Rotating flows with free-surface vortices can be found in many engineering applications, such as pump and turbine intakes, vessels, and nuclear reactors. The need to address rather different flow regions existing in such flows, such as Ekman and Stewartson layers and the line vortex zone, in a coupled manner, makes modeling of free-surface rotating flows very challenging. In this work, the flow field of a free-surface vortex, created in a rotating cylinder with a drain hole in its bottom, is investigated numerically and analytically. Above the drain hole of the cylinder, a free-surface vortex, accompanied by axial velocity, is created. This axial velocity profile is governed by the Ekman boundary layer far from the axis and by the drainage in its proximity. The experiments of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)] on the so-called bathtub vortex are numerically modeled with the finite volume method. The simulations are validated with the available measurements from the experiments. Using the simulation results, self-similar and non-self-similar models, describing the velocity fields in the Ekman boundary layer, are compared and tested. It is shown that the self-similar model is more accurate than the non-self-similar model. It is also demonstrated that the analytical model of Andersen et al. [“Anatomy of a bathtub vortex,” Phys. Rev. Lett. 91(10), 104502 (2003a); “The bathtub vortex in a rotating container,” J. Fluid Mech. 556, 121–146 (2006)], when modified as suggested in the present study, is capable of predicting the free-surface profile for low rotation rates. However, for high rotation rates, only the numerical simulation can predict the relation between the flow field within the liquid and the free-surface profile.

1.
Andersen
,
A.
,
Bohr
,
T.
,
Stenum
,
B.
,
Rasmussen
,
J. J.
, and
Lautrup
,
B.
, “
Anatomy of a bathtub vortex
,”
Phys. Rev. Lett.
91
(
10
),
104502
(
2003a
).
2.
Andersen
,
A.
,
Bohr
,
T.
,
Stenum
,
B.
,
Rasmussen
,
J. J.
, and
Lautrup
,
B.
, “
The bathtub vortex in a rotating container
,”
J. Fluid Mech.
556
,
121
146
(
2006
).
3.
Andersen
,
A.
,
Lautrup
,
B.
, and
Bohr
,
T.
, “
An averaging method for nonlinear laminar Ekman layers
,”
J. Fluid Mech.
487
,
81
90
(
2003b
).
4.
ANSYS
,
ICEM CFD Academic (Version R1)
(
ANSYS Inc
.,
Canonsburg, PA
,
2019
).
5.
Batchelor
,
G. K.
,
An Introduction to Fluid Dynamics
(
Cambridge University Press
,
1967
).
6.
Bergmann
,
R.
,
Andersen
,
A.
,
van der Meer
,
D.
, and
Bohr
,
T.
, “
Bubble pinch-off in a rotating flow
,”
Phys. Rev. Lett.
102
(
20
),
204501
(
2009
).
36.
Brackbill
,
J. U.
,
Kothe
,
D. B.
, and
Zemach
,
C.
, “
A continuum method for modelling surface tension
,”
J. Comput. Phys.
100
(
20
),
335
354
(
1992
).
7.
Burgers
,
J. M.
, “
A mathematical model illustrating the theory of turbulence
,” in
Advances in Applied Mechanics
, edited by
R.
Von Mises
and
T.
Von Kármán
(
Elsevier
,
1948
), Vol.
1
, pp.
171
199
.
8.
Duinmeijer
,
A.
,
Oldenziel
,
G.
, and
Clemens
,
F.
, “
Experimental study on the 3D-flow field of a free-surface vortex using stereo PIV
,”
J. Hydraul. Res.
58
(
1
),
105
119
(
2020
).
9.
Einstein
,
H. A.
and
Li
,
H.
, “
Steady vortex flow in a real fluid
,” in
Proceedings of Heat Transfer and Fluid Mechanics Institute
(
Stanford University Press
,
1951
), pp.
33
43
.
10.
Ekman
,
V. W.
, “
On the influence of the Earth's rotation on ocean-currents
,”
Ark. Mat., Asron. Fys.
2
(
11
),
1
53
(
1905
).
11.
Goller
,
H.
and
Ranov
,
T.
, “
Unsteady rotating flow in a cylinder with a free surface
,”
J. Basic Eng.
90
(
4
),
445
452
(
1968
).
12.
Greenspan
,
H. P.
,
The Theory of Rotating Fluids
(
Cambridge University Press
,
1968
).
13.
Greenspan
,
H. P.
and
Howard
,
L. N.
, “
On a time-dependent motion of a rotating fluid
,”
J. Fluid Mech.
17
(
3
),
385
404
(
1963
).
14.
Hirt
,
C. W.
and
Nichols
,
B. D.
, “
Volume of fluid (VOF) method for the dynamics of free boundaries
,”
J. Comput. Phys.
39
(
1
),
201
225
(
1981
).
15.
Khoshkalam
,
N.
,
Najafi
,
A. F.
,
Rahimian
,
M. H.
, and
Magagnato
,
F.
, “
Numerical study on air-core vortex: Analysis of generation mechanism
,”
Arch. Appl. Mech.
90
(
1
),
1
16
(
2020
).
16.
Knauss
,
J.
,
Swirling Flow Problems at Intakes. Hydraulic structure design manual
(
Taylor & Francis
,
London and New York
,
1987
).
17.
Lewellen
,
W. S.
, “
A solution for three-dimensional vortex flows with strong circulation
,”
J. Fluid Mech.
14
(
3
),
420
432
(
1962
).
18.
Liu
,
S.
,
Zhou
,
J.
,
Huang
,
W.
, and
Cheng
,
X.
, “
Study of stationary vortex with a free surface at the bottom of the orifice flow
,”
J. Nucl. Sci. Technol.
55
(
1
),
79
89
(
2018
).
19.
Lugt
,
H. J.
,
Introduction to Vortex Theory
(
Vortex Flow Press
,
1996
).
20.
Lugt
,
H. J.
,
Vortex Flow in Nature and Technology
(
Wiley-Interscience
,
New York
,
1983
).
21.
Lundgren
,
T. S.
, “
The vortical flow above the drain-hole in a rotating vessel
,”
J. Fluid Mech.
155
,
381
412
(
1985
).
22.
Mondal
,
R. K.
and
Kumar
,
P.
, “
Experimental study of entrained air-core structures induced by a pump intake vortex
,”
Phys. Fluids
34
(
5
),
052116
(
2022
).
23.
Odgaard
,
A. J.
, “
Free‐surface air core vortex
,”
J. Hydraul. Eng.
112
(
7
),
610
620
(
1986
).
24.
Owen
,
J. M.
,
Pincombe
,
J. R.
, and
Rogers
,
R. H.
, “
Source–sink flow inside a rotating cylindrical cavity
,”
J. Fluid Mech.
155
,
233
265
(
1985
).
25.
Rashkovan
,
A.
,
Amar
,
S. D.
,
Bieder
,
U.
, and
Ziskind
,
G.
, “
Analysis of polygonal vortex flows in a cylinder with a rotating bottom
,”
Appl. Sci.
11
(
3
),
1348
(
2021
).
26.
Rogers
,
M. H.
and
Lance
,
G. N.
, “
The rotationally symmetric flow of a viscous fluid in the presence of an infinite rotating disk
,”
J. Fluid Mech.
7
(
4
),
617
631
(
1960
).
27.
Rott
,
N.
, “
On the viscous core of a line vortex
,”
J. Appl. Math. Phys.
9
(
5–6
),
543
553
(
1958
).
28.
Sohn
,
C. H.
,
Son
,
J. H.
, and
Park
,
I. S.
, “
Numerical analysis of vortex core phenomenon during draining from cylinder tank for various initial swirling speeds and various tank and drain port sizes
,”
J. Hydrodyn.
25
(
2
),
183
195
(
2013
).
29.
Stepanyants
,
Y. A.
and
Yeoh
,
G. H.
, “
Stationary bathtub vortices and a critical regime of liquid discharge
,”
J. Fluid Mech.
604
,
77
98
(
2008
).
30.
Takahashi
,
M.
,
Inoue
,
A.
,
Aritomi
,
M.
,
Takenaka
,
Y.
, and
Suzuki
,
K.
, “
Gas entrainment at free surface of liquid (I)
,”
J. Nucl. Sci. Technol.
25
(
2
),
131
142
(
1988
).
31.
Tan
,
D.
,
Li
,
L.
,
Yin
,
Z.
,
Li
,
D.
,
Zhu
,
Y.
, and
Zheng
,
S.
, “
Ekman boundary layer mass transfer mechanism of free sink vortex
,”
Int. J. Heat Mass Transfer
150
,
119250
(
2020
).
32.
Tenchine
,
D.
,
Fournier
,
C.
, and
Dolias
,
Y.
, “
Gas entrainment issues in sodium cooled fast reactors
,”
Nucl. Eng. Des.
270
,
302
311
(
2014
).
33.
Vanyo
,
J. P.
,
Rotating Fluids in Engineering and Science
(
Courier Corporation
,
2001
).
34.
Von Kármán
,
T.
, “
Über laminare und turbulente Reibung
,”
Z. Angew. Math. Mech.
1
(
4
),
233
252
(
1921
).
35.
Youngs
,
D. L.
, “
Time-dependent multi-material flow with large fluid distortion
,” in
Numerical Methods for Fluid Dynamics
, edited by K. W. Morton and M. J. Baibes (Academic Press, New York, 1982), pp. 273–285.
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