R.
Batish
,
J.
Weber
,
S.
Hendrickson
,
L.
Yoder
,
H.
Do
,
T.
Magat
,
R.
Nayar
,
D.
Gupta
,
T.
DaSilva
,
A.
Sangal
,
S.
Kothari
,
V.
Kaul
, and
P.
Vashi
, “
Liquid nitrogen spray cryotherapy for dysphagia palliation in patients with inoperable esophageal cancer
,”
Gastrointest. Endosc.
88
(
3
),
447
455
(
2018
).
5.
Z.
Wang
,
H.
Cheng
, and
B.
Ji
, “
Numerical investigation of condensation shock and re-entrant jet dynamics around a cavitating hydrofoil using a dynamic cubic nonlinear subgrid-scale model
,”
Appl. Math. Modell.
100
,
410
431
(
2021
).
6.
Y.
Long
,
C.
Han
,
X.
Long
,
B.
Ji
, and
H.
Huang
, “
Verification and validation of delayed detached eddy simulation for cavitating turbulent flow around a hydrofoil and a marine propeller behind the hull
,”
Appl. Math. Modell.
96
,
382
401
(
2021
).
7.
E. B.
Flint
and
K. S.
Suslick
, “
The temperature of cavitation
,”
Science
253
(
5026
),
1397
1399
(
1991
).
8.
A.
Wei
,
L.
Yu
,
R.
Gao
,
W.
Zhang
, and
X.
Zhang
, “
Unsteady cloud cavitation mechanisms of liquid nitrogen in convergent-divergent nozzle
,”
Phys. Fluids
33
(
9
),
092116
(
2021
).
9.
S.
Dong
,
J.
Duan
, and
T.
Sun
, “
Dynamic response and acoustic characteristics of composite hydrofoil under cavitation-induced vibration
,”
Phys. Fluids
35
(
1
),
013302
(
2023
).
10.
G. T.
Bokman
,
L.
Biasiori-Poulanges
,
B.
Lukić
,
C.
Bourquard
,
D. W.
Meyer
,
A.
Rack
, and
O.
Supponen
, “
High-speed x-ray phase-contrast imaging of single cavitation bubbles near a solid boundary
,”
Phys. Fluids
35
(
1
),
013322
(
2023
).
11.
J.
Nahon
,
M.
Zangeneh
,
T.
Tsuneda
,
M.
Nohmi
,
H.
Watanabe
, and
A.
Goto
, “
Experimental analysis of shock smoothing design strategy for reducing cavitation erosion aggressiveness
,”
Phys. Fluids
35
(
1
),
013331
(
2023
).
12.
F.
Denner
and
S.
Schenke
, “
Modeling acoustic emissions and shock formation of cavitation bubbles
,”
Phys. Fluids
35
(
1
),
012114
(
2023
).
13.
B.
Ji
,
X. W.
Luo
,
R. E. A.
Arndt
,
X.
Peng
, and
Y.
Wu
, “
Large eddy simulation and theoretical investigations of the transient cavitating vortical flow structure around a NACA66 hydrofoil
,”
Int. J. Multiphase Flow
68
,
121
134
(
2015
).
14.
J.
Zhu
,
S.
Wang
, and
X.
Zhang
, “
Influences of thermal effects on cavitation dynamics in liquid nitrogen through venturi tube
,”
Phys. Fluids
32
(
1
),
012105
(
2020
).
15.
L.
Wang
,
P.
Wang
,
Z.
Chang
,
B.
Huang
, and
D.
Wu
, “
A Lagrangian analysis of partial cavitation growth and cavitation control mechanism
,”
Phys. Fluids
34
(
11
),
113329
(
2022
).
16.
Y.
Zhi
,
R.
Huang
,
R.
Qiu
,
Y.
Wang
, and
C.
Huang
, “
LES investigation into the cavity shedding dynamics and cavitation–vortex interaction around a surface-piercing hydrofoil
,”
Phys. Fluids
34
(
12
),
123314
(
2022
).
17.
Y.
Utturkar
,
J.
Wu
,
G.
Wang
, and
W.
Shyy
, “
Recent progress in modeling of cryogenic cavitation for liquid rocket propulsion
,”
Prog. Aerosp. Sci.
41
(
7
),
558
608
(
2005
).
18.
J.
Hord
, “
Cavitation in liquid cryogens. III. Ogives
,”
NASA Contract Report No. CR-2242
,
1973
.
19.
R.
Xue
,
Y.
Ruan
,
X.
Liu
,
L.
Chen
,
X.
Zhang
,
Y.
Hou
, and
S.
Chen
, “
Experimental study of liquid nitrogen spray characteristics in atmospheric environment
,”
Appl. Therm. Eng.
142
,
717
722
(
2018
).
20.
T.
Chen
,
H.
Chen
,
W.
Liu
,
B.
Huang
, and
G.
Wang
, “
Unsteady characteristics of liquid nitrogen cavitating flows in different thermal cavitation mode
,”
Appl. Therm. Eng.
156
,
63
76
(
2019
).
21.
C. C.
Tseng
and
W.
Shyy
, “
Modeling for isothermal and cryogenic cavitation
,”
Int. J. Heat Mass Transfer
53
(
1–3
),
513
525
(
2010
).
22.
E.
Goncalvès
, “
Modeling for non isothermal cavitation using 4-equation models
,”
Int. J. Heat Mass Transfer
76
,
247
262
(
2014
).
23.
A.
Wei
,
L.
Yu
,
L.
Qiu
, and
X.
Zhang
, “
Cavitation in cryogenic fluids: A critical research review
,”
Phys. Fluids
34
(
10
),
101303
(
2022
).
24.
J.
Ishimoto
and
K.
Kamijo
, “
Numerical study of cavitating flow characteristics of liquid helium in a pipe
,”
Int. J. Heat Mass Transfer
47
(
1
),
149
163
(
2004
).
25.
F.
Petitpas
,
J.
Massoni
,
R.
Saurel
,
E.
Lapebie
, and
L.
Munier
, “
Diffuse interface model for high speed cavitating underwater systems
,”
Int. J. Multiphase Flow
35
(
8
),
747
759
(
2009
).
26.
A.
Zein
,
M.
Hantke
, and
G.
Warnecke
, “
Modeling phase transition for compressible two-phase flows applied to metastable liquids
,”
J. Comput. Phys.
229
(
8
),
2964
2998
(
2010
).
27.
M. G.
Rodio
,
M. G.
De Giorgi
, and
A.
Ficarella
, “
Influence of convective heat transfer modeling on the estimation of thermal effects in cryogenic cavitating flows
,”
Int. J. Heat Mass Transfer
55
(
23–24
),
6538
6554
(
2012
).
28.
E.
Goncalvès
and
R. F.
Patella
, “
Numerical study of cavitating flows with thermodynamic effect
,”
Comput. Fluids
39
(
1
),
99
113
(
2010
).
29.
S.
Clerc
, “
Numerical simulation of the homogeneous equilibrium model for two-phase flows
,”
J. Comput. Phys.
161
(
1
),
354
375
(
2000
).
30.
S.
Barre
,
J.
Rolland
,
G.
Boitel
,
E.
Goncalves
, and
R. F.
Patella
, “
Experiments and modeling of cavitating flows in venturi: Attached sheet cavitation
,”
Eur. J. Mech. B/Fluids
28
(
3
),
444
464
(
2009
).
31.
A.
Hosangadi
and
V.
Ahuja
, “
Numerical study of cavitation in cryogenic fluids
,”
J. Fluids Eng.
127
(
2
),
267
281
(
2005
).
32.
J.
Zhu
,
S.
Wang
,
L.
Qiu
,
X.
Zhi
, and
X.
Zhang
, “
Frequency characteristics of liquid hydrogen cavitating flow over a NACA0015 hydrofoil
,”
Cryogenics
90
,
7
19
(
2018
).
33.
X. B.
Zhang
,
L. M.
Qiu
,
H.
Qi
,
X. J.
Zhang
, and
Z. H.
Gan
, “
Modeling liquid hydrogen cavitating flow with the full cavitation model
,”
Int. J. Hydrogen Energy
33
(
23
),
7197
7206
(
2008
).
34.
X. B.
Zhang
,
L. M.
Qiu
,
Y.
Gao
, and
X. J.
Zhang
, “
Computational fluid dynamic study on cavitation in liquid nitrogen
,”
Cryogenics
48
(
9–10
),
432
438
(
2008
).
35.
X.
Long
,
Q.
Liu
,
B.
Ji
, and
Y.
Lu
, “
Numerical investigation of two typical cavitation shedding dynamics flow in liquid hydrogen with thermodynamic effects
,”
Int. J. Heat Mass Transfer
109
,
879
893
(
2017
).
36.
B.
Huang
,
Q.
Wu
, and
G.
Wang
, “
Numerical investigation of cavitating flow in liquid hydrogen
,”
Int. J. Hydrogen Energy
39
(
4
),
1698
1709
(
2014
).
37.
S.
Zhang
,
X.
Li
,
B.
Hu
,
Y.
Liu
, and
Z.
Zhu
, “
Numerical investigation of attached cavitating flow in thermo-sensitive fluid with special emphasis on thermal effect and shedding dynamics
,”
Int. J. Hydrogen Energy
44
(
5
),
3170
3184
(
2019
).
38.
B.
Zoph
,
G.
Brain
, and
J.
Shlens
, “
Learning transferable architectures for scalable image recognition
,” in
Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition
(
IEEE
,
2018
), pp.
8697
8710
.
39.
M. C.
Kenton
,
L.
Kristina
, and
J.
Devlin
, “
BERT: Pre-training of deep bidirectional transformers for language understanding
,” arXiv:1810.04805 (
2018
).
40.
O.
Ronneberger
,
K.
Tunyasuvunakool
,
R.
Bates
,
A.
Žídek
,
A. J.
Ballard
,
A.
Cowie
,
B.
Romera-paredes
,
S.
Nikolov
,
R.
Jain
,
J.
Adler
,
T.
Back
,
S.
Petersen
,
D.
Reiman
,
E.
Clancy
,
M.
Zielinski
,
M.
Steinegger
,
M.
Pacholska
,
T.
Berghammer
,
S.
Bodenstein
,
D.
Silver
,
O.
Vinyals
,
A. W.
Senior
, and
K.
Kavukcuoglu
, “
Highly accurate protein structure prediction with AlphaFold
,”
Nature
596
(
7873
),
583
589
(
2021
).
41.
Z.
Zhang
,
X.
Dong Song
,
S.
Ran Ye
,
Y.
Wei Wang
,
C.
Guang Huang
,
Y.
Ran An
, and
Y.
Song Chen
, “
Application of deep learning method to Reynolds stress models of channel flow based on reduced-order modeling of DNS data
,”
J. Hydrodyn.
31
(
1
),
58
65
(
2019
).
42.
M.
Xu
,
H.
Cheng
, and
B.
Ji
, “
RANS simulation of unsteady cavitation around a Clark-Y hydrofoil with the assistance of machine learning
,”
Ocean Eng.
231
,
109058
(
2021
).
43.
D.
Kochkov
,
J. A.
Smith
,
A.
Alieva
,
Q.
Wang
,
M. P.
Brenner
, and
S.
Hoyer
, “
Machine learning-accelerated computational fluid dynamics
,”
Proc. Natl. Acad. Sci.
118
(
21
),
e2101784118
(
2021
).
44.
Z.
Wang
,
K.
Luo
,
D.
Li
,
J.
Tan
, and
J.
Fan
, “
Investigations of data-driven closure for subgrid-scale stress in large-eddy simulation
,”
Phys. Fluids
30
(
12
),
125101
(
2018
).
45.
N.
Kovachki
,
Z.
Li
,
B.
Liu
,
K.
Azizzadenesheli
,
K.
Bhattacharya
,
A.
Stuart
, and
A.
Anandkumar
, “
Neural operator: Learning maps between function spaces
,” arXiv:2108.08481 (
2021
).
46.
L.
Lu
,
P.
Jin
,
G.
Pang
,
Z.
Zhang
, and
G. E.
Karniadakis
, “
Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators
,”
Nat. Mach. Intell.
3
(
3
),
218
229
(
2021
).
47.
C.
Lin
,
Z.
Li
,
L.
Lu
,
S.
Cai
,
M.
Maxey
, and
G. E.
Karniadakis
, “
Operator learning for predicting multiscale bubble growth dynamics
,”
J. Chem. Phys.
154
(
10
),
104118
(
2021
).
48.
Z.
Li
,
N.
Kovachki
,
K.
Azizzadenesheli
,
B.
Liu
,
K.
Bhattacharya
,
A.
Stuart
, and
A.
Anandkumar
, “
Fourier neural operator for parametric partial differential equations
,” arXiv:2010.08895 (
2020
).
49.
G.
Wen
,
Z.
Li
,
K.
Azizzadenesheli
,
A.
Anandkumar
, and
S. M.
Benson
, “
U-FNO—An enhanced Fourier neural operator-based deep-learning model for multiphase flow
,”
Adv. Water Resour.
163
,
104180
(
2022
).
50.
A.
Karpatne
,
G.
Atluri
,
J. H.
Faghmous
,
M.
Steinbach
,
A.
Banerjee
,
A.
Ganguly
,
S.
Shekhar
,
N.
Samatova
, and
V.
Kumar
, “
Theory-guided data science: A new paradigm for scientific discovery from data
,”
IEEE Trans. Knowl. Data Eng.
29
(
10
),
2318
2331
(
2017
).
51.
G. E.
Karniadakis
, “
Physics-informed machine learning
,”
Nat. Rev. Phys.
3
(
6
),
422
440
(
2021
).
52.
S.
Cai
,
Z.
Mao
,
Z.
Wang
,
M.
Yin
, and
G. E.
Karniadakis
, “
Physics-informed neural networks (PINNs) for fluid mechanics: A review
,”
Acta Mech. Sin.
37
(
12
),
1727
1738
(
2022
).
53.
M.
Raissi
and
G. E.
Karniadakis
, “
Hidden physics models: Machine learning of nonlinear partial differential equations
,”
J. Comput. Phys.
357
,
125
141
(
2018
).
54.
M.
Raissi
,
A.
Yazdani
, and
G. E.
Karniadakis
, “
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
,”
Science
367
(
6481
),
1026
1030
(
2020
).
55.
S.
Cai
,
Z.
Wang
,
S.
Wang
,
P.
Perdikaris
, and
G. E.
Karniadakis
, “
Physics-informed neural networks for heat transfer problems
,”
J. Heat Transfer.
143
(
6
),
060801
(
2021
).
56.
C.
Methods
,
A.
Mech
,
Z.
Mao
,
A. D.
Jagtap
, and
G. E.
Karniadakis
, “
Physics-informed neural networks for high-speed flows
,”
Comput. Methods Appl. Mech. Eng.
360
,
112789
(
2020
).
57.
X.
Jin
,
S.
Cai
,
H.
Li
, and
G. E.
Karniadakis
, “
NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
,”
J. Comput. Phys.
426
,
109951
(
2021
).
58.
R.
Laubscher
, “
Simulation of multi-species flow and heat transfer using physics-informed neural networks
,”
Phys. Fluids
33
(
8
),
087101
(
2021
).
59.
V.
Kag
,
K.
Seshasayanan
, and
V.
Gopinath
, “
Physics-informed data based neural networks for two-dimensional turbulence
,”
Phys. Fluids
34
(
5
),
055130
(
2022
).
60.
H.
Eivazi
,
M.
Tahani
,
P.
Schlatter
, and
R.
Vinuesa
, “
Physics-informed neural networks for solving Reynolds-averaged Navier-Stokes equations
,”
Phys. Fluids
34
(
7
),
075117
(
2022
).
61.
R.
Qiu
,
R.
Huang
,
X.
Yao
, and
Z.
Zhang
, “
Physics-informed neural networks for phase-field method in two-phase flow
,”
Phys. Fluids
34
(
5
),
052109
(
2022
).
62.
N.
Zobeiry
and
K. D.
Humfeld
, “
A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications
,”
Eng. Appl. Artif. Intell.
101
,
104232
(
2021
).
63.
S.
Pawar
,
O.
San
,
B.
Aksoylu
,
A.
Rasheed
, and
T.
Kvamsdal
, “
Physics guided machine learning using simplified theories
,”
Phys. Fluids
33
(
1
),
011701
(
2021
).
64.
J.
Zhu
,
D.
Zhao
,
L.
Xu
, and
X.
Zhang
, “
Interactions of vortices, thermal effects and cavitation in liquid hydrogen cavitating flows
,”
Int. J. Hydrogen Energy
41
(
1
),
614
631
(
2016
).
65.
I.
Venturi
, “
Cavitation in liquid cryogens. II. Hydrofoil
,”
NASA Contract Report No. NASA-CR-21
,
1973
.
66.
M. G.
De Giorgi
,
A.
Ficarella
, and
M.
Tarantino
, “
Evaluating cavitation regimes in an internal orifice at different temperatures using frequency analysis and visualization
,”
Int. J. Heat Fluid Flow
39
,
160
172
(
2013
).
67.
J.
Zhu
,
Y.
Chen
,
D.
Zhao
, and
X.
Zhang
, “
Extension of the Schnerr-Sauer model for cryogenic cavitation
,”
Eur. J. Mech. B/Fluids
52
,
1
10
(
2015
).
68.
C. L.
Merkle
,
J. Z.
Feng
, and
P.
Buelow
, “
Computational modeling of dynamics of sheet cavitation
,” in
Proceedings of the 3rd International Symposium on Cavitation
(
1998
).
69.
M.-H.
Guo
,
Z.-N.
Liu
,
T.-J.
Mu
, and
S.-M.
Hu
, “
Beyond self-attention: External attention using two linear layers for visual tasks
,” arXiv:2105.02358 (
2021
).
70.
J.
Hord
, “
Cavitation in liquid cryogens. II. Hydrofoil
,”
NASA Contract Report No. CR-2156
,
1973
.
71.
C. E.
Brennen
,
Cavitation and Bubble Dynamics
(
Cambridge University Press
,
2014
).
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