The purpose of this work is twofold: to present a computational strategy to simulate the dynamics of a rigid sphere during water sloshing and to validate the model with original experimental data. The numerical solution is obtained through the coupling between a two-fluid Navier-Stokes solver and a rigid solid dynamics solver, based on a Newton scheme. A settling sphere case reported in the literature is first analyzed to validate the numerical strategy by ascertaining the settling velocity. In addition, an experiment is carried out based on a sphere submerged into a communicating vessel subjected to sloshing. Experimental data are captured using image processing and statistically treated to provide sphere dynamics quantitative information. The effects of different classical models used to describe drag coefficients, added mass, and wall effects are considered in the study to evaluate their influence on the results. The numerical model provides results that are consistent with the physical data, and the trajectory analysis shows good agreement between the simulations and the experiments.

1.
J.
Orona
,
S.
Zorrilla
, and
J.
Peralta
, “
Sensitivity analysis using a model based on computational fluid dynamics, discrete element method and discrete phase model to study a food hydrofluidization system
,”
J. Food Eng.
237
,
183
193
(
2018
).
2.
J.
Yang
and
F.
Stern
, “
Sharp interface immersed-boundary/level-set method for wave–body interactions
,”
J. Comput. Phys.
228
,
6590
6616
(
2009
).
3.
A.
Calderer
,
S.
Kang
, and
F.
Sotiropoulos
, “
Level set immersed boundary method for coupled simulation of air/water interaction with complex floating structures
,”
J. Comput. Phys.
277
,
201
227
(
2014
).
4.
B.
Ducassou
,
J.
Nuñez
,
M.
Cruchaga
, and
S.
Abadie
, “
A fictitious domain approach based on a viscosity penalty method to simulate wave/structure interaction
,”
J. Hydraul. Res.
55
,
847
862
(
2017
).
5.
H. H.
Hu
,
N.
Patankar
, and
M.
Zhu
, “
Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique
,”
J. Comput. Phys.
169
,
427
462
(
2001
).
6.
M. A.
Storti
,
L.
Garelli
, and
R. R.
Paz
, “
A finite element formulation satisfying the discrete geometric conservation law based on averaged Jacobians
,”
Int. J. Numer. Methods Fluids
69
,
1872
1890
(
2012
).
7.
A. J.
Lew
and
G. C.
Buscaglia
, “
A discontinuous-Galerkin-based immersed boundary method
,”
Int. J. Numer. Methods Eng.
76
,
427
454
(
2008
).
8.
R.
Mittal
and
G.
Iaccarino
, “
Immersed boundary methods
,”
Annu. Rev. Fluid Mech.
37
,
239
261
(
2005
).
9.
S.
Tao
,
Z.
Guo
, and
L.-P.
Wang
, “
Numerical study on the sedimentation of single and multiple slippery particles in a Newtonian fluid
,”
Powder Technol.
315
,
126
138
(
2017
).
10.
S.
Dash
and
T.
Lee
, “
Two spheres sedimentation dynamics in a viscous liquid column
,”
Comput. Fluids
123
,
218
234
(
2015
).
11.
A.
Ten Cate
,
C. H.
Nieuwstad
,
J. J.
Derksen
, and
H. E. A.
Van den Akker
, “
Particle imaging velocimetry experiments and lattice-Boltzmann simulations on a single sphere settling under gravity
,”
Phys. Fluids
14
,
4012
4025
(
2002
).
12.
C.
Diaz-Goano
,
P.
Minev
, and
K.
Nandakumar
, “
A fictitious domain/finite element method for particulate flows
,”
J. Comput. Phys.
192
,
105
123
(
2003
).
13.
M. A.
Cruchaga
,
C. M.
Muñoz
, and
D. J.
Celentano
, “
Simulation and experimental validation of the motion of immersed rigid bodies in viscous flows
,”
Comput. Methods Appl. Mech. Eng.
197
,
2823
2835
(
2008
).
14.
C.
Choi
,
H. S.
Yoon
, and
M. Y.
Ha
, “
Flow structure around a square cylinder impacting a wall
,”
Phys. Fluids
26
,
013602
(
2014
).
15.
S. D.
Costarelli
,
L.
Garelli
,
M. A.
Cruchaga
,
M. A.
Storti
,
R.
Ausensi
, and
S. R.
Idelsohn
, “
An embedded strategy for the analysis of fluid structure interaction problems
,”
Comput. Methods Appl. Mech. Eng.
300
,
106
128
(
2016
).
16.
A.
Pathak
and
M.
Raessi
, “
A 3D, fully Eulerian, VOF-based solver to study the interaction between two fluids and moving rigid bodies using the fictitious domain method
,”
J. Comput. Phys.
311
,
87
113
(
2016
).
17.
O. M.
Faltinsen
,
O. F.
Rognebakke
, and
A. N.
Timokha
, “
Transient and steady-state amplitudes of resonant three-dimensional sloshing in a square base tank with a finite fluid depth
,”
Phys. Fluids
18
,
012103
(
2006
).
18.
J. B.
Frandsen
, “
Sloshing motions in excited tanks
,”
J. Comput. Phys.
196
,
53
87
(
2004
).
19.
O. M.
Faltinsen
,
R.
Firoozkoohi
, and
A. N.
Timokha
, “
Steady-state liquid sloshing in a rectangular tank with a slat-type screen in the middle: Quasilinear modal analysis and experiments
,”
Phys. Fluids
23
,
042101
(
2011
).
20.
S.
Elgeti
and
H.
Sauerland
, “
Deforming fluid domains within the finite element method: Five mesh-based tracking methods in comparison
,”
Arch. Comput. Methods Eng.
23
,
323
361
(
2016
).
21.
M.
Cruchaga
,
L.
Battaglia
,
M.
Storti
, and
J.
D’Elía
, “
Numerical modeling and experimental validation of free surface flow problems
,”
Arch. Comput. Methods Eng.
23
,
139
169
(
2016
).
22.
M. A.
Cruchaga
,
R. S.
Reinoso
,
M. A.
Storti
,
D. J.
Celentano
, and
T. E.
Tezduyar
, “
Finite element computation and experimental validation of sloshing in rectangular tanks
,”
Comput. Mech.
52
,
1301
1312
(
2013
).
23.
E. L.
Grotle
,
H.
Bihs
, and
V.
Æsøy
, “
Experimental and numerical investigation of sloshing under roll excitation at shallow liquid depths
,”
Ocean Eng.
138
,
73
85
(
2017
).
24.
L.
Battaglia
,
M.
Cruchaga
,
M.
Storti
,
J.
D’Elía
,
J.
Nuñez Aedo
, and
R.
Reinoso
, “
Numerical modelling of 3D sloshing experiments in rectangular tanks
,”
Appl. Math. Modell.
59
,
357
378
(
2018
).
25.
P. A.
Caron
,
M. A.
Cruchaga
, and
A. E.
Larreteguy
, “
Study of 3D sloshing in a vertical cylindrical tank
,”
Phys. Fluids
30
,
082112
(
2018
).
26.
P.
Joshi
,
D. M.
Escrivá
, and
V.
Godoy
,
OpenCV by Example
(
Packt Publishing Ltd.
,
2016
).
27.
G.
Bueno Garcia
,
O.
Deniz Suarez
,
J. L.
Espinosa Aranda
,
J.
Salido Tercero
,
I.
Serrano Gracía
, and
N.
Vállez Enano
,
Learning Image Processing with OpenCV
(
Packt Publishing Ltd.
,
2015
).
28.
C.
Sanderson
and
R.
Curtin
, “
A user-friendly hybrid sparse matrix class in C++
,” in
Mathematical Software–ICMS 2018
, edited by
J. H.
Davenport
,
M.
Kauers
,
G.
Labahn
, and
J.
Urban
(
Springer International Publishing
,
2018
), pp.
422
430
.
29.
C.
Sanderson
and
R.
Curtin
, “
Armadillo: A template-based C++ library for linear algebra
,”
J. Open Source Software
1
,
26
27
(
2016
).
30.
J. C.
Russ
,
The Image Processing Handbook
(
CRC Press
,
2016
).
31.
F.
Kurugollu
,
B.
Sankur
, and
A. E.
Harmanci
, “
Color image segmentation using histogram multithresholding and fusion
,”
Image Vision Comput.
19
,
915
928
(
2001
).
32.
See http://www.cimec.org.ar/petscfem for PETSc-FEM, A general purpose, parallel, multi-physics FEM program,
2019
.
33.
A. N.
Brooks
and
T. J.
Hughes
, “
Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
,”
Comput. Methods Appl. Mech. Eng.
32
,
199
259
(
1982
).
34.
T. E.
Tezduyar
, “
Finite elements in fluids: Stabilized formulations and moving boundaries and interfaces
,”
Comput. Fluids
36
,
191
206
(
2007
).
35.
D.
Liu
,
W.
Tang
,
J.
Wang
,
H.
Xue
, and
K.
Wang
, “
Comparison of laminar model, RANS, LES and VLES for simulation of liquid sloshing
,”
Appl. Ocean Res.
59
,
638
649
(
2016
).
36.
P.
Caron
,
M.
Cruchaga
, and
A.
Larreteguy
, “
Sensitivity analysis of finite volume simulations of a breaking dam problem
,”
Int. J. Numer. Methods Heat Fluid Flow
25
,
1718
1745
(
2015
).
37.
L.
Battaglia
,
M. A.
Storti
, and
J.
D’Elía
, “
Simulation of free surface flows by a finite element interface capturing technique
,”
Int. J. Comput. Fluid Dyn.
24
,
121
133
(
2010
).
38.
L.
Battaglia
,
M. A.
Storti
, and
J.
D’Elía
, “
Bounded renormalization with continuous penalization for level set interface-capturing methods
,”
Int. J. Numer. Methods Eng.
84
,
830
848
(
2010
).
39.
J.
Strnadel
,
M.
Simon
, and
I.
Machač
, “
Wall effects on terminal falling velocity of spherical particles moving in a Carreau model fluid
,”
Chem. Pap.
65
,
177
184
(
2011
).
40.
H.
Schlichting
and
K.
Gersten
,
Boundary-Layer Theory
(
Springer
,
2016
).
41.
H.
Brenner
, “
The slow motion of a sphere through a viscous fluid towards a plane surface
,”
Chem. Eng. Sci.
16
,
242
251
(
1961
).
42.
S.-Y.
Lin
and
J.-F.
Lin
, “
Numerical investigation of lubrication force on a spherical particle moving to a plane wall at finite Reynolds numbers
,”
Int. J. Multiphase Flow
53
,
40
53
(
2013
).
43.
C. H.
Ataíde
,
F. A. R.
Pereira
, and
M. A. S.
Barrozo
, “
Wall effects on the terminal velocity of spherical particles in Newtonian and non-Newtonian fluids
,”
Braz. J. Chem. Eng.
16
,
387
394
(
1999
).
44.
A.
Goldman
,
R.
Cox
, and
H.
Brenner
, “
Slow viscous motion of a sphere parallel to a plane wall—II Couette flow
,”
Chem. Eng. Sci.
22
,
653
660
(
1967
).
45.
M. E.
O’Neill
, “
A sphere in contact with a plane wall in a slow linear shear flow
,”
Chem. Eng. Sci.
23
,
1293
1298
(
1968
).
46.
N.
Lukerchenko
,
Y.
Kvurt
,
I.
Keita
,
Z.
Chara
, and
P.
Vlasak
, “
Drag force, drag torque, and Magnus force coefficients of rotating spherical particle moving in fluid
,”
Part. Sci. Technol.
30
,
55
67
(
2012
).
47.
T.
Kray
,
J.
Franke
, and
W.
Frank
, “
Magnus effect on a rotating sphere at high Reynolds numbers
,”
J. Wind Eng. Ind. Aerodyn.
110
,
1
9
(
2012
).
48.
R.
Mei
and
J.
Klausner
, “
Shear lift force on spherical bubbles
,”
Int. J. Heat Fluid Flow
15
,
62
65
(
1994
).
49.
A. R.
Harris
and
C. I.
Davidson
, “
Particle resuspension in turbulent flow: A stochastic model for individual soil grains
,”
Aerosol Sci. Technol.
42
,
613
628
(
2008
).
50.
D.
Neill
,
D.
Livelybrooks
, and
R. J.
Donnelly
, “
A pendulum experiment on added mass and the principle of equivalence
,”
Am. J. Phys.
75
,
226
229
(
2007
).
51.
A. A.
Kendoush
,
A. H.
Sulaymon
, and
S. A.
Mohammed
, “
Experimental evaluation of the virtual mass of two solid spheres accelerating in fluids
,”
Exp. Therm. Fluid Sci.
31
,
813
823
(
2007
).
52.
J.
Almedeij
, “
Drag coefficient of flow around a sphere: Matching asymptotically the wide trend
,”
Powder Technol.
186
,
218
223
(
2008
).
53.
F. F.
Abraham
, “
Functional dependence of drag coefficient of a sphere on Reynolds number
,”
Phys. Fluids
13
,
2194
2195
(
1970
).
54.
F. M.
White
,
Viscous Fluid Flow
(
McGraw-Hill
,
1991
).
55.
Y.
Kim
and
C. S.
Peskin
, “
A penalty immersed boundary method for a rigid body in fluid
,”
Phys. Fluids
28
,
033603
(
2016
).
56.
M.
Rahmani
and
A.
Wachs
, “
Free falling and rising of spherical and angular particles
,”
Phys. Fluids
26
,
083301
(
2014
).
57.
R.
Ortega
,
J. C.
García Orden
,
M.
Cruchaga
, and
C.
García
, “
Energy-consistent simulation of frictional contact in rigid multibody systems using implicit surfaces and penalty method
,”
Multibody Syst. Dyn.
41
,
275
295
(
2017
).
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