Proceeding of ’99 Korea-Japan Joint Seminar
on Particle Image Velocimetry
pp132-139
A PIV Study on the Self-induced Sloshing Phenomena in
a Rectangular Tank
Hui HU1), Toshio KOBAYASHI1) ,Tetsuo SAGA1), Shigeki SEGAWA1) and Tomomasa UEMURA2)
1. Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Tokyo 106, Japan
2. Department of Mechanical Engineering, Kansai University, Suita, Osaka, 564-8680, Japan
ABSTRACT
The flow patterns in a rectangular tank and the evolution of the votical and turbulent structures in the rectangular tank
when the self-induced sloshing phenomena occurring had been investigated experimentally by using Particle Imaging
Velocimetry (PIV) technique. It was found that three different flow patterns could be found in the test tank for the same
system parameter setting, which are reverse vortex flow pattern, first sloshing mode flow pattern and second sloshing
mode flow pattern. The evolution of the vortical and turbulent structures in the test tank was found to be quite different
from each other for the three flow patterns. Based on the PIV measurement results, the resonated oscillation was
suggested to be the mechanism of the self-induced sloshing phenomena. The fluctuation of the inlet plane jet and the
periodically shedding of the unsteady vortices were conjectured to be the excitation source of the self-induced sloshing.
Keywords: self-induced sloshing, vortex shedding, resonated oscillation, PIV technique
INTRODUCTION
Self-induced sloshing is a natural oscillation phenomenon that had been paid great attention by many researchers
in the fields of civil engineering, petroleum industry and nuclear energy engineering. For example, in the Liquid Metal
Fast Breeder Reactor (LMFBR), which is one of the major energy plants in the near future, the self-induced sloshing of
the sodium coolant may occur in reactor vessels. Such kind of self-induced sloshing will result in very high thermal
stresses on the vessel walls, which may do severe damage to the vessel structures (Okamoto et al. 1998). The fluid
dynamic mechanism of the self-induced sloshing phenomena should be understood clearly In order to have an optimum
safety design of these reactor vessels to prevent the occurring of the self-induced sloshing of the high temperature
sodium coolant in the reactor vessels.
Since the first systemical study on the self-induced sloshing in a rectangular tank reported by Okamoto et al.
(1991), several investigations on the self-induced sloshing in a rectangular tank with circulating flow had been
conducted experimentally and numerically in the past ten years. Okamoto et al. (1993) reported that the self-induced
sloshing was found to occur in a certain region of the flow rate and water level in a rectangular tank, and the frequency
of the free surface sloshing equaled to the eigenvalue of the water in the test tank. Based on the superposition of the
steady circulating flow in the test tank being represented by an ideal flow with a vortex, Madarame et al. (1992)
proposed that the oscillation energy of the sloshing be supplied by pressure fluctuations caused by the interaction
between the circulating flow and the sloshing motion. Fukaya et al. (1996) reported that two kinds of sloshing mode
were observed under the certain geometrical condition of a rectangular tank, and proposed that the self-induced sloshing
be caused by the interaction of the plane jet flow with the free surface.
Numerical simulation of the self-induced sloshing was firstly carried out by Takizawa et al. (1992a). They solved
two-dimensional Navier-Stokes equations with Physical Component Boundary Fitted Coordinate (PCBFC) (Takizawa et
al. 1992b). Based on the analyzing of their numerical result, they suggested that the oscillation energy of the selfinduced sloshing be supplied by the surface potential, which was varied by the secondary flow due to the flow
circulating. The recent work of Saeki et al.(1997, 1998) used a Boundary Fitted Coordinate (BFC) method with height
function to conduct two-dimensional numerical simulation. They reported that their numerical result agreed with an
experiment result very well and also suggested that the self-induced sloshing was mainly dependent on the inlet jet
fluctuation.
Although many important results had been got through these previous investigations, much work still needs to be
done to understand the fluid dynamic mechanism of the self-induced sloshing more clearly. In the present study, a highresolution PIV system was used to study the flow pattern in the fluid tank and to reveal the evolution of the vortical and
the turbulent structures in the flow field when the self-induced sloshing phenomena is occurring. By using the
instantaneous PIV velocity vector fields, time-average values and the phase-average results of the PIV measurement,
the characteristics of the vortical and the turbulent structures in the test tank were studied, and then the mechanism of
the self-induced sloshing was suggested based on the PIV measurement results.
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EXPERIMENT SETUP
Figure 1 shows the experimental setup used in the present study. The flow in the test loop was supplied from a head
tank, which was continuously pump-filled from a lower tank. The water level in the head tank was maintained in
constant by an overflow system in order to eliminate the effect of the pump vibration on the inlet condition of the test
tank. The flow rate of the loop, which was used to calculate the representative velocity and Reynolds numbers, was
measured by a flow meter. Honeycomb structures and a convergent section were installed in the upstream of the inlet of
the test tank to insure the uniform flow entrance. A valve was installed at the downstream of the test tank exit to adjust
the water level of the free surface in the test tank.
overflow
head tank
laser sheet
twin Nd:Yag lasers
(15Hz,20mJ/Pulse)
flowmeter
pump
honeycomb sturcture
valve
test section
Synchronizer
lower tank
cross-correlation
CCD Camera
(1008 by 1018)
PC computer
(RAM 1GB,HD 20GB)
Figure 1. The schematic of the experiment setup
Figure 2 shows the schematic view of the thin rectangular test tank. Water flowed horizontally into the test tank
and flowed out at a bottom centered vertical outlet. During the experiment, the water level in the test tank was about
H=160mm. The flow rate of the test loop was set about 20 liter/min, which corresponded to the average velocity at the
inlet of the test tank being about 0.333 m/s, and Reynolds number about 6,700 based on the height of the inlet
(b=20mm). Since the test tank was designed to insure the flow field in the test tank to be two-dimensional, PIV
measurement was mainly conducted at the middle section of the test tank (Z=25mm section) in the present study. (Our
measurement results also proved that the flow field in the test tank was almost two dimensional along the Z direction
except the regions near two walls).
free surface
inlet
H=160mm
b=20mm
Y
L=110mm
X
outlet
Z
S=150mm
T=50mm
E=60mm
W=300mm
Figure 2. The schematic of the test tank
For the PIV measurement, twin pulsed Nd:YAG Lasers were used to supply pulsed laser sheet (thickness of the
sheet is about 1.0 mm) with the frequency of 7.5 Hz and power of 20 mJ/pulse to illuminated the measured flow field.
Polystyrene particles (diameter of the particles is about 20-50 m, density is 1.02kg/l) were seeded in the flow loop as
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PIV tracers. A 1008 by 1016 pixels Cross-Correlation CCD array camera (PIVCAM 10-30) was used to capture the
images. The twin Nd:YAG Lasers and the CCD camera were controlled by a Synchronizer Control System. The PIV
images captured by the CCD camera were digitized by an image processing board, then transferred to a workstation
(host computer, CPU450MHz, RAM 1024MB, HD20GB) for image processing and displayed on a PC monitor.
Rather than tracking individual particle, the cross correlation method (Willert et al., 1991) was used in the present
study to obtain the average displacement of the ensemble particles. The PIV images (Figure. 3) were divided into 32 by
32 pixel interrogation windows, and 50% overlap grids were employed for the PIV image processing. The time interval
between the two pulses was set at 4ms to have the average displacement of the tracer particles in the inlet jet is about 4
to 6 pixels. The post-processing procedures which including sub-pixel interpolation (Hu et al., 1998) and velocity
outliner deletion (Westerweel, 1994) were used to improve the accuracy of the PIV result.
a. t=t0
a. t=t0+4ms
Figure.3 A pair of PIV images used in the present study
RESULTS AND DISCUSSIONS
1.The flow patterns in the test tank
During the experiment, it was found that three different flow patterns could be observed in the test tank for the same
system parameter setting, i.e. water level H=160mm, inlet jet velocity V=0.333m/s or flow rate of the test loop
Q=20l/min. They are Pattern A (Reverse Vortex Flow Pattern), Pattern B (First Sloshing Mode Flow Pattern) and
Pattern C (Second Sloshing Mode Flow Pattern). The schematic of these three flow patterns was shown on Figure 4.
free surface
free surface
free surface
Unsteady vortex shedding
a. flow pattern A
(reverse vortex flow pattern)
b. flow pattern B
(first sloshing mode flow pattern)
Figure 4. The schematic of the three flow patterns
Unsteady vortex shedding
c. flow pattern C
(second sloshing mode flow pattern)
It was found that, when the flow field in the test tank was in reverse vortex flow pattern (Fig.4(a)), the self-induced
sloshing phenomena dose not happen. The water free surface level in the test tank was found to keep in constant. The
inlet jet was also found to be stable, which direction is upward to water free surface and then turned down. There is just
one big clockwise steady vortex in the test tank, none unsteady vortex shedding can be found in the flow field. Figure 5
shows the PIV measurement result of the reverse vortex flow pattern.
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200
Spanwise Vorticity ( Z-direction )
Y mm
200
-30.00 -24.00 -18.00 -12.00 -6.00
0.00
6.00
0.5 m/s
VELOCITY
0.3985
0.3721
0.3458
0.3194
0.2931
0.2667
0.2404
0.2140
0.1877
0.1613
0.1350
0.1086
0.0823
0.0559
0.0296
0.0032
12.00 18.00 24.00 30.00
150
150
Y mm
Re =6,700
Uin = 0.33 m/s
100
100
50
0
0
U out
50
-50
-50
0
50
100
X mm
150
200
250
0
300
100
200
300
X mm
a. instantanous result
b. time average result
Figure 5. PIV measurement results of the reverse vortex flow pattern
However, when the flow field in the test tank was in Pattern B or Pattern C, the water free surface in the test tank was
found to oscillate periodically. The self-induced sloshing phenomena occurred. Figure 6 shows the water level changes
of the free surface by water level detecting sensors when flow in the test tank is in flow pattern B occurred. It can be
seen that the oscillating mode of the free surface is very stable when the self-induced sloshing is occurring.
Figure 5. The oscillating water level of the first sloshing mode flow pattern
200
Spanwise Vorticity ( Z-direction )
Y mm
200
-30.00 -24.00 -18.00 -12.00 -6.00
0.00
6.00
0.5 m/s
150
150
Y mm
Re =6,700
Uin = 0.33 m/s
100
100
50
0
0
U out
50
-50
-50
VELOCITY
0.3896
0.3639
0.3381
0.3124
0.2866
0.2609
0.2352
0.2094
0.1837
0.1579
0.1322
0.1065
0.0807
0.0550
0.0292
0.0035
12.00 18.00 24.00 30.00
0
50
100
150
X mm
200
250
0
300
100
200
X mm
a. instantanous result
b. time average result
Figure 6. PIV measurement results of the first sloshing mode flow pattern
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300
When the self-induced sloshing phenomena occurring, the inlet jet was also found to fluctuate up and down
periodically. The differences between the pattern B and Pattern C are the oscillation frequency and oscillating wave
mode of the water free surface in the test tank. For the pattern B, i.e. first sloshing mode flow pattern, the free surface in
the test tank at right side and left side oscillate up and down alternately. The oscillation phase angle of the free surface
at the left side and right is out of phase (Fig. 4(b)). For the pattern C, i.e., second sloshing mode flow pattern, the water
free surface in the test tank at right side and left side move up and down simultaneously. I.e. the oscillation phase angles
of the free surface at the left side and right of the test tank are in phase. Besides these, the oscillation frequencies of the
self-induced sloshing for pattern B and pattern C were also found to be different, which will be discussed in the
following section. The PIV measurement results of the flow pattern B and flow pattern C were shown on the Figure 6
and Figure 7.
200
250
Spanwise Vorticity (Z-direction)
-20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00
Y mm
150
Y (mm)
VELOCITY
0.3799
0.3548
0.3297
0.3046
0.2795
0.2544
0.2293
0.2042
0.1791
0.1540
0.1289
0.1038
0.0787
0.0536
0.0285
0.0034
150
200
Uin =0.333m/s
100
100
50
50
0
0
0
-50
-50
0
50
100
150
200
250
100
200
300
X mm
300
X (mm)
a. instantanous result
b. time average result
Figure 7. PIV measurement results of the second sloshing mode flow pattern
2.The velocity power spectrum profiles for the three flow patterns
By using the continuos instantaneous PIV measurement results, the velocity power spectrum of the flow field can be
calculated by using FFT transformation. The velocity power spectrum profiles at the same point in the flow field
(X=100mm, Y=100mm, Z=25mm) for the three flow patterns were shown on Figure 8. For the flow pattern A (Fig.8(a)),
the velocity power spectrum profile was found to be in random. There is not any characteristic frequency that can be
found in the velocity power spectrum. However, when the flow field was in pattern B or pattern C (Fig. 8(b) and Fig.
8(c)), an obvious characteristic frequency can be identified in each velocity power spectrum profile, which are 1.6Hz
and 2.3Hz for the flow pattern B and pattern C respectively. In the meanwhile, the oscillation frequency of the water
free surface can also be got from the measurement of the free surface water level changing shown on figure 5. It was
found that the characteristic frequencies in the velocity power spectrum profiles are just the frequencies of the selfinduced sloshing of the free water surface in the test tank.
a. pattern A
b. pattern B
Figure 8. Typical velocity power spectrum profiles for the three flow patterns
c. pattern C
The theoretical analysis of the Lamb (1932) had suggested that the eigenvalues of the water in a rectangular could be
expressed as:
1 nπg
nπH
(1)
fn =
tanh(
)
2π W
W
where n is the mode number, W and H are the width and heigth of the water in the tank.
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For the present study case, i.e. H=160mm and W=300mm, the eigenvalues of the water in the test tank can be
calculated, which are listed in the Table 1.
Table 1. the eigenvalues of the water in the test tank
n-th mode
Eigenvalues (Hz)
1
1.6
2
2.3
3
2.8
4
3.2
5
3.6
From the table 1, it can be seen that, the characteristic frequencies in the velocity power spectrum profiles of the selfinduced sloshing flow field for pattern B (f0=1.6Hz) and Pattern C (f0=2.3Hz) are just the 1st mode eigenvalue and 2nd
mode eigenvalue of the water in the test tank. This is reason why the flow pattern B is called first sloshing mode pattern
and the pattern C is called second sloshing mode flow pattern in the present paper.
3.The evolution of the vortical structures in the test tank
It can be found that the characteristics of the vortical and turbulent structures in the test tank are quite different from
each other for the three flow patterns. For the flow reverse vortex flow pattern A (Fig. 4(a)), just one big clockwise
vortex can be found in the flow field as mentioned above. However, when the flow was in pattern B or Pattern C, two
counter rotating vortices can be found from the PIV time average results, which were at the right side and the left lower
corner of the test tank. During the experiment, it was also found that when the flow was sloshing (pattern B or pattern
C), the inlet jet fluctuated up and down and some smalle scale unsteady vortices can also be found to shed from the inlet
jet periodically.
Spanwise Vorticity ( Z-direction )
Spanwise Vorticity ( Z-direction )
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
Y mm
200
Y mm
200
10.00 15.00 20.00 25.00
water free surface
150
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
water free surface
150
Re =6,700
Re =6,700
Uin = 0.33 m/s
Uin = 0.33 m/s
50
50
0
0
-50
-50
0
50
100
U out
100
U out
100
X mm
150
200
250
300
-50
-50
0
50
Y mm
Y mm
5.00
250
300
Spanwise Vorticity ( Z-direction )
200
0.00
200
(b). situation 2 ( = /2)
Spanwise Vorticity ( Z-direction )
-25.00 -20.00 -15.00 -10.00 -5.00
X mm
150
(a). situation 1 ( =0)
200
100
10.00 15.00 20.00 25.00
water free surface
150
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
water free surface
150
Re =6,700
Re =6,700
Uin = 0.33 m/s
Uin = 0.33 m/s
50
50
0
0
-50
-50
0
50
100
150
U out
100
U out
100
X mm
200
250
300
-50
-50
0
50
100
150
(c). situation 3 ( = )
(d). situation 4 ( =3 /2)
Figure 9. The phase average flow field of PIV result for the flow pattern B
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X mm
200
250
300
In order to investigate the relationship of the periodically shedding of the unsteady vortices and the inlet jet
fluctuation with the self-induced sloshing phenomena, the PIV phase average measurement had also been conducted in
the present study. During the PIV phase average measurements, the water level of the free surface in the test tank was
detected by a water level sensor. The water level sensor can generate a signal to trig the synchronizer of the PIV system
when the free surface in the test tank arrive at the pre-set water level.
The phase average measurement results for the flow pattern B were shown on Figure 9. The phase angles of these
results are θ = 0 , /2, and 3 /2. The water levels of the free surface at the left side (inlet side) of the test tank
were at its highest position, middle level (the free surface level is decreasing), lowest position and middle level (the free
surface level is increasing) corresponding to these four phase angles respectively.
Unlike the time average results, which can just reveal two steady vortices in the sloshing flow field, the unsteady
vortex can also be revealed clearly in the flow field from the phase average measurement result. The unsteady vortex
was found to change its position with the changing of the phase angle. When the phase angle increasing from 0 to , i.e.,
the free surface water level at the left side of the test tank decreased from its highest position to its lowest position (from
Fig.9(a), Fig.9(b) to Fig.9(c)); the unsteady vortex shed from the inlet jet and moved downstream. When the phase
angle increasing from to 2 , i.e, the free surface water level at the left side of the test tank began to increased from its
lowerst position to its highest position (from Fig. 9(c), Fig. 9(d) to Fig. 9(a)), the unsteady vortex was engulfed by the
large steady vortex at the right side of the test tank, and another new vortex was found to rollen up from the inlet of the
test tank. Then, another new self-induced sloshing cycle began. As the phase angle changing, the inlet jet flow was also
found to fluctuated up and down periodically.
Since the flow field in the test tank was found to be oscillating (sloshing) periodically when the flow field in the test
tank was in pattern B or pattern C. The oscillating movement component was extracted by subtracting the PIV time
average result from the PIV phase average results in order to reveal the oscillating movement more clearly. Typical
results of the oscillating movement for the pattern B and pattern C were shown on Fig.10. The evolution of the vortical
and turbulent structures and the form of the oscillating wave can be seen clearly in these figures.
spanwise vorticity (Z-direction)
spanwise vorticity ( Z-direction )
200
-12.0 -9.6
-7.2
-4.8
-2.4
0.0
2.4
4.8
7.2
9.6
200
12.0
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
0.1 m/s
150
0.1 m/s
Y (mm)
Y mm
150
100
100
50
50
0
0
-50
0
100
200
300
X mm
0
100
200
X (mm)
Figure 10. The oscillation movement of the first sloshing mode pattern and the second sloshing mode pattern
4.The mechanism of the self-induced sloshing
As mentioned above, for the same test tank and system parameter setting, i. e. water level H=160mm, inlet jet
velocity U=0.333m/s or flow rate Q=20L/min, three completely different flow patterns can be observed in the test tank.
All of these three flow patterns are found to be stable. The transformation of the flow pattern was found to be
determined by the inlet jet condition. Although the representative velocity of the inlet jet flow or the flow rate of the test
loop is same, the velocity distribution, turbulence intensity level and momentum boundary layer thickness of the inlet
jet flow may affect the flow pattern, oscillating mode and frequency of the self-induced sloshing.
From the above discussions, it can be seen that when the flow field in the test tank is sloshing, (pattern B and pattern
C), the frequencies of the self-induced sloshing always equal to the eigenvalues of the water in the test tank. The same
results were also found from the Sierras studies of Okamaoto et al. (1991,1998). These mean that the self-induced
sloshing has a very close relationship with the oscillation characteristics of the water in the test tank. So the resonated
oscillation may be suggested to be the mechanism for the self-induced sloshing.
It was well known that there are two necessary factors for the resonated oscillation, which are excitation source and
frequency coupling of the excitation source with the eigenvalue of the oscillating body. From the above PIV
measurement result, it can be seen that during the self-induced sloshing, the fluctuation of the inlet jet and the
periodically shedding of the unsteady vortex played a key role on the distribution of the vortical and turbulence
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30
structures in the test tank. The frequency of jet fluctuation and the perodically shedding of the unsteady vortex just
equal to the frequency of the self-induced sloshing (which means it can be coupled with the eigenvalue of the water in
the test tank). So the inlet jet fluctuation and the periodical shedding of the unsteady vortex may be suggested to be the
excitation source of the resonated oscillation (self-induced sloshing).
In the meanwhile, it should be mentioned that the inlet jet fluctuation of the flow pattern B or Pattern C has very
similar characteristics to the well known phenomenon of “edged tone” (Rockwell et al. 1979). “Edged tone”
phenomenon is also a self-sustained jet oscillation phenomenon and caused by the interaction between the jet instability
and the edge. So the relationship between the self-induced sloshing and “edged tone” phenomenon will be studied in the
future work.
CONCLUSION
The self-induced sloshing in a rectangular tank was investigated experimentally by using PIV technique. The
instantaneous flow field, time average value and phase average result of the PIV measurement were used to investigate
the evolution of vortical structures in the self-induced sloshing flow field. Based on the experimental results, resonated
oscillation was suggested to be the mechanism of the self-induced sloshing. Periodic fluctuation of the inlet jet and the
periodically shedding of the unsteady vortices from the inlet jet were conjectured to be the excitation source of the selfinduced sloshing.
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