Analysis on the Self-Induced Sloshing
using Particle Image Velocimetry
Koji OKAMOTO, Souichi SAEKI and Haruki MADARAME
Nuclear Engineering Research Laboratory
The University of Tokyo
Tokai-mura, Ibaraki, JAPAN
Email: okamoto@tokai.t.u-tokyo.ac.jp
Hui HU, Tetsuo SAGA and Toshio KOBAYASHI
Institute of Industrial Science
The University of Tokyo
Roppongi, Tokyo, JAPAN
Abstract
A self-induced sloshing is excited by flow without any other external force.
In a rectangular tank having a horizontal plane jet, the 1st mode sloshing grow
in a certain condition of the inlet velocity and the water level. In this study,
high resolution velocity vector distributions were measured with high accuracy
by means of Particle Image Velocimetry. Using the double pulsed Nd:YAG
Laser and high resolution CCD camera, phase averaged velocity vectors were
measured. The sloshing energy supply was analyzed using the measured velocity
vectors based on the Saeki’s theory. The calculated energy distributions agreed
well with the theory. The overall physical oscillation mechanism of the selfinduced sloshing was clarified using the experimental data.
INTRODUCTION
In the Fast Breeder Reactor, advanced nuclear power plant, the flow with the free surface
exists in the reactor vessel. The interaction between the high speed vortical flow and the free
surface may cause undesirable surface phenomena to occur, e.g. “Self-Induced Sloshing”.
These oscillating phenomena must be prevented from breaking out under any operating
condition. Therefore, the mechanisms and governing parameters of these phenomena should
be evaluated.
A self-induced sloshing is the natural oscillation of fluid in a tank, which is excited by flow
without any other external force. The self-induced sloshing was discovered in a rectangular
tank having a horizontal plane jet, as shown by Fig.1. [3] It was reported that the 1st
or 2nd mode self-induced sloshing was excited, according to the inlet velocity, water level
1
W
D
Surface
Inlet
U0
H
b
B
Outlet
S
s
Figure 1: Test tank geometry
Table 1.
Case
Previous
Present
GEOMETRICAL PARAMETERS (unit: mm)
W
D
B
S
1000 100 250 500
300 50 110 150
b
100
20
s Ref.
100 [3]
60 [1]
and tank geometry [4] A few numerical simulations also were performed (e.g., [9],[5]) The
numerical results agreed qualitatively with the experimental ones.
To understand the self-induced sloshing, the mechanism of the excitation process is very
important. Several growth mechanisms of this phenomenon were proposed (e.g., [3],[2]) in
which the oscillation energy was supplied by the interaction between the circulating flow
and the free surface wave. However, these mechanism could not satisfactly explain the
experimental nor numerical results.
Saeki et al. [6] proposed an oscillation growth mechanism based on the numerical simulation results and experiments. In the mechanism, the sloshing excitation energy was
supplied by the non-linear interaction between the sloshing motion and the jet fluctuation.
The Saeki’s model succesfully explained the occurence of self-induced sloshing in the numerical simulation. However, his model was only confirmed by the numerical self-induced
sloshing. In the numerical simulation, the high resolution velocity distributions could be
obtained high accurately. While in the experiment, the velocity vector information were
relatively sparse with low accuracy. Therefore, it was difficult to confirm the model using
the experimental data.
Hu et al. [1] measured the self-induced sloshing in the smaller vessel using the high
resolution Particle Image Velocimetry technique. While the vessel size of the Saeki’s experiment is 1m in width, it is 0.3m in Hu’s experiment. They measured the high resolution
velocity information during the self-induced slosing using the Twin Nd:YAG Laser with
1k×1k cross-correlation CCD camera.
The object of the present study was to confirm the oscillation growth mechanism using
2
Figure 2: Phase averaged velocity distribution measured by PIV
the high resolution experimental velocity data. Then, the mechanism will be supported
not only numerically, but also experimentally.
SELF-INDUCED SLOSHING
Experimental Set Up and Procedure
Figure 1 schematically shows the rectangular test tank having an inlet at the left side wall
and an outlet at the bottom. The test tank was made of a transparent acrylic board to
visualize the flow. The parameters of the tank were the tank width W , the tank depth from
the front D, the inlet height from the bottom of the tank B, the outlet location from the
left side wall S, the inlet width b, the outlet width s. The size of the present experimental
tank was about 1/3 of the tank of previous studies. Table 1 shows the differences of the
representative test tank geometries [3],[1]. The self-induced oscillation was observed under
the certain inlet velocity U0 and the water level H.
In a certain condition of U0 and H according to a tank geometry, the free surface was
oscillated periodically with the 1st or 2nd mode sloshing. The both ends of the surface
oscillated in the reverse phase with a node or in the same phase with two nodes, causing
the sloshing to be the 1st or 2nd mode sloshing, respectively. The measured frequencies
were almost equal to the theoretical eigenvalues of respective sloshing modes.
U0 was given by the inlet flow rate H could be controlled by an overflow tank gate.
In order to evaluate the flow pattern in a test tank, the unsteady velocity profiles under the sloshing condition was investigated using the Particle Image Velocimetry (PIV)
3
[1]. The oscillation in the tank was visualized by seeding plastic particle tracers, of which
the size and the specific gravity was 30µm in diameter and about 1.0, respectively. The
twin Nd:YAG pulse laser was used as the illumination. With synchronizing the laser pulse
and camera, two serial images with short time interval were captured. The 1000×1000
high resolution camera was applied for the measurement to obtain the whole field images
with high resolution. Using the two serial images, the two-dimensional velocity distributions were calculated using the cross-correlation technique. They were obtained in 15 Hz
periodically.
This sloshing phenomenon was the oscillation of the free surface with its eigenvalue
frequency. Since the test tank was smaller than the previous studies, the sloshing frequency
is about 1.6 and 2.3 Hz, for 1st and 2nd eigenfrequency, respectively.
In this study, the velocity distributions should be obtained with every sloshing phase.
Although the phenomenon itself could be theoretically defined, the actual oscillation frequency was affected by the turbulence or non-linear interaction of free surface. Therefore,
the sloshing frequency was fluctuated around the theoretical one. In the frequency analysis,
sharp peak could be detected. However, for the measurement, the small fluctuation may
be accumulated to be the larger error. To obtain the phase averaged velocity distributions,
the phase information should be correctly detected. In this study, the water level detector
was set at the left side of the test tank. The signal of the water level detector was taken
as the trigger for the laser pulse. Then, a certain delay was applied to capture the phase
averaged information. Varying the delay, the phase averaged velocity distributions could
be measured. In this study, the averaged velocity was obtained for every 30 degree (π/6).
Then 12 phase averaged vectors were obtained by the average of the 250 instantaneous
velocity distributions in each phase.
Figure 2 shows the phase averaged results of the 1st mode self-induced sloshing. With
the free surface oscillation, the jet fluctuated. The interaction between the jet oscillation
and the sloshing motion may cause the oscillation to excite. The vortexes along the jet
motion were clearly seen.
Sloshing Energy
Saeki et al. [6] proposed the sloshing energy, which was based on the feedback mechanism
shown in Fig. 3 The sloshing eneregy was calculated as interaction between the sloshing
potential and fluid momentum.
Sloshing Potential In the self-induced sloshing, the frequency almost agreed with that of
the natural sloshing. The natural sloshing was assumed to be expressed by the potential
flow. The 1st mode sloshing potential was expressed as,
φs (x, y, t) = a
ωs W cosh π(y + H)/W
sin(πx/W ) sin(ωs t + δ),
π
sinh πH/W
(1)
where ωs , W and H denote the sloshing frequency, the tank width and the water level,
respectively. The amplitude (a) was determined by the averaged value of the measured
surface fluctuation. In this study, the phase delay (δ) was set to zero, since the phase
averaged data were measured based on the surface phase information.
4
Free Surface Oscillation
Sloshing Motion
(Potential Solution)
En > 0
En < 0
Positive Feedback
Negative Feedback
Oscillation Energy E n
Fluid Force Variation
Unsteady Flow
Figure 3: Feedback mechanism
Fluid Forces In Saeki’s feedback mechanism [6], the fluid force caused by jet fluctuation
was calculated by means of the momentum theory, which showed the volume-integrated
form of Navier-Stokes equation (F unst ). He evaluated the contribution strength of the
right hand side terms on the sloshing energy, showing that the convection term mainly
contributed to the sloshing energy. Also, the pressure data could not be measured in the
experiment. In this study, the convection term was calculated as the sloshing energy, i.e.,
F unst ≈ F con .
d
dt
F unst
ρ udV = (−
V
ρ u(u · n) dS)
ρ ν∇2 u dV ,
(grad p + ρ gy) dV ) +
+ (−
S
F con
V
F press
V
(2)
F diss
where S and V denote a control surface and a volume in a control surface, respectively.
And n and g denote normal vector on the control surface and gravity acceleration vector,
respectively.
In the calculation of the fluid forces, the measurement grid point was taken as the fixed
control volume.
Oscillation Energy The oscillation energy in a control volume during a unit time (∆E(x, y, t))
was calculated as an inner product between the fluid force (Fcon (x, y, t)) and the sloshing
velocity (grad φs (x, y, t)) in a control volume.
5
Figure 4: Sloshing energy distributions
∆E(x, y, t) = Fcon (x, y, t) · grad φs (x, y, t),
(3)
The local oscillation energy (∆E(x, y)) during a natural sloshing period (Ts ) was expressed as
∆E(x, y) =
Ts
∆E(x, y, t) · dt.
(4)
The local oscillation energy (∆E(x, y)) was space-integrated for the whole field, resulting
in the oscillation energy (E) supplied for the sloshing motion, as follows.
1
∆E(x, y) · dS,
(5)
W Ha2 Stank
where Stank denotes the area in the tank. The oscillation energy was converted into the
energy per unit mass (W H = Stank ) and the square of the amplitude (a2 ).
Figure 4 shows the calculated sloshing energy distributions (∆E(x, y)) using the measured velocity distributions (Fig. 2) The positive energy (+) and negative energy (−) area
distributed along the jet. The integrated energy (E) was calculated as 0.9×10−3 J/m2 ,
which was positive value. When the sloshing energy was positive, the sloshing may excited
because of the positive feed back effect, according to Saeki’s mechanism. In another words, the non-linear interaction between jet motion to sloshing supply enough energy to the
surface to oscillate.
In the sloshing energy distributions, there are exactly two pair of the positive/negative
(+/−) energy pattern along the jet from inlet to outlet. This distribution also supports
the Saeki’s mechanism, in which the number of the (+/−) energy pattern pair along the
jet should be integer value to excite the sloshing. Also, the two pattern means that the
oscillation was in the “2nd stage”.
With these calculated results with measured PIV data, the Saeki’s mechanism was experimentally confirmed.
E=
6
4
0.33m/s
3.5
3rd stage
3
2.5
St s
2nd stage
2
2nd
1.5
mod
e
1st stage
1
1st mo
de
0.5
0
0
50000
100000
150000
200000
Re
Figure 5: Oscillating condition map for present tank geometry
Modified Strouhal Number
It is well known that the jet is intrinsically unstable. The jet features were explained
generally using Strouhal number St = f l/U0 . Saeki et al. [7] proposed the governing
parameter of the self-induced sloshing, i.e. the modified Strouhal number Sts , in order
to evaluate a specific phase state of jet fluctuation. They proposed the modified Strouhal
number as,
fs L
Sts =
κ
U0
L + 2b
b
(κ ≈ 2/3) .
(6)
√
where L is the distance between inlet center to outlet center, i.e., L = B 2 + S 2 , (representative length). b is the inlet nozzle width. fs and U0 are sloshing frequency and the
averaged inlet jet velocity. κ is the constant derived from the turbulent jet theory.
Saeki et al. [7] reported that the self-induced sloshing was excited, when Sts satisfied the
following equation,
m − 0.25 < Sts < m + 0.25 (m = 1 or 2 · · ·) .
(7)
The above equation indicated that the self-induced sloshing occurred when Sts was about
1.0, 2.0, · · ·. This means that the wave number of the winding jet had to be about 1.0
“1st Stage Sloshing” or 2.0 “2nd Stage Sloshing” from inlet to outlet. The excitation of
self-induced sloshing was arranged well by Sts , in spite of different tank geometries and
multi-mode sloshing. They confirmed the relationship in several larger tank geometries.
In this experiment, the modified Strouhal Number was calculated according to the test
tank geometry. The geometrical parameters are L = 186mm, b = 20mm, fs = 1.6Hz,
U = 333mm/s. With substituting above parameters into Equation (6), Sts = 1.98. This
result means that the sloshing was excited in the 2nd stage. The experimental result
7
agrees well with the theory. Especially, Fig. 5 shows that two energy sources exist along
the winding jet. The modified Strouhal number agrees well with the number of energy
sources.
Saeki et al. [7] also proposed the oscillating condition map based on the Modified Strouhal
number and Reynolds number. When L and U0 are taken as the representative length and
velocity, the relationship between Sts and Re is inverse propotional, i.e., Sts = constRe−1 .
Figure 5 shows the oscillating condition map for the present tank geometry. The curved
line shows the above inverse propotional relations for sloshing eigen frequencies, 1st and
2nd mode. The dotted stripes show the oscillation criteria described in Eq. (7). Therefore,
the self-induced oscillation could occur under the crossing condition of the curved line and
dotted stipes. Present experimental condition (1st mode / 2nd stage) was well displaied
in the map. According to the map, with increasing the inlet velocity, the 2nd mode/ 2nd
stage sloshing may occur.
The oscillating condition map is also useful to discuss the nessesary condition of the
self-induced sloshing.
Second mode sloshing
Using the same experimental test tank, the second mode sloshing was observed under the
almost simillar condition in the different institute. The major difference from the above
experiment was the inlet jet condition. The inlet nozzle was directly connected by the
pump, resulting in the inlet condition to be ununiform. Saga et al. [8] reported that the
inlet condition highly affects on the sloshing condition. The sloshing occurence condition
depended on the uniformity of the inlet jet. Further examination will be carried out for
the 2nd mode sloshing.
CONCLUSION
The growth mechanism of the self-induced sloshing excited by a horizontal plane jet was
experimentally investigated. With the phase averaged PIV technique, the high accurate
velocity distributions in various phases were measured experimentally. They have enough
spatial and temporal resolutions for the evaluation of the sloshing energy. Applying the
Saeki’s model, the sloshing energy distributions were calculated, showing the good agreement with the theory and experiment. The experimental condition was also evaluated by
the modified Strouhal Number Sts , indicating the sloshing condition.
With the high resolution and high accurate PIV measurement, the sloshing growth mechanism was confirmed, which was represented by a simple feedback loop focused on the jetsurface interaction. The overall oscillation system was clarified by the feedback mechanism.
References
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The 2nd Pacific Symposium on Flow Visualization and Image Processing, PF-152
8
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