9TH. INTERNATIONAL SYMPOSIUM ON FLOW VISUALIZATION, 2000
ABSTRACT
The flow patterns in a rectangular tank and the evolution of the vortex 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 jet and the periodically shedding of the unsteady vortices from the inlet jet were conjectured to be the excitation source of the self-induced sloshing.
1. 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[1]).
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)[2], 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) [3] 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
Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi,
Minato-ku, Tokyo 106-8558, Japan
Corresponding author: Tetsuo SAGA or Hui HU
Email: saga@iis.u-tokyo.ac.jp or huhui@iis.u-tokyo.ac.jp
Paper number 259 259-1

Tetsuo SAGA, Hui HU, Toshio KOBAYASHI, Shigeki SEGAWA and Nobuyuki TANIGUCHI et al. (1992) [4] 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) [5] 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.
(1992) [6]. They solved two-dimensional Navier-Stokes equations with Physical Component
Boundary Fitted Coordinate (PCBFC) [7]. Based on the analyzing of their numerical result, they suggested that the oscillation energy of the self-induced 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) [8][9]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 high-resolution 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.
2 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 pump flowmeter twin Nd:Yag lasers
(15Hz,20mJ/Pulse) honeycomb sturcture valve test section lower tank
Synchronizer cross-correlation
CCD Camera
(1008 by 1018)
Figure 1. The schematic of the experiment set-up
PC computer
(RAM 1GB,HD 20GB)
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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Research on the Self-induced Sloshing Phenomena in a Rectangular Tank free surface inlet b=20mm H=160mm
L=110mm
Y
Z
X outlet
S=150mm
T=50mm
E=60mm
W=300mm
Figure 2. The schematic of the test tank
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).
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-
50µm, density is 1.02kg/l) were seeded in the flow loop as 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.
a. t=t0 a. t=t0+4ms
Figure.3. An example of PIV image pair used in the present study
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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Tetsuo SAGA, Hui HU, Toshio KOBAYASHI, Shigeki SEGAWA and Nobuyuki TANIGUCHI
Rather than tracking individual particle, the cross correlation method [10] 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 postprocessing procedures which including sub-pixel interpolation [11] and velocity outliner deletion
[12] were used to improve the accuracy of the PIV result.
3. RESULTS AND DISCUSSIONS
3.1. 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. 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 f ree surface free surface
Unsteady vortex Unsteady vortex
a. flow pattern A
(reverse vortex flow pattern) b. flow pattern B c. flow pattern C
(First sloshing mode flow pattern) (Second sloshing mode flow pattern)
Figure 4. The schematic of the three flow patterns
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 firstly 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.
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, i. e., 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. It can be seen that the oscillating mode of the free surface is very stable when the self-induced sloshing is occurring.
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. It means. 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 Fig.7 and Fig. 8.
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-4

Research on the Self-induced Sloshing Phenomena in a Rectangular Tank
200
50
0
-50
-50
200
150
100
Re =6,700
Uin = 0.33 m/s
Spanwise Vorticity ( Z-direction )
-30.00 -24.00 -18.00 -12.00 -6.00
0.00
6.00
12.00 18.00 24.00 30.00
0
150
100
50
0
0.5 m/s
X mm
250 300
0 100
50 100 150 200 X mm a. instantanous result b. time average result
Figure 5. PIV measurement results of the reverse vortex flow pattern
200 left side m iddle right side
166
165
164
163
162
161
160
159
158
157
156
0 0.5
1 1.5
tim e (s)
2 2.5
3
Figure 6. The oscillating water level of the first sloshing mode flow pattern
300
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
200
50
0
-50
-50
200
150
100
Re =6,700
Uin = 0.33 m/s
Spanwise Vorticity ( Z-direction )
-30.00 -24.00 -18.00 -12.00 -6.00
0.00
6.00
12.00
18.00
24.00
30.00
150
100
50
0
0.5 m/s
0
X mm
250 300
0 100 200
50 100 150 200 X mm a. instantanous result b. time average result
Figure 7. PIV measurement results of the first sloshing mode flow pattern
300
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
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-5

Tetsuo SAGA, Hui HU, Toshio KOBAYASHI, Shigeki SEGAWA and Nobuyuki TANIGUCHI
200
250
200
150
100
Uin =0.333m/s
Spanwise Vorticity (Z-direction)
-20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00
20.00
50
0
150
100
50
0
-50
-50 0
0 100
X mm b. time average result
200
50 100 150
X (mm)
200 250 300 a. instantanous result
Figure 8. PIV measurement results of the second sloshing mode flow pattern
300
3.2. The velocity power spectrum profiles for the three flow patterns
By using the continuous 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 9. For the flow pattern A (Fig.9(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. 9(b) and
Fig.9(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 6. It was found that the characteristic frequencies in the velocity power spectrum profiles are just the frequencies of the self-induced sloshing of the free water surface in the test tank.
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0.000
0.00
1.00
0.025
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
0.00
1.6
0.020
0.015
0.010
0.005
0.000
2.3
2.00
Frequency (H z)
3.00
4.00
1.00
2.00
Frequency (Hz)
3.00
4.00
0 2 4 6
Frequency (Hz) a. pattern A b. pattern B c. pattern C
Figure 9. Typical velocity power spectrum profiles for the three flow patterns
The theoretical analysis of the Lamb (1932) [13] had suggested that the eigenvalues of the water in a rectangular could be expressed as: f n =
1
2 π n
π
W g tanh( n
π
W
H
)
(1)
where n is the mode number, W and H are the width and heigth of the water in the tank.
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. From the Table 1, it can be seen that, the characteristic frequencies in the velocity power spectrum profiles of the sef-induced sloshing flow field for pattern B (f
0
=1.6Hz) and Pattern C (f
0
=2.3Hz) are just the 1 st mode eigenvalue and 2 nd
8
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
10
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-6

Research on the Self-induced Sloshing Phenomena in a Rectangular Tank 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.
Table 1. the eigenvalues of the water in the test tank n-th mode Eigenvalues (Hz)
1
2
3
4
5
1.6
2.3
2.8
3.2
3.6
1.
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 larger scale 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.
200
S panwis e Vo rtic ity ( Z-dire c tio n )
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
wa te r fre e s urfac e
200
S panwis e Vo rtic ity ( Z-dire c tio n )
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
wa te r fre e s urfac e
150
100
Re =6 ,7 0 0
Uin = 0.3 3 m/s
150
100
Re =6 ,7 0 0
Uin = 0.3 3 m/s
50
0
-50
-50 0
50
50
0
100 150 200
X mm
250 300
-50
-50
(a). situation 1 ( θ =0)
0 50 100 150
(b). situation 2 ( θ = π /2)
200
X mm
250 300
50
0
200
150
100
Re =6 ,7 0 0
Uin = 0.3 3 m/s
S panwis e Vo rtic ity ( Z-dire c tio n )
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
wa te r fre e s urfac e
-50
-50
200
150
100
Re =6 ,7 0 0
Uin = 0.3 3 m/s
50
S panwis e Vo rtic ity ( Z-dire c tio n )
-25.00 -20.00 -15.00 -10.00 -5.00
0.00
5.00
10.00 15.00 20.00 25.00
wa te r fre e s urfac e
0
0
X mm
50 100 150 200 250 300
-50
-50 0 50 100 150 200
(c). situation 3 ( θ = π ) (d). situation 4 ( θ =3 π /2)
Figure 10. The phase average flow field of PIV result for the flow pattern B
X mm
250 300
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-7

Tetsuo SAGA, Hui HU, Toshio KOBAYASHI, Shigeki SEGAWA and Nobuyuki TANIGUCHI
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 10. The phase angles of these results are θ = 0 , π /2, π and 3 π /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 π , 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.10(a),
Fig.10(b) to Fig.10(c)); the unsteady vortex shed from the inlet jet and moved downstream. When the phase angle increasing from π to 2 π , 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. 10(c), Fig. 10(d) to Fig. 10(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 roll 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.11. The evolution of the vortical and turbulent structures and the form of the oscillating wave can be seen clearly in these figures.
200
150 0.1 m/s spanwise vorticity (Z-direction)
-12.0 -9.6
-7.2
-4.8
-2.4
0.0
2.4
4.8
7.2
9.6
12.0
200
150
0.1 m/s spanwise vorticity ( Z-direction )
-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
100
100
50
50
0
0
-50
0 100
X mm
200 300 0 100
X (mm)
200
Figure 11. The oscillation movement of the first sloshing mode pattern and the second sloshing mode pattern
30
3. 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.
9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-8

Research on the Self-induced Sloshing Phenomena in a Rectangular Tank
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 continous studies of Okamaoto et al. [1-5][14]. 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 selfinduced 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 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” [15].
“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.
4. CONCLUSION
The self-induced sloshing in a rectangular tank was investigated experimentally by using PIV technique. The instantaneous flow field, time averaged value and phase averaged 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 self-induced sloshing.
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9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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Tetsuo SAGA, Hui HU, Toshio KOBAYASHI, Shigeki SEGAWA and Nobuyuki TANIGUCHI
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9 th
International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
259-10