A PIV Study on the Self-induced Sloshing in a Tank with Circulating Flow

Proceeding of PSFVIP-2 May 16-19, 1999, Honolulu, USA PF152 A PIV Study on the Self-induced Sloshing in a Tank with Circulating Flow Hu Hu Toshio Kobayash Tetsuo Saga, Shigeki Segawa and obuyuki Taniguchi Institute of Industrial Science, University of Tokyo, 7-22-1 Roppong Tokyo 16, Japan E-mail: huhui@cc.iis.u-tokyo.ac.jp Masaho agoshi Fluid Measuring Instrument Division, KAOMAX JAPA,IC. 3-18-2 ishishinjyuku, Tokyo16, Japan Koji Okamoto uclear Engineering Research Laboratory, University of Tokyo, Tokai-mura, Ibarak 319-11, Japan The self-induced sloshing in a rectangular tank with circulating flow had been investigated experimentally by using Particle Imaging Velocimetry (PIV) technique. The instantaneous flow fields, time average values and phase average results of the PIV measurement were used to analyze the mechanism of the self-induced sloshing of the water free surface in the test tank. Besides the two steady vortices in the left lower corner and right side of the test tank reported by Okamoto et al. (1993a), a new unsteady vortex was found in the test tank shedding periodically from the inlet plane jet. The shedding frequency of this unsteady vortex was found to be the frequency of the self-induced sloshing, which also equals the eigenvalue of the water in the test tank. The flow parttern in the test tank changes drastically with the periodic shedding of this unsteady vortex. Based on the experimental result, resonated oscillation was suggested to be the mechanism of the self-induced sloshing, and the instability of the inlet plane jet was conjectured to be the excitation source of the self-induced sloshing. Keyword: self-induced sloshing, vortex shedding, resonated oscillation, and PIV technique ITRODUCTIO 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 the reactor vessel. Such selfinduced sloshing can result in high thermal stresses on the vessel walls, which may do severe damage to the vessel structures (Okamoto et al. 1998). So, the mechanism of the self-induced sloshing should be understood clearly in order to do an optimum safety design to prevent the self-induced sloshing of the sodium coolant in the reactor vessel. Since the first systemical research 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. (1991) 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 ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

by the interaction of the plane jet flow with the free surface. umerical simulation of the self-induced sloshing was firstly carried out by Takizawa et al. (1992a). They solved two-dimensional avier-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 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) 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. Such as, the evolution of the vortices and the turbulent structures in the test tank were not fully researched yet. Meanwhile, most of the previous experimental researches were mainly based on the qualitative flow visualization. Although a primary PIV measurement had been conducted by Okamoto et al. (1993b), the resolution of their measurement was not fine enough to reveal small-scale vortices in the test tank due to the limitation of their PIV hardware. In the present study, a high-resolution PIV system was used to study the self-induced sloshing in a rectangular tank instantaneously. By using the instantaneous PIV results, time-average values and the phase-average result of the PIV measurement, the evolution of the vortices, 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 result. overflow head tank laser sheet pump flowmeter twin d:yag lasers (15Hz,2mJ/Pulse) honeycomb sturcture valve test section lower tank cross-correlation CCD Camera (18 by 118) Synchronizer PC computer (RAM 52MB,HD 12GB) Figure 1. The schematic of the experiment setup EXPERIMET SETUP Figure 1 shows the experimental setup used in the present research. 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. 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=16mm. The flow rate of the test loop was about 2 liter/min, which corresponded to the average velocity at the inlet of the test tank being about.333 m/s, and Reynolds number about 6,7 based on the height of the inlet (b=2mm). Since the test tank was designed to let the flow field in the test tank to be two dimensional (our measurement result also showed that the flow field in the test tank was ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

almost two dimensional along the Z direction except the regions near the two walls), PIV measurement was just conducted at the middle section of the test tank (Z=25mm section) in the present study. free surface inlet L=11mm Y b=2mm H=16mm Z X S=15mm outlet E=6mm T=5mm W=3mm Figure 2. The schematic of the test tank The pulsed laser sheet (thickness of the sheet is about 1.5 mm and the life per pulse is 6ns) used for PIV measurement was supplied by a Twin d:yag Laser with the frequency of 15 Hz and power of 2 mj/pulse. Polystyrene particles (diameter of the particles is about 2-5 m, density is 1.2) were used as PIV tracers in the flow field. A 18 by 118 pixels Cross-Correlation CCD array camera (PIVCAM 1-3) was used to capture the images. The Twin d:yag Laser 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 PC computer (host computer, RAM 512MB, HD12GB) for image processing and displayed on a PC monitor. To obtain fluid velocities by using PIV technique, two or more images of seeded flow fields are captured at successive points in time, and comparison of these images allows the velocity fields to be constructed. In the present study, rather than tracking individual particle, the cross correlation method (Willert et al., 1991) was used to obtain the average displacement of the ensemble particles. The images were divided into 32 by 32 pixel interrogation windows, and 5% overlap grids were employed for the PIV image processing. 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. RESULTS AD DISCUSSIOS 1. Oscillation frequency of the self-induced sloshing Figure 3 shows the oscillation of the free surface with time at three positions, i. e., left side (inlet side), center and left side of the test tank. The frequency of the self-induced sloshing can be calculated from these signals, which is about 1.6Hz. According to the definition of Fukaya et al.(1996), the self-induced sloshing mode of the present study is 1 st mode. Okamoto et al. (1991) had suggested that the frequency of the self-induced sloshing equal to the eigenvalue of the water in the test tank, which can be expressed as: f n 1 = 2π nπg nπh tanh( ) W W ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

Water level (mm) 166 165 164 163 162 161 16 159 158 157 156 left side middle right side.5 1 1.5 2 2.5 3 3.5 4 4.5 time (s) Figure 3. The oscillating water level of the free surface By using above equation, the 1 st mode eigenvalue of the water in the test tank for the present study case can be calculated, which is 1.56Hz. So, the difference between the frequency of the self-induced sloshing (f=1.6hz) measured in present study and the 1 st mode eigenvalue of the water in the test tank is within the 3% range, which verified the propose of Okamoto et al.(1991) once again. 2. Instantaneous flow field As mentioned above, since the frequency of the illumination pulse supplied by the d:yag Laser was settled as 15 Hz, the instantaneous PIV velocity field can be got at the rate of 7.5Hz (two pulses for one velocity field). According to the above measurement result, the frequency of the self-induced sloshing is 1.6Hz for the present study condition. This means about 5 instantaneous velocity fields can be obtained for every cycle of the self-induced sloshing by using present PIV system. Figure 4 shows the 6 continuous instantaneous PIV measurement results. The time interval between the Fig. 4(a) and Fig.4(f) is about one period of the self-induced sloshing. The evolutions of the vortices and turbulent structures in the test tank can be seen clearly from these instantaneous flow fields. It can be seen that, beside the two large vortices at the right side and left lower corner of the test tank (which are almost steady vortices), there is another smaller vortex (which is unsteady) shedding from the inlet plane jet. Since there is only one unsteady vortex shedding can be found for one cycle of the self-induced sloshing, the shedding frequency of this smaller unsteady vortex is equal to the frequency of the self-induced sloshing. The whole flow pattern in the test tank, such as the form of the inlet plane jet, the size of the two steady vortices, and the flow direction at the exit of the test tank, changed very drastically with the evolution of this unsteady vortex. Figure 5 shows two typical power spectrum profiles of the velocity (u and v components) in the test tank obtained by PIV measurement. Since the PIV instantaneous flow field was got at the rate of 7.5Hz, the spectrum range of the PIV measurement result is less than 3.75 Hz. From the figure, it can be seen that there is an obvious peak at the frequency value of 1.6Hz in the power spectrum profiles for both u- component and v-component. This means that the velocity field in the test tank was affected mainly by the periodic shedding of the unsteady vortex. 3. Time average result In the present study, time average value of the PIV measurement result was also used to research the self-induced sloshing phenomena. 1 PIV instantaneous velocity fields were used to calculate the mean values of the velocity field and turbulent intensity field. Since the PIV instantaneous velocity fields were got at the rate of 7.5Hz by using the present PIV system, the sampling time interval for getting 1 instantaneous PIV velocity fields was about 13 seconds, and it was about the life of 21 cycles of the selfinduced sloshing. ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

-3. -24. -18. -12. -6.. 6. 12. 18. 24. 3. 15 2 2-3. -24. -18. -12. -6.. 6. 12. 18. 24. 3. 15 Re =6,7 Re =6,7 Uin =.33 m/s Uin =.33 m/s 5 5-5 -5 (a). 1 15 2 25-5 -5 3 at time t=t (b). -3. -24. -18. -12. -6.. 6. 12. 18. 24. 3. 15 5 1 15 2 25 3 at time t=t+1/7.5 s 2 2 5 1 1-3. -24. -18. -12. -6.. 6. 12. 18. 24. 3. 15 Re =6,7 Re =6,7 Uin =.33 m/s Uin =.33 m/s 5 5-5 -5 (c). 1 15 2 25-5 -5 3 at time t=t+2/7.5 s 2. 6. 1 15 2 25 3 12. 18. 24. 3. 15-3. -24. -18. -12. -6. 5 (d). at time t=t+3/7.5s 2 5 1 1-3. -24. -18. -12. -6.. 6. 12. 18. 24. 3. 15 Re =6,7 Re =6,7 Uin =.33 m/s Uin =.33 m/s 5 5-5 -5 5 1 15 1 1 2 25-5 -5 3 (e). at time t=t+4/7.5 s 5 1 15 2 25 (f). at time t=t+5/7.5 s (new cycle ) Figure 4. The instantaneous velocity and spanwise voriticity distributions 3

Power Spectrum.1.8.6.4.2 1.6...5 1. 1.5 2. 2.5 3. 3.5 4. Frequency (Hz) (a). u-component Power Spectrum.4.3.3.2.2.1.1. 1.6..5 1. 1.5 2. 2.5 3. 3.5 4. Frequency (Hz) (b). v-component Figure 5. The power spectrum of the velocity at point (X=1mm,Y=1mm, Z=25mm) by using PIV 2 15 1 5.5 m/s VELOCITY.3896.3639.3381.3124.2866.269.2352.294.1837.1579.1322.165.87.55.292.35 2 15 1 5.5 m/s Turbulent Intensity.1274.1194.1114.133.953.873.792.712.632.551.471.391.31.23.15.69 1 2 3 1 2 3 (a). mean velocity (b). Turbulent intensity Figure 6. The time average flow field of the PIV result Figure 6 shows the time average result of the flow field in the test tank, which included the mean velocity field (Fig.6(a)) and the mean turbulent intensity field (Fig.6(b)). The mean velocity (U j and V j ) and the mean turbulent intensity T ij shown on these figures were defined as: u t v t U j = V j = t = 1 t = 1 ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

T j = = t = 1 ( u 2 t + v 2 t 2 ( u t U j ) + t = 1 i = 1 In which =1, u t and v t are the instantaneous velocities in the X and Y direction, while u t and v t are the instantaneous fluctuating components. From the figures it can be seen that, just as the results reported by Okamoto et al.(1993b), only two steady vortices can be found at the right side and the left lower corner of the test tank from the time average result. The unsteady vortex revealed in the above instantaneous flow field, which shed periodically from the inlet plane jet, cannot be found in the time average results. However, the high turbulent intensity region can be found along the shedding path of the unsteady vortex. In order to verify the present PIV measurement results, the LDV measurement had also been conducted along the central lines of the inlet and outlet of the test tank. The comparison of the PIV and LDV measurement result was shown in Figure 7. In general, the PIV measurement result agrees with the LDV result well for both the mean velocity (U and V) and mean turbulent intensity (STD(u)and STD(v)). However, some local disagreement (always in the high shear region) between the PIV and LDV result can also be found in the profiles. This may be explained by the fact that, since the cross-correlation method was used in the present study to do PIV image processing, the velocity field got by PIV measurement is the average velocity of the ensemble particles in interrogation windows. So, the disagreement was expected to exist between the LDV and PIV measurement, especially in the high shear region. The gap between the PIV and LDV results is smaller in the mean velocity (U and V) profiles than that in the mean turbulent intensity (STD(u) and STD(v)) profiles. This may be explained by the following fact: according to the research of Ullum et al. (1997), the necessary sampling number to calculate the mean turbulent intensity should be the square of the number to calculate the mean velocity in order to get the same level of the standard deviation error. So, the standard deviation error by using the same sample number (1 instantaneous velocity fields) to calculate the mean turbulent intensity was expected to be bigger than that to calculate the mean velocity filed. Velocity (m/s).5.4.3.2.1 -.1 -.2 U-piv V-piv U-ldv V-ldv 5 1 15 2 25 3 X (mm) ) Turbulent intensity (m/s) ( v.14.12.1.8.6.4.2. t V ) 2 j (a). along the inlet central line (Y=11mm, Z=25 mm) STD(u)-piv STD(v)-piv STD(u)-ldv STD(v)-ldv 5 1 15 2 25 3 X (mm) Velocity (m/s).2.15 U-piv V-piv.1 U-ldv.5 V-ldv -.5 -.1 -.15 -.2 2 4 6 8 1 12 14 Y (mm) (b). along the outlet central line (X=15mm, Z=25 mm) Figure 7. The comparison of the PIV and LDV results Turbulent Intensity (m/s).12.1.8.6.4.2. STD(u)-piv STD(v)-piv STD(u)-ldv STD(v)-ldv 2 4 6 8 1 12 14 Y (mm) ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

4. Phase average result Since the self-induced sloshing oscillated periodically with a frequency of 1.6Hz, the phase average measurement had also been conducted in the present research. During the measurement, the free surface water level at the left side (inlet side) was detected by a water-level detecting system, which can sent a signal to the Synchronizer Control System to trigger the laser pulses and CCD camera. The phase average results shown on Figure 8 were obtained by the average of the 25 instantanous velocity fields at four phase angles ( θ =, π, π and 3π ). The free surface water levels at the left side (inlet side) of the test 2 2 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 phase angles, respectively..5 m/s.5 m/s 2 15 1 5 water surface VELOCITY.4236.3958.3679.34.3122.2843.2564.2286.27.1728.145.1171.892.614.335.56 2 15 1 5 water surface VELOCITY.438.424.3739.3455.3171.2886.262.2318.233.1749.1465.118.896.612.327.43 1 2 3 1 2 3 (a). situation 1 (θ =) (b). situation 2 (θ = 2 π ).5 m/s.5 m/s 2 15 1 5 water surface VELOCITY.387.3615.3361.316.2851.2597.2342.287.1833.1578.1324.169.814.56.35.51 2 15 1 5 water surface VELOCITY.3931.3671.3412.3152.2893.2633.2374.2114.1855.1595.1335.176.816.557.297.38 1 2 3 1 2 3 3π (c). situation 3 (θ =π ) (d). situation 4 (θ = ) 2 Figure 8. The phase average flow field of PIV result Unlike the above time average result which can just reveal two steady vortices in the flow field, the unsteady vortices can also be found clearly in the flow field from the phase average measurement result. Besides the two steady vortices at the left lower corner and right side of the test tank, the unsteady vortex was found to change its position with the change of the phase angle. When the phase angle increased from 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.8(a), Fig.8(b) to Fig.8(c)); the ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

unsteady vortex shed from the inlet plane jet and moved downstream. When the phase angle increased from π to 2π, 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. 8(c), Fig. 8(d) to Fig. 8(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. 5.The mechanism of the self-induced sloshing As mentioned above, Okamoto et al. (1991) had proposed that the frequency of the self-induced sloshing always equaled to the eigenvalue of the water in the test tank, and the measurement result of the present research also verified this conclusion. That means the self-induced sloshing in the test tank has a very close relationship with the oscillation characteristics of the water in the test tank. Since the resonated oscillation just vibrated with the frequency decided by the eigenvalue of the oscillating body, the "resonance model" was conjectured in the present paper to be the reason for the self-induced sloshing in the test tank. It was well known that there are two necessary factors for the resonated oscillation. The first is excitation source and the second is that the frequency of the excitation should be coupled with the eigenvalue of the oscillating body. From the above PIV measurement result, it can be seen that during the self-induced sloshing of the water free surface, the periodic shedding of the unsteady vortex from the inlet plane jet played a key role on the distribution of the flow pattern in the test tank. The shedding frequency of this unsteady vortex just equaled 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 periodical shedding of the unsteady vortex can be suggested to be the excitation source of the resonated oscillation (self-induced sloshing). However, where does this unsteady vortex come from? Since this unsteady vortex was originated from the inlet plane jet, the instability of the inlet plane jet may be conjectured to be the source of this unsteady vortex. It was well known that, according to the linear instability theory (Michalke (1965) and Ho et al. (1984)), the most unstable mode in the plane jet should satisfy the equation: fθ St = =.17 U θ For the present research case, the velocity at the inlet of the test tank was about.333m/s, and the momentum thickness at the inlet of the test tank was about.88mm according to the LDV measuremnt. So the most unstable mode of the inlet plane jet is about 6.4Hz, which equals 4 times of the self-induced sloshing (eigenvalue of the water in the test tank). So the instability of the inlet plane jet may be suggested to be real excitation source of the self-induced sloshing in the test tank. When the unstable mode of the inlet plane jet is coupled with the eigenvalue of the water in the test tank, the resonated oscillation (self-induced sloshing) is expected to occur. (a). U componet at point (b). V componet at point (X=1mm,Y=11mm, Z=25mm) (X=9mm,Y=11mm, Z=25mm) Figure 9. The power spectrum profiles by LDV measurement Figure 9 shows the power spectrum of the velocity at two typical points in the test tank by LDV measurement. From the figure, it can be seen that, besides the peak at the frequency of self-induced sloshing (also the eigenvalue of the water in the test tank, f =1.6Hz), the sub-peaks can also be found at the frequencies of 2f, 3f and 4f. This means that due to the frequency toning of the water in test tank ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

(oscillating body), only the some modes, whose frequency can be coupled with the eigenvalue of the oscillating body, can have advantageous growing rate. This also verified the above analysis. Okamoto et al.(1991) had reported that the self-induced sloshing in the test tank occurred just in a certain range of the flow rate and water level in the test tank. The "resonance model" suggested by the present paper could also be used to explain such kind of phenomena. The flow rate changes in the test loop may cause the change of the inlet plane jet condition, which will result in the change of the instability of the inlet plane jet. This may cause the couple or uncouple of the unstable mode in the inlet plane jet with the eigenvalue of the water in the test tank, so the self-induced sloshing can just be observed in a certain range of the flow rate in the test loop. The water level change in the test tank will cause the change of the eigenvalue of the water in the test tank, which may also result in the couple or uncouple of the unstable mode of the inlet plane tank with the eigenvalue of the water in the test tank. So, the self-induced sloshing was just observed in the certain range of the water level in the test tank. COCLUSIO The self-induced sloshing in a rectangular tank with circulating flow 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 mechanism of the self-induced sloshing. Besides the two steady vortices in the left lower side and right side of the test tank reported by Okamoto et al. (1993a), a new unsteady vortex was found in the test tank which was shedding periodically from the inlet plane jet. The shedding frequency of this unsteady vortex equals to the frequency of the self-induced sloshing, which also equals to the eigenvalue of the water in the test tank. When the water level of the free surface at the left side of the test tank increased from its lowest position to its highest position, the unsteady vortex was found to be generated and grown up. When the water level of the free surface at the left side of the test tank decreased from its highest position to its lowest position, the unsteady vortex moved downstream and then engulfed by the large steady vortex in the right side of the test tank. The flow pattern in the test tank changed seriously with the evolution of the unsteady vortex. Based on the above experimental result, resonated oscillation was suggested to be the mechanism of the self-induced sloshing, and the instability of the inlet planet jet was conjectured to be the excitation source of the self-induced sloshing. ACKOWLEGEMET The authors would like to thank the "PIV-STD" project for providing some of the experimental facility for present research. The research fellowship provided by Japan Society for Promotion of Science (JSPS) to the first author, and the support for the Original Industrial Technology R&D Promotion Program from the ew Energy and Industrial Technology Development Organization (EDO) of Japan are also acknowledged. REFERECE Fukaya, M. Madarame, H. and Okamoto, K., "Growth Mechanism of Self-Induced Sloshing Caused by Jet in Rectangular Tank (2 nd Report, Multimode Sloshing Caused by Horizontal Rectangular Jet)" Trans. of JSME, (B), Vol.62, o.599. p64-71, 1996. Madarame, H., Okamoto, K. and Hagiwara, T. "Self-induced Sloshing in a Tank With Circulating Flow", PVP-Vol.232, Fluid-Structure Vibrations and Sloshing, 1992. Michalke, A. "On Spatially Growing Disturbance in an Invisid Shear Layer." J. Fluid Mech., Vol.23, p521-544, 1965. Ho, C. M. and Patrick, H. "Perturbed Free Shear Layers ", Ann. Rev. of Fluid Mech. Vol. 16, p365-424, 1984. Hu, H., Saga, T., Kobayash T., Okamoto, K. and Taniguch.," Evaluation of the Cross Correlation Method by Using PIV Standard Images", Journal of Visualization, Vol.1, o.1, pp87-94, 1998. Okamoto, K., Madrarame, H. and Hagiwara, T. "Self-induced Oscillation of Free Surface in a Tank with Circulating Flow ", C416/92, IMechE, p539-545, 1991 Okamoto, K., Fukaya, M. and Madarame, H., "Self-induced Sloshing Caused by Flow in a Tank", PVP- Vol.258, Flow-Induced Vibration and Fluid-Structure Interaction, ASME, 1993 (a). Okamoto, K., Madrarame, H. and Fukaya, M. "Flow Pattern and Self-induced Oscillation in a Thin Rectangular Tank with Free Surface", Journal of Faculty of Engineering, The University of Tokyo. Vol. XLII, o. 2, 1993(b). Â ÂÃÄÅÆÇÈÉÂÃÄÄÄÂÅÆÂÇÈÉÇ

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