A toroidal bubble is a type of buoyant vortex
rings moving in a liquid, where gas is used as a buoyant substance [1-3]. A
distinctive feature of
buoyant
vortex rings
compared to homogeneous ones is the presence of a buoyant force acting on them,
which has a significant effect on their dynamics. A large number of works have
been devoted to homogeneous vortex rings, where various optical methods are
actively used. Thus, the use of optical velocimetry made it possible to study
in detail the structure of vortex rings, especially inside the core [4–6]. By
PIV method the presence of a vortex ring has been established during direct
fuel injection in internal combustion engines [7]. The general state of the
human heart was shown in [8] to depend on whether the vortex ring is formed as
a result of left ventricle diastole of the heart. Much less attention has been
paid to inhomogeneous vortex rings [9–11], including toroidal bubbles. The
regularities obtained for toroidal bubbles can be attributed to inhomogeneous
vortex rings, although they have distinctive features. One of them is the gas
mass conservation, which, under the action of centrifugal forces, is
concentrated near the torus circular axis. The peculiar ring structure of
toroidal bubbles, together with the rotation of the liquid around the torus,
could possibly be useful for intensifying the gas-liquid mass transfer [12],
which is encountered in many industrial applications, such as the removal of
harmful gases and the capture of
. However, the calculation of the bubble parameters as a function of
time according to the available theoretical models [2, 11] does not fully agree
with the experiment.
As noted in [13], the dependence of the bubble
radius on time, if the bubble is created by injecting an air jet into water, is
greater than that obtained experimentally. This may indicate the presence of a
drag force acting on the toroidal bubble. It is known that the
drag
force acting on a bubble in the form of a
spherical segment leads to stabilization of its velocity [14, 15]. In the case
of homogeneous vortex rings, the drag force does not significantly affect the
motion [16]. The question of the influence of drag on the dynamics of a
toroidal bubble remains open.
It should be noted that in [13] a model was
proposed taking into account the drag force. However, no comparison with
experiment was made, and the calculation of the drag coefficient was performed
for only one bubble volume. Thus, the main goal of this study is to test the
theoretical model taking into account the drag force, as well as to determine
the values of the empirical drag coefficient included into the model for
toroidal bubbles with different initial parameters.
The model applied in this paper considers a
toroidal bubble rising vertically upward in a gravitational field. The
following assumptions are introduced: the bubble has the shape of a torus with
parameters
and
,
is the radius of the torus,
is the radius of the cross section of the torus, and
<<
; the flow around the bubble is irrotational with constant circulation
; the volume of the torus is constant and equal to
. According to the assumptions made, the expressions obtained in [2,
17] for the vortex momentum
and the rise velocity
are used:
|
|
(1.1)
|
|
|
(1.2)
|
where
is the density of the surrounding liquid.
The differential equation, which can be used
to obtain the time dependence of the radius of a toroidal bubble, is derived
from the dynamic equation of motion, where, similarly to [13], along with the
buoyancy force, the drag force is taken into account in the form:
|
|
(1.3)
|
where
is the drag coefficient.
Without drag force,
that is, with
, from equation (1.3) it is easy to obtain the dependence of the bubble
radius on time:
|
|
(1.4)
|
which is also
applicable for inhomogeneous vortex rings with a small density difference
between the ring and the environment [11]. In this study, formula (1.4) is used
to compare models with and without taking into account the drag force with the
experiment.
The length, time, and circulation parameters
used below are dimensionless on
,
,
respectively, where
is equal to the radius of the ideal sphere containing the same volume
of air as a toroidal bubble.
In dimensionless variables, the Cauchy problem
with respect to equation (1.3), taking into account (1.1), (1.2) and the
constant volume of the bubble
, takes the form:
|
|
(1.5)
|
where
,
,
.
Figure
1. Toroidal bubble ascent, t and z are dimensionless
(
).
The experiments were carried out in a
water-filled Plexiglas tank 0.5 m long, 0.5 m wide and 1.5 m high. The method
for generating toroidal bubbles is as follows. First, compressed air, passing
through the pressure regulator, was supplied to the solenoid valve. The open
time of the solenoid valve was determined by a programmable logic controller.
Moving on, the compressed air flow passed through a mechanical valve, where,
pushing the cup against the spring tension, it was injected into the tank. The
diameter of the mechanical valve nozzle was 0.4 cm. After injection, the bubble
acquired a toroidal shape during the ascent. In the upper part of the tank, the
toroidal bubble was captured by a funnel-shaped device for determining the
volume of air
W. This method for determining the volume allows us to
avoid errors associated with perturbations of the bubble surface during
formation and movement compared to measurements from images. The volume
measurement accuracy is 0.75 ml, which made it possible to measure the volume
with an accuracy of 10% for the smallest volume to 1.1% for the largest. The
volume of released air was regulated by the pressure of compressed air
and the duration of the open state of the solenoid valve. The pressure
took values of 3, 4, 5 and 6 bar, the duration of the open state of the
electromagnetic valve varied from 14 to 50 ms. The distance from the nozzle to
the lower edge of the funnel-shaped device was 1 m. In the experiments, the
volume varied from 7.3 to 70
.
The
shadow registration of the toroidal bubble against the background of a light
matte screen was carried out by a high-speed video camera in the direction
perpendicular to the motion trajectory with a frequency of 60 fps, an exposure
of 750 µs, and a resolution of 1504 × 1128 pixels. When using the shadow
method, the bubble border on the image is darkened, and the contrast in
comparison with the background is increased, which allows using software
processing algorithms to determine the ring parameters on each frame. The
resulting videos were processed frame by frame in the Matlab environment using
a method similar to that developed earlier and presented in [18]. The main goal
of the algorithm from [18] is to find a bubble among all objects in the image
and determine the coordinates of the points belonging to it with high accuracy.
Figure 1 shows the ring images combination as it moves, where the white lines
mark the doubled maximum transverse size of the torus
found using the algorithm from [18] and its
position along the
axis. The
-coordinate of the bubble, corresponding to its
traveled path from the nozzle, was measured as the arithmetic mean of the
coordinates of the points lying on the bubble boundary. To determine the bubble
radius
, first half of the maximum transverse size of the
torus
was measured in the image.
Then
and
are calculated in terms of
and
by solving a system of two equations:
,
. First, we define the boundaries of the range in
which physically correct solutions lie. So, from geometrical considerations,
the value of the main radius of the torus
must be greater than the cross-sectional radius
, but less than the maximum cross-sectional
dimension
, i.e.
. Further, in order to find the only correct
solution, we turn to the analysis of the function
. The maximum point is located at
and
, and the minimum point is at
and
. Then according to the Bolzano-Cauchy theorem,
there is a unique solution on the interval
. Thus, the solution of the cubic equation for
is the required one. The measurement error for
did not exceed 6%. The experimental rate of ascent
is obtained as the ratio of the measured
coordinates after a time interval of 1/12 to this interval. This interval is
much shorter than the observation time and is sufficient to achieve a
measurement accuracy of at least 7% and eliminate sharp fluctuations in the
measured velocity. Having determined the experimental rate of rise and the
dimensions of the toroidal bubble, the circulation was calculated from
expression (1.2):
|
|
(1.6)
|
Taking into account
that the circulation does not change with time, its calculation was carried out
at a reference point.
The differential equation (1.5) is solvable
for any value of the initial radius
. Therefore, the reference point can be chosen at an arbitrary place
where the bubble has a toroidal shape. In the experiments performed, the
reference point was set at a distance of
from the nozzle. At this distance, in all
launches, the bubbles had the torus shape.
Having obtained the dependence of the radius
on time and the circulation for one bubble, an estimate of its drag coefficient
can be found from the solution of equation (1.5) by comparing the solution with
the experimental dependence. Comparison of solutions with different
coefficients
, which ranged from 0 to 1 with a step of 0.001, was carried out using
the least squares method. The best result, that is,
such that the discrepancy between solution (1.5) and the experiment
was the smallest, was assumed to be the drag coefficient of the given bubble.
Note that the proposed method does not provide information on the time
dependence of the drag coefficient.
Figure 2. (a) Dependence of radius on time (
): dashed line - calculation by formula (1.4),
∇
- experimental dependence. (b) Averaged over
all experiments
, where the dashed line is the average value
at time equal to
, the confidence interval of 95% is
highlighted in color.
Figure 3. (a) Dependence of radius on time (
): solid line - calculation by formula (1.5)
with
,
∇
- experimental dependence. (b) Averaged over all experiments
with
, determined by the least squares method,
where the dashed line is the average value at time equal to
, the confidence interval of 95% is
highlighted in color.
Figures 2(a) and 3(a) show the characteristic
experimental points of the core radius against time (∇),
at 2(a)
data calculated by formula (1.4) (dashed line), at 3(a) data calculated by
formula (1.5) with
, found by the least square’s method (solid line) at
and volume
. It can be seen that the calculation by formula (1.4) lies above the
experimental points, which was observed in all experiments performed. An
estimate of the systematic deviation of the calculation according to formulas
(1.4) and (1.5) from the experimental points for all bubbles is plotted in
figure 2(b) and 3(b). The y-axis shows the ratio of the modules of the
difference between theory and experiment with respect to the experiment at the
corresponding time t. The density of points distribution at time t is presented
through a 95% confidence interval, highlighted in color, where the dashed line
means the average value of the relative deviation of theory from experiment.
The limits of the confidence interval and the average value at time t were
calculated from data from more than 100 bubbles. The comparatively large
deviation at t from 0 to 25 relative to the later time is due to the error in
determining the radii and the bubble boundaries fluctuation. With time t from
25 onwards, the average and maximum deviation in figure 2(b) is 8-9% and
12-13%, and in figure 3(b) is 0.5-1% and 2-3%, respectively. Thus, including
the drag with a constant
in the model allows to increase the accuracy of the theoretical
calculation several times. It should be noted that the drag coefficient is only
assumed to be constant, but in general, it may depend on time. However, taking
into account this dependence will not significantly change the calculations
accuracy. Since the term responsible for the drag in (1.5) decreases with
increasing radius much faster than the buoyancy force. Figure 4 shows the
calculated drag coefficients for the corresponding volumes of toroidal bubbles
(a) and Froude numbers (b). There is no clear correlation between coefficients
and volume, but the drag coefficients seem to correlate with Froude numbers.
We define the Froude number in dimensional
terms as:
|
|
(1.7)
|
in this case, in
dimensionless quantities, the Froude number coincides with the squared value of
the translational velocity. The toroidal bubble velocity changes as it moves
therefore, to compare different experiments, the Froude numbers are calculated
at the same distance from the nozzle. The Froude number in figure 4(b) was
calculated by (1.7) at the reference point.
Figure 4. Drag coefficient at corresponding
bubble volumes (a) and Froude numbers (b).
The ascent of toroidal bubbles is studied for
various values of the initial parameters: volume, radius, and circulation.
Toroidal bubbles were created by injecting a jet of air vertically into water.
The reference point was set at a distance of 15 radii of the sphere, the volume
of which is equal to the toroidal bubble volume. It is shown that in this case
the dependence of the torus radius on time without taking into account the drag
force lies above the experimental points. On a dimensionless time fragment from
0 to 25, the average value of the relative deviation grows to 8-9% and remains
constant further. The statistics were collected from measurements of the
parameters of more than 100 toroidal bubbles.
A theoretical model was tested, where it is
assumed that, in addition to the buoyancy force, a drag force acts on the
toroidal bubble. Taking into account the drag force with greater accuracy
allows the theoretical calculation to be consistent with the experimental data.
The average relative deviation at times 0 to 25 does not exceed 4% and is
0.5-1% further. Thus, we can conclude that, in addition to the buoyancy force,
a drag force acts on the toroidal bubble, and it should be taken into account
in tasks requiring high accuracy.
The drag coefficient used in the calculations
was determined by searching for the best agreement between the calculation and
experiment using the least squares method. No clear correlation between drag
coefficient and bubble volume is shown. However, it can be seen that an increase
in the Froude numbers leads to a decrease in the drag coefficients.
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