spohn1993

Experiments in Fluids 14, 70 77 (1993)
Experiments in Fluids
9 Springer-Verlag 1993
Observations of vortex breakdown in an open
with a rotating bottom
A. Spohn*, M. Mory and E.J. Hopfinger
Institut de M~canique de Grenoble B.P. 53 X, F-38041 Grenoble Cedcx, France
cylindrical container
Abstract. A laser induced fluorescent dye technique was used to
visualize the steady-state flow driven by a rotating bottom in an
open, cylindrical container. The flow behaviour and the vortex
breakdown conditions were studied as a function of the container
aspect ratio H/R and the Reynolds number Re = f2 R2/v. Like in the
closed container configuration, previously studied by Vogel (1968)
and Escudier (1984), vortex breakdown occurs in a certain parame-
ter range (H/R, Re). However, in the free surface configuration vor-
tex breakdown conditions as well as the forms of the breakdown
bubbles differ notably from what is observed in the closed container
configuration. In particular, it is found that as Re is increased, the
breakdown bubbles get attached to the free surface and grow in
diameter.
1 Introduction
The swirling flow generated in a closed cylindrical container
by a rotating disc gives, under certain conditions, rise to the
formation of one or more recirculation bubbles, centered on
the cylinder axis. Each of these recirculation bubbles has a
stagnation point on its upstream side and thus satisfies the
definition of vortex breakdown given by Hall (1972). How-
ever, two major characteristics distinguish the vortex break-
down in the container flow from those generally observed in
vortex tubes or in trailing vortices. Firstly, the Reynolds
number is lower and the swirl number is higher. Secondly,
the phenomenon may be observed under laminar and steady
conditions. These differences and the possibility to study the
phenomenon without the influence of turbulence, make the
container flow a useful configuration for obtaining further
insight into vortex breakdown.
Most of the previous studies used closed cylindrical con-
tainers with one rotating end wall, which recirculates the
fluid inside the container. The flow structure is determined
by the container aspect ratio H/R and the Reynolds number
Re = f2 RZ/v, H and R being, respectively, the height and the
inner radius of the container, Q the angular velocity of the
* Present address: Laboratoire d'Electrostatique et des Mat6riaux
Di61ectriques, 25 avenue des Martyrs, B.P. 166 X, Grenoble Cedex,
France
rotating end wall, and v the kinematic viscosity of the work-
ing fluid. The visualizations by Vogel (1968) and the LDA
measurements by Ronnenberg (1977), demonstrated the oc-
currence of one breakdown bubble in the range Re < 3,000
and H/R < 2.5. More recently, Escudier (1984) enlarged the
investigated aspect ratio range up to H/R < 3.5. He found
flow regimes with up to three separated breakdown bubbles.
The number and locations of the observed breakdown bub-
bles have been confirmed by several numerical simulations
(e.g. Lugt and Haussling, 1982; Lugt and Abboud, 1987;
Neitzel, 1988; Daube and Sorensen, 1989; Lopez, 1990).
Roesner (1990) extended the previous investigations to the
case of two corotating discs, one at the top and the other at
the bottom of the container. Although his results show that
the parameter combinations (H/R, Re) for which vortex
breakdown is observed depend on the angular velocity of the
second rotating disc, the location and the structure of the
breakdown bubbles are similar to the case of one rotating
disc in a closed container.
In all studies mentioned above the rigid end walls impose
the no-slip condition on the flow and thus force the forma-
tion of three-dimensional boundary layers, similar to Ekman
layers, which interact with the swirling motion in the inner
part of the container. In this way the Reynolds number and
the swirl number of the vortex flow are determined by the
boundary conditions, and by changing them it is possible to
act on vortex breakdown conditions. The aim of the present
study was to investigate the consequences on the flow struc-
ture and on the conditions of vortex breakdown, when the
upper boundary no-slip condition, used in previous studies,
is modified. The upper rigid end wall of the cylindrical con-
tainer was therefore replaced by a free surface. The resulting
flow regimes were studied in the range Re<4,000 and
H/R ~< 4. As in the closed container configuration, these two
non-dimensional parameters proved to be the most relevant.
The Froude number Fr = ~2 Re/gH was negligibly small.
Studies of rotating flows in containers with a free surface
are scarce. Published results mainly deal with the transient
flow in open cylindrical containers, which rotate about their

71
axis of symmetry (e.g. Goller and Ranov, 1968). An exception
are the numerical simulations of an open container flow with
a rotating bottom made by Hyun (1985). However, his calcu-
lations for Re = 100 and H/R = 0.5 give no indications of the
existence of vortex breakdown.
2 Experimental arrangement
Figure 1 shows schematically the experimental set-up. A ver-
tical circular perspex tube of 55 mm inner radius, fixed on a
metal table, encloses the test section. The rotating disc of
radius 54 mm is made of stainless steel and is driven by an
electric motor via a transmission belt. The angular velocity
of the disc is adjustable with a precision better than ___ 0.1%.
The upper end of the cylinder is left open. By changing the
quantity of working fluid inside the test section, it was pos-
sible to vary H/R continuously. A mixture of glycerine-water
with 60% glycerine was used as working fluid. The cinematic
viscosity of this mixture varied, depending on the tempera-
ture of the mixture, between 6 x 10 -6 and 10 x 10 -6 m2/s.
The cylinder itself is placed inside a rectangular water tank
made of transparent perspex of dimensions 46 cm x 46 cm x
60 cm. The water in this tank was recirculated through a
constant temperature bath. By means of the control of the
recirculating water temperature, the temperature of the glyc-
erine-water mixture in the test section was kept constant to
within _ 0.2 ~ In order to prevent heat losses across the
free surface, the air in contact with the free surface was
separated from the air in the laboratory by a styrofoam
cover.
The steady state flow structure was visualized by the laser
induced fluorescent dye technique. A small quantity of the
same glycerine-water mixture as used in the test section, was
coloured with fluorescent dye in a small reservoir. This res-
ervoir communicated with the free surface in the test section
through a plastic tube of 1 mm i.d. The end of the plastic
tube was equipped with a little piece of foam. By adjusting
a small difference of height between the free surfaces in the
reservoir and the free surface in the test section, it was pos-
sible to release the coloured mixture into the container with-
out any visible perturbation. The radial distance of the emis-
sion tube from the cylinder axis varied between 0 and 3 cm,
depending on the observed flow structure. Once released, the
distribution of the dye tracer inside the container was ob-
served in a diametral plane with the help of a sheet of light
of 1 mm thickness, produced by expanding the light beam of
a 4 Watt Argon laser by means of an oscillating mirror.
[ 9 .,,i----
r- , i .... ...... __ ~-~/~-- RECTANGULAR
1
-~ R r'-- IIit]1 ~CYLINDRICAL
| ! IIIV CONTA 1 NER
! URFACE
I ~ ~/~ DYE RESERVOIR
h
I I
Fig. l. Schematic diagram of the experimental arrangement
cal properties, different from those of the bulk inside the
container (see, e.g., Davis and Rideal, 1963). This may cause
constraints imposed by the surface film on the flow. Strictly
speaking the container flow with a free surface, therefore,
also depends on the chemical composition of the interface.
However, the observed flow structures were reproducible,
even when the working fluid and the visualization technique
were changed (see Spohn, 1991). This suggests indirectly
that, although the chemical composition of the free surface
is not known in the present experiments, any possible fluctu-
ations of this chemical composition had no detectable influ-
ence on the flow behaviour. In order to get similar initial
conditions before starting the fluid motion from rest, surface
films which might have formed before the experiments on
the interface at rest, were destroyed by agitating the working
fluid 15 min before starting the experiments. During all ex-
periments the free surface showed an azimuthal velocity
component as indicated by the movement of small floating
dust particles. Complementary velocity measurements near
the free surface with water as working fluid indicated no
measurable azimuthal stresses and thus suggest the absence
of a significant surface viscosity (see Spohn, 1991).
The existence of a free surface makes the open container
flow very sensitive to shocks. Flow separations on the side
boundaries caused by vibrations were observed to modify
the whole flow structure. Therefore, special care was taken to
prevent any kind of shocks in the laboratory during the
experiments.
Finally, it should be emphasized that the rotation of the
fluid causes a deformation h of the free surface proportional
to the Froude number Fr = 0 2 R2/oH. For the smallest as-
pect ratio and the highest Reynolds number considered in
this study h/H did not exceed 0.1.
3 Free surface conditions
The existence of a free surface causes some particular exper-
imental difficulties which are hard to overcome. First of all,
surface-active contaminations of the free surface may give
rise to the formation of a surface film with special mechani-
4 Experimental procedure
To study the flow behaviour, the whole H/R range between
0.5 and 4 was subdivided into discrete values, separated by
steps of 0.25. At each H/R value a series of measurements

72
Fig. 2a-g. Visualization of the evolution of the central flow struc-
ture with increasing
f2RZ/v for H/R=2; a ~R2/y ~ 1130; b 1160; e
1180; d 1210; e 1420; f 1740; g 2200
was carried out for different Reynolds numbers. Before and
after a series of experiments the temperature of the glycerine-
water mixture, the water in the surrounding tank, and the air
2 cm above the free surface, was checked with a digital ther-
mometer (Chateau Arnoux, sensor Pt 100 for liquids, respec-
tively gases). The measured temperature difference between
the air and the glycerine-water mixture varied from _+ 0.5 ~
at the beginning of a series of experiments to _+ 1 ~ at the
end of a series. At the end of a series of experiments the
viscosity of the glycerine-water mixture was checked with a
viscosimeter (Haake-Kugelfall Pr/izisionsausftihrung, Geb.
Haake K.G., Berlin, Germany).
At each flow condition a small quantity of the dyed fluid
was released, typically 3-5 min after the angular velocity of
the disc was regulated and a steady flow state at a fixed
Reynolds number was recovered. Once the dyed fluid was
convected through the whole central plane, the flow struc-
ture visualized by the dye was photographed, using a stan-
dard Nikon F2 camera, objective 200mm which was
equipped with a black and white film (125 ASA ILFORD
FP4). Due to the presence of the rectangular water tank the
optical distortion caused by the circular container walls was
negligible up to a radius of r = 0.9 R. The optical distortion
due to the lenses of the camera did not exceed 6%. Light
reflexions at the free surface of the expanded light beam
produced near the free surface an intensification of light and
agitation of the fluorescein molecules. Since these effects
varied with dye concentration, flow structure and aspect
ratio it was not possible to optimize the exposure time for
every flow configuration. Therefore some of the photographs
are over-exposed near the free surface.
in order to observe breakdown bubbles and their loca-
tions it is most convenient to make successive observations
for one fixed aspect ratio
H/R and different values of Re,
without changing the fluid. However, after one or two series
of experiments the fluid became too contaminated and the
whole working fluid had to be replaced.
Finally it should be emphasized that the installation was
tested with a rigid cover instead of a free surface (see Spohn,

73
Fig. 3 a-e. Visualization of the evolution of the central flow structure with increasing f2 R2/v for H/R = 2.75;
a ORZ/v=1720; b 1760; e 1820; d 1985; e 2175
1991). The results were in good agreement with those of
Escudier (1984).
5 Results
To illustrate the evolution of the breakdown region, photo-
graphs for five selected aspect ratios
(H/R = 1, 1.5, 2, 2.75,
and 3.25) are presented. The case
H/R = 2, shown in Fig. 2,
is typical of intermediate aspect ratios. At low Reynolds
numbers (I2
R2/v = 1,130, Fig. 2 a), the coloured fluid released
at the center of the free surface follows the axis of symmetry
in the direction of the rotating disc without any visible per-
turbation. A slight increase in
Re to ~2R2/v = 1,160 causes
the formation of spiral structures (Fig. 2 b). Beginning at the
downstream spirals (the axial flow is from top to the bot-
tom), a further increase in Reynolds number to
(2R2/v=
1,180 leads to a more and more pronounced radial stretching
of the spiral structure (Fig. 2 c). However, no central stagna-
tion point may be identified in Fig. 2c. Finally
at ~2RZ/v =
1,210 the spirals completely disappear, giving place to a fully
developed recirculation bubble (Fig. 2d). At the upstream
side of the bubble the coloured fluid diverges, flows around
the recirculation zone and moves, at the back stream side of
the bubble, partly upwards, i.e., upstream towards the free
surface, and partly downstream, i.e., in direction of the rotat-
ing disc. Since this photograph was made after changing the
fluid, the absence of dye inside the two axisymmetric elliptical
regions of the bubble indicates the existence of a closed
recirculation torus without through flow. At still greater
Reynolds numbers the position of the whole breakdown
bubble moves upstream towards the free surface, and attach-
es to it when
f2R2/v = 1,420 (Fig. 2e). Once adjacent to the
free surface, the diameter of the closed recirculation torus
begins to grow as
Re is further increased (cf. Fig. 2 f and g).
While for
H/R <2.75 only a single stagnation point is
observed, two successive breakdowns occur in a small range
of the parameter space
(H/R, Re) for H/R > 2.75. The photo-
graphs of Fig. 3 illustrate the evolution for
H/R = 2.75. At
f2RZ/v = 1,720 (Fig. 3 a) two separated divergence regions of
the streamlines can be distinguished. Inside the upper one,
spirals begin to form. At
OR2/v= 1,760 a fully developed
breakdown bubble replaces the spirals, while in the lower
divergence region a bubble just begins to emerge (Fig. 3 b).
When further increasing the Reynolds number to
ORZ/v=
1,820 (Fig. 3 c), the two breakdown bubbles merge and give
rise to an elongated bubble with two well distinguishable
recirculation zones, but only one stagnation point on the
axis of symmetry. At even greater Reynolds numbers
(f2R2/v=l,985, Fig. 3d) this elongated bubble moves up-
stream towards the free surface, and becomes finally also
adjacent to the free surface
(f2R2/v=2,175, Fig. 3e). With
the exception of the coexistence of two successive fully
developed breakdown bubbles in a small range of parame-
ters
(H/R, Re), the described evolution also holds for H/R = 3.
However, for H/R>3.25 the sequence of formation of

74
Fig. 4a-d. Visualization of the
evolution of the central flow
structure with increasing f2
R2/v
for H/R = 3.25; a O R2/v = 1980;
b 2200; c 2245; d
2325
of the central flow structure with increasing f2 R /v for H/R = 1.5; Fig. 5a-d. Visualization of the evolution 2
a f2R2/v=740; b 770; e 975; d 2095
the two breakdown bubbles is inverted (Fig. 4): the lower
recirculation bubble appears first
(Q R2/v-- 2,200, Fig. 4b)
and is followed by the formation of a second breakdown
bubble upstream of the first one
(f2RZ/v=2,245, Fig. 4c).
The two bubbles merge
(f2R2/v=2,325, Fig. 4d) and the
whole long vortex breakdown region moves again towards
the free surface and becomes finally adjacent to it.
When
H/R is decreased below H/R--2 the breakdown
bubble becomes attached to the free surface at smaller and
smaller Reynolds numbers. Figure 5 shows the evolution for
H/R = 1.5. While the flow behaviour is qualitatively the same
as before, the bubble gets already attached at
Re = 975. Fig-
ure 5 d
(QR2/v = 2,095) displays the form of the breakdown
bubble attached to the free surface when the bubble diameter
has already increased. The coloured fluid released at the free
surface 3 cm outside of the axis of symmetry flows around
the recirculation torus, visible by the absence of dye traces,
converges towards the axis of symmetry and finally passes
downwards in the direction of the rotating disc. This be-
haviour becomes even more perceptible for
H/R = 1 (Fig. 6).
Once adjacent to the free surface the diameter of the recircu-
lation torus grows as the Reynolds number is increased,
while the cross-section of the torus gets smaller and has a
more pointed form (cf. Fig. 6 a and b).