In this paper, we present a numerical approach to solve the Navier–Stokes equations for arbitrary vessel geometries by combining a Fourier-spectral method with a direct-forcing immersed boundary method, which one allows to consider solid–fluid interactions. The approach is applied to a paradigmatic setup motivated by the precession dynamo experiment currently under construction at Helmholtz–Zentrum Dresden–Rossendorf. The experiment consists of a fluid-filled cylinder rotating about two axes, which induces a precession-driven flow inside the cavity. The cylinder is also equipped with baffles at the end caps with adjustable penetration depth to impact the flow. The numerical details and simulation results for the spin-up and precession-driven flow in a circular cylinder with additional baffles are presented. The results provide a first confirmation that the use of such baffles in the precession dynamo experiment is a useful way of influencing the flow, allowing more efficient driving without changing the known flow structure too much.
References
1.
P.
Meunier
, “Geoinspired soft mixers
,” J. Fluids Mech.
903
, A15
(2020
).2.
H.
Poincaré
, “Sur la précession des corps déformables
,” Bull. Astron. I
27
, 321
–356
(1910
).3.
R. R.
Kerswell
, “The instability of precessing flow
,” Geophys. Astrophys. Fluid Dyn.
72
, 107
–144
(1993
).4.
R. R.
Kerswell
, “Secondary instabilities in rapidly rotating fluids: Inertial wave breakdown
,” J. Fluid Mech.
382
, 283
–306
(1999
).5.
S.
Goto
, A.
Matsunaga
, M.
Fujiwara
, M.
Nishioka
, S.
Kida
, M.
Yamato
, and S.
Tsuda
, “Turbulence driven by precession in spherical and slightly elongated spheroidal cavities
,” Phys. Fluids
26
, 055107
(2014
).6.
Y.
Lin
, P.
Marti
, and J.
Noir
, “Shear-driven parametric instability in a precessing sphere
,” Phys. Fluids
27
, 046601
(2015
).7.
J.
Noir
, P.
Cardin
, D.
Jault
, and J.-P.
Masson
, “Experimental evidence of non-linear resonance effects between retrograde precession and the tilt-over mode within a spheroid
,” Geophys. J. Int.
154
, 407
–416
(2003
).8.
R.
Hollerbach
and R. R.
Kerswell
, “Oscillatory internal shear layers in rotating and precessing flows
,” J. Fluid Mech.
298
, 327
–339
(1995
).9.
S.
Lorenzani
and A.
Tilgner
, “Fluid instabilities in precessing spheroidal cavities
,” J. Fluid Mech.
447
, 111
–128
(2001
).10.
F. H.
Busse
, “Steady fluid flow in a precessing spheroidal shell
,” J. Fluid Mech.
33
, 739
–751
(1968
).11.
W. V. R.
Malkus
, “Precession of the earth as the cause of geomagnetism
,” Science
160
, 259
–264
(1968
).12.
J. P.
Vanyo
, “A geodynamo powered by luni-solar precession
,” Geophys. Astrophys. Fluid Dyn.
59
, 209
–234
(1991
).13.
A.
Tilgner
, “Precession driven dynamos
,” Phys. Fluids
17
, 034104
(2005
).14.
C. A.
Dwyer
, D. J.
Stevenson
, and F.
Nimmo
, “A long-lived lunar dynamo driven by continuous mechanical stirring
,” Nature
479
, 212
–214
(2011
).15.
J.
Noir
and D.
Cébron
, “Precession-driven flows in non-axisymmetric ellipsoids
,” J. Fluids Mech.
737
, 412
–439
(2013
).16.
B. P.
Weiss
and S. M.
Tikoo
, “The lunar dynamo
,” Science
346
, 1198
(2014
).17.
C.-C.
Wu
and P.
Roberts
, “On a dynamo driven by topographic precession
,” Geophys. Astrophys. Fluid Dyn.
103
, 467
–501
(2009
).18.
C.
Nore
, J.
Léorat
, J.-L.
Guermond
, and F.
Luddens
, “Nonlinear dynamo action in a precessing cylindrical container
,” Phys. Rev. E
84
, 016317
(2011
).19.
O.
Goepfert
and A.
Tilgner
, “Dynamos in precessing cubes
,” New J. Phys.
18
, 103019
(2016
).20.
Y.
Lin
, P.
Marti
, J.
Noir
, and A.
Jackson
, “Precession-driven dynamos in a full sphere and the role of large scale cyclonic vortices
,” Phys. Fluids
28
, 066601
(2016
).21.
D.
Cébron
, R.
Laguerre
, J.
Noir
, and N.
Schaeffer
, “Precessing spherical shells: Flows, dissipation, dynamo and the lunar core
,” Geophys. J. Int.
219
, S34
–S57
(2019
).22.
F.
Stefani
, S.
Eckert
, G.
Gerbeth
, A.
Giesecke
, T.
Gundrum
, C.
Steglich
, T.
Weier
, and B.
Wustmann
, “DRESDYN—A new facility for MHD experiments with liquid sodium
,” Magnetohydrodynamics
48
, 103
–114
(2012
).23.
F.
Stefani
, T.
Albrecht
, G.
Gerbeth
, A.
Giesecke
, T.
Gundrum
, J.
Herault
, C.
Nore
, and C.
Steglich
, “Towards a precession driven dynamo experiment
,” Magnetohydrodynamics
51
, 275
–284
(2015
).24.
R. F.
Gans
, “On the precession of a resonant cylinder
,” J. Fluid Mech.
41
, 865
–872
(1970
).25.
R. F.
Gans
, “On hydromagnetic precession in a cylinder
,” J. Fluid Mech.
45
, 111
–130
(1971
).26.
A.
Giesecke
, T.
Vogt
, T.
Gundrum
, and F.
Stefani
, “Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action
,” Phys. Rev. Lett.
120
, 024502
(2018
).27.
A.
Giesecke
, T.
Vogt
, T.
Gundrum
, and F.
Stefani
, “Kinematic dynamo action of a precession-driven flow based on the results of water experiments and hydrodynamic simulations
,” Geophys. Astrophys. Fluid Dyn.
113
, 235
–255
(2019
).28.
J.
Herault
, T.
Gundrum
, A.
Giesecke
, and F.
Stefani
, “Subcritical transition to turbulence of a precessing flow in a cylindrical vessel
,” Phys. Fluids
27
, 124102
(2015
).29.
J.
Léorat
, P.
Lallemand
, J.
Guermond
, and F.
Plunian
, “Dynamo action, between numerical experiments and liquid sodium devices
,” in Dynamo and Dynamics, a Mathematical Challenge
, edited by P.
Chossat
, D.
Ambruster
, and I.
Oprea
(Springer
, Dordrecht, The Netherlands
, 2001
), Vol. 26
, pp. 25
–33
.30.
J.
Léorat
, “Large scales features of a flow driven by precession
,” Magnetohydrodynamics
42
, 143
–151
(2006
).31.
J. M.
Lopez
and F.
Marques
, “Nonlinear and detuning effects of the nutation angle in precessionally forced rotating cylinder flow
,” Phys. Rev. Fluids
1
, 023602
(2016
).32.
T.
Albrecht
, H. M.
Blackburn
, J. M.
Lopez
, R.
Manasseh
, and P.
Meunier
, “On triadic resonance as an instability mechanism in precessing cylinder flow
,” J. Fluid Mech.
841
, R3
(2018
).33.
J. P.
Vanyo
and P. W.
Likins
, “Use of baffles to suppress energy dissipation in liquid-filled precessing cavities
,” J. Spacecr. Rockets
10
, 627
(1973
).34.
D. S.
Zimmerman
, S. A.
Triana
, H.-C.
Nataf
, and D. P.
Lathrop
, “A turbulent, high magnetic Reynolds number experimental model of Earth's core
,” J. Geophys. Res. (Solid Earth)
119
, 4538
–4557
, (2014
).35.
K.
Finke
and A.
Tilgner
, “Simulations of the kinematic dynamo onset of spherical Couette flows with smooth and rough boundaries
,” Phys. Rev. E
86
, 016310
(2012
).36.
R.
Monchaux
, M.
Berhanu
, M.
Bourgoin
, M.
Moulin
, P.
Odier
, J.-F.
Pinton
, R.
Volk
, S.
Fauve
, N.
Mordant
, F.
Pétrélis
, A.
Chiffaudel
, F.
Daviaud
, B.
Dubrulle
, C.
Gasquet
, L.
Marié
, and F.
Ravelet
, “Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium
,” Phys. Rev. Lett.
98
, 044502
(2007
).37.
S.
Miralles
, N.
Bonnefoy
, M.
Bourgoin
, P.
Odier
, J.-F.
Pinton
, N.
Plihon
, G.
Verhille
, J.
Boisson
, F.
Daviaud
, and B.
Dubrulle
, “Dynamo threshold detection in the von Kármán sodium experiment
,” Phys. Rev. E
88
, 013002
(2013
).38.
A.
Giesecke
, C.
Nore
, F.
Stefani
, G.
Gerbeth
, J.
Léorat
, W.
Herreman
, F.
Luddens
, and J.-L.
Guermond
, “Influence of high-permeability discs in an axisymmetric model of the Cadarache dynamo experiment
,” New J. Phys.
14
, 053005
(2012
).39.
C.
Nore
, J.
Léorat
, J.-L.
Guermond
, and A.
Giesecke
, “Mean-field model of the von Kármán sodium dynamo experiment using soft iron impellers
,” Phys. Rev. E
91
, 013008
(2015
).40.
S.
Kreuzahler
, D.
Schulz
, H.
Homann
, Y.
Ponty
, and R.
Grauer
, “Numerical study of impeller-driven von Kármán flows via a volume penalization method
,” New J. Phys.
16
, 103001
(2014
).41.
S.
Kreuzahler
, Y.
Ponty
, N.
Plihon
, H.
Homann
, and R.
Grauer
, “Dynamo enhancement and mode selection triggered by high magnetic permeability
,” Phys. Rev. Lett.
119
, 234501
(2017
).42.
C.
Nore
, D.
Castanon Quiroz
, L.
Cappanera
, and J.-L.
Guermond
, “Direct numerical simulation of the axial dipolar dynamo in the Von Kármán sodium experiment
,” Europhys. Lett.
114
, 65002
(2016
).43.
F.
Pizzi
, A.
Giesecke
, and F.
Stefani
, “Ekman boundary layers in a fluid filled precessing cylinder
,” AIP Adv.
11
, 035023
(2021
).44.
F.
Pizzi
, A.
Giesecke
, J.
Šimkanin
, and F.
Stefani
, “Prograde and retrograde precession of a fluid-filled cylinder
,” New J. Phys.
23
, 123016
(2021
).45.
J. W.
Cooley
and J. W.
Tukey
, “An algorithm for the machine calculation of complex Fourier series
,” Math. Comput.
19
, 297
–301
(1965
).46.
C.
Canuto
, M. Y.
Hussaini
, A.
Quarteroni
, and T. A.
Zang
, Spectral Methods: Fundamentals in Single Domains
(Springer Science and Business Media
, 2007
).47.
48.
G.
Patterson
, Jr. and S. A.
Orszag
, “Spectral calculations of isotropic turbulence: Efficient removal of aliasing interactions
,” Phys. Fluids
14
, 2538
–2541
(1971
).49.
C.-W.
Shu
and S.
Osher
, “Efficient implementation of essentially non-oscillatory shock-capturing schemes
,” J. Comput. Phys.
77
, 439
–471
(1988
).50.
S.
Gottlieb
, C.-W.
Shu
, and E.
Tadmor
, “Strong stability-preserving high-order time discretization methods
,” SIAM Rev.
43
, 89
–112
(2001
).51.
A. J.
Chorin
, “Numerical solution of the Navier-Stokes equations
,” Math. Comput.
22
, 745
–762
(1968
).52.
R.
Temam
, “Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II
,” Arch. Ration. Mech. Anal.
33
, 377
–385
(1969
).53.
R.
Temam
, Navier-Stokes Equations: Theory Numerical Analysis
(American Mathematical Soc.
, 2001
), Vol. 343
.54.
C. S.
Peskin
, “Numerical analysis of blood flow in the heart
,” J. Comput. Phys.
25
, 220
–252
(1977
).55.
W.-X.
Huang
and F.-B.
Tian
, “Recent trends and progress in the immersed boundary method
,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
233
, 7617
–7636
(2019
).56.
D.
Goldstein
, R.
Handler
, and L.
Sirovich
, “Modeling a no-slip flow boundary with an external force field
,” J. Comput. Phys.
105
, 354
–366
(1993
).57.
D.
Goldstein
, R.
Handler
, and L.
Sirovich
, “Direct numerical simulation of turbulent flow over a modeled riblet covered surface
,” J. Fluid Mech.
302
, 333
–376
(1995
).58.
E.
Saiki
and S.
Biringen
, “Numerical simulation of a cylinder in uniform flow: Application of a virtual boundary method
,” J. Comput. Phys.
123
, 450
–465
(1996
).59.
M.-C.
Lai
and C. S.
Peskin
, “An immersed boundary method with formal second-order accuracy and reduced numerical viscosity
,” J. Comput. Phys.
160
, 705
–719
(2000
).60.
G.
Iaccarino
and R.
Verzicco
, “Immersed boundary technique for turbulent flow simulations
,” Appl. Mech. Rev.
56
, 331
–347
(2003
).61.
R.
Mittal
and G.
Iaccarino
, “Immersed boundary methods
,” Annu. Rev. Fluid Mech.
37
, 239
–261
(2005
).62.
E.
Arquis
, J.
Caltagirone
et al, “Sur les conditions hydrodynamiques au voisinage d'une interface milieu fluide-milieu poreux: Applicationa la convection naturelle
,” C.R. Acad. Sci. Paris II
299
, 1
–4
(1984
).63.
G. H.
Keetels
, H.
Clercx
, and G.
van Heijst
, “Fourier spectral solver for the incompressible Navier-Stokes equations with volume-penalization
,” in Proceedings of the International Conference on Computational Science
(Springer
, 2007
), pp. 898
–905
.64.
C.
Jause-Labert
, F. S.
Godeferd
, and B.
Favier
, “Numerical validation of the volume penalization method in three-dimensional pseudo-spectral simulations
,” Comput. Fluids
67
, 41
–56
(2012
).65.
E.
Fadlun
, R.
Verzicco
, P.
Orlandi
, and J.
Mohd-Yusof
, “Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations
,” J. Comput. Phys.
161
, 35
–60
(2000
).66.
A.
Cristallo
and R.
Verzicco
, “Combined immersed boundary/large-eddy-simulations of incompressible three dimensional complex flows
,” Flow Turbul. Combust.
77
, 3
–26
(2006
).67.
T.
Engels
, D.
Kolomenskiy
, K.
Schneider
, F.-O.
Lehmann
, and J.
Sesterhenn
, “Bumblebee flight in heavy turbulence
,” Phys. Rev. Lett.
116
, 028103
(2016
).68.
J. A.
Morales
, W. J.
Bos
, K.
Schneider
, and D. C.
Montgomery
, “Magnetohydrodynamically generated toroidal and poloidal velocities in confined plasma
,” APS Div. Plasma Phys. Meet. Abstr.
2013
, TO5–003
.69.
J. A.
Morales
, M.
Leroy
, W. J.
Bos
, and K.
Schneider
, “Simulation of confined magnetohydrodynamic flows with Dirichlet boundary conditions using a pseudo-spectral method with volume penalization
,” J. Comput. Phys.
274
, 64
–94
(2014
).70.
K.
Schneider
, “Immersed boundary methods for numerical simulation of confined fluid and plasma turbulence in complex geometries: A review
,” J. Plasma Phys.
81
, 435810601
(2015
).71.
J.
Mohd-Yusof
, “Combined immersed boundary/b-spline method for simulations of flows in complex geometries
,” in CTR Annual Research Briefs
(NASA AMES/Stanford University
, 1997
).72.
R.
Verzicco
, J.
Mohd-Yusof
, P.
Orlandi
, and D.
Haworth
, “LES in complex geometries using boundary body forces
,” in Proceedings of the Summer Program
(Center for Turbulence Research Stanford
, CA
, 1998
), pp. 171
–186
.73.
E.
Balaras
, “Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations
,” Comput. Fluids
33
, 375
–404
(2004
).74.
A.
Gilmanov
, F.
Sotiropoulos
, and E.
Balaras
, “A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids
,” J. Comput. Phys.
191
, 660
–669
(2003
).75.
A.
Gilmanov
and F.
Sotiropoulos
, “A hybrid Cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies
,” J. Comput. Phys.
207
, 457
–492
(2005
).76.
L.
Ge
and F.
Sotiropoulos
, “A numerical method for solving the 3d unsteady incompressible Navier–Stokes equations in curvilinear domains with complex immersed boundaries
,” J. Comput. Phys.
225
, 1782
–1809
(2007
).77.
I.
Borazjani
, L.
Ge
, and F.
Sotiropoulos
, “Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies
,” J. Comput. Phys.
227
, 7587
–7620
(2008
).78.
79.
80.
A.
Narkhede
and D.
Manocha
, “Fast polygon triangulation based on seidel's algorithm
,” in Graphics Gems V
(Elsevier
, 1995
), pp. 394
–397
.81.
S.
Woop
, C.
Benthin
, and I.
Wald
, “Watertight ray/triangle intersection
,” J. Comput. Graph. Tech. (JCGT)
2
(1
), 65
–82
(2013
), available at http://jcgt.org/published/0002/01/05/.82.
T.
Ikeno
and T.
Kajishima
, “Finite-difference immersed boundary method consistent with wall conditions for incompressible turbulent flow simulations
,” J. Comput. Phys.
226
, 1485
–1508
(2007
).83.
T.-L.
Horng
, P.-W.
Hsieh
, S.-Y.
Yang
, and C.-S.
You
, “A simple direct-forcing immersed boundary projection method with prediction-correction for fluid-solid interaction problems
,” Comput. Fluids
176
, 135
–152
(2018
).84.
R.
Rannacher
, “On Chorin's projection method for the incompressible Navier-Stokes equations
,” in The Navier-Stokes Equations II—Theory and Numerical Method
(Springer, 1992), pp. 167
–183.
85.
J.-L.
Guermond
, P.
Minev
, and J.
Shen
, “An overview of projection methods for incompressible flows
,” Comput. Methods Appl. Mech. Eng.
195
, 6011
–6045
(2006
).86.
J.
Van Kan
, “A second-order accurate pressure-correction scheme for viscous incompressible flow
,” SIAM J. Sci. Stat. Comput.
7
, 870
–891
(1986
).87.
K.
Goda
, “A multistep technique with implicit difference schemes for calculating two-or three-dimensional cavity flows
,” J. Comput. Phys.
30
, 76
–95
(1979
).88.
L. J.
Timmermans
, P. D.
Minev
, and F. N.
Van De Vosse
, “An approximate projection scheme for incompressible flow using spectral elements
,” Int. J. Numer. Methods Fluids
22
, 673
–688
(1996
).89.
J.
Guermond
and J.
Shen
, “On the error estimates for the rotational pressure-correction projection methods
,” Math. Comput.
73
, 1719
–1737
(2004
).90.
D.
Pekurovsky
, “P3dfft: A framework for parallel computations of Fourier transforms in three dimensions
,” SIAM J. Sci. Comput.
34
, C192
–C209
(2012
).91.
N.
Li
and S.
Laizet
, “2decomp and FFT—A highly scalable 2d decomposition library and FFT interface
,” in Proceedings of the Cray User Group 2010 Conference
(2010
), pp. 1
–13
.92.
M.
Pippig
, “PFFT: An extension of FFTW to massively parallel architectures
,” SIAM J. Sci. Comput.
35
, C213
–C236
(2013
).93.
K. G.
Larkin
, M. A.
Oldfield
, and H.
Klemm
, “Fast Fourier method for the accurate rotation of sampled images
,” Opt. Commun.
139
, 99
–106
(1997
).94.
L.
Dalcin
, M.
Mortensen
, and D. E.
Keyes
, “Fast parallel multidimensional FFT using advanced MPI
,” J. Parallel Distrib. Comput.
128
, 137
–150
(2019
).95.
Message Passing Interface Forum, MPI: A Message-Passing Interface Standard Version 4.0,
2021
.96.
M.
Frigo
and S. G.
Johnson
, “The design and implementation of FFTW3
,” Proc. IEEE
93
, 216
–231
(2005
).97.
D.
Krause
, “Juwels: Modular tier-0/1 supercomputer at the jülich supercomputing centre
,” J. Large-Scale Res. Facil.
5
, A135
–A135
(2019
).98.
E.
Benton
and A.
Clark
, Jr., “Spin-up
,” Annu. Rev. Fluid Mech.
6
, 257
–280
(1974
).99.
P.
Duck
and M.
Foster
, “Spin-up of homogeneous and stratified fluids
,” Annu. Rev. Fluid Mech.
33
, 231
–263
(2001
).100.
J. S.
Park
and J. M.
Hyun
, “Review on open-problems of spin-up flow of an incompressible fluid
,” J. Mech. Sci. Technol.
22
, 780
–787
(2008
).101.
H.
Greenspan
and L.
Howard
, “On a time-dependent motion of a rotating fluid
,” J. Fluid Mech.
17
, 385
–404
(1963
).102.
H. P.
Greenspan
, The Theory of Rotating Fluids
, Cambridge Monographs on Mechanics and Applied Mathematics (Cambridge University Press
, 1968
).103.
E.
Wedemeyer
, “The unsteady flow within a spinning cylinder
,” J. Fluid Mech.
20
, 383
–399
(1964
).104.
G.
Venezian
, “Spin-up of a contained fluid
,” California Institute of Technology, Division of Engineering and Applied Science Report No. 97-17, Pasadena, CA, 1969
.105.
A.
Giesecke
, T.
Albrecht
, T.
Gundrum
, J.
Herault
, and F.
Stefani
, “Triadic resonances in nonlinear simulations of a fluid flow in a precessing cylinder
,” New J. Phys.
17
, 113044
(2015
).106.
F.
Burmann
and J.
Noir
, “Effects of bottom topography on the spin-up in a cylinder
,” Phys. Fluids
30
, 106601
(2018
).© 2022 Author(s). Published under an exclusive license by AIP Publishing.
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