numerical analysis of suppression effect of asymmetric...

Numerical Analysis of Suppression Effect of Asymmetric Slit on Cavitation Instabilities

Hiroki Kobayashi1, Ryosuke Hagiwara1, Satoshi Kawasaki2, Masaharu Uchiumi2, Kazuyuki Yada2 and Yuka Iga3

Abstract In the present study, numerical analysis carried out around the cyclic flat-plate cascade with a

asymmetric slit, so as to examine the suppressing or controlling effect of the slit for cavitation instabilities such as a cavitation oscillation which is similar to cavitation surge, and rotating cavitation. These instabilities cause various problem for the turbomachinery, for example, rotating cavitation causes an asynchronous axial vibration, and cavitation oscillation causes a pulsation of working fluid by resonance phenomenon of the system. Especially, in liquid propellant rocket engine, suppression process for these instabilities bring increase in cost of the launch. Therefore, it is thought that to find effective suppression technique is significant for turbomachinery. In this paper, two types of the flat-plate three blades cascade which have an asymmetric slit on each blade were analyzed, and compare with the result of cascade without slit. As a result, the cavitation oscillation is perfectly suppressed in both of two types cascade with an asymmetric slit. Also, in the one case of them, sub-synchronous rotating cavitation and rotating stall cavitation are suppressed. Furthermore, super-synchronous rotating cavitation which is not observed in single-stage cascade, is detected by arranging an asymmetric slit. These results indicate the possibility of suppressing cavitation instabilities or controlling the type of the cavitation instabilities by the arrangement of the slit. Moreover, the head performance at the almost same cavitation number is equal or slightly increasing by arranging the asymmetric slit.

Keywords Homogeneous model — Cascade — Inducer — Rotating cavitation — Cavitation oscillation

1Graduate School of Engineering, Tohoku University, Sendai, Japan, 2JAXA Kakuda Space Center, Miyagi, Japan 3Institute of Fluid Science, Tohoku University, Sendai, Japan

1. INTRODUCTION

Cavitation is a phenomenon which liquid evaporates and vapor bubbles occur in the low pressure region such as a pressure of liquid is roughly less than vapor pressure. Usually, this phenomenon is observed in the high-speed fluid machinery, for example, pumps and hydraulic machinery. Also, the cavitation exerts a bad influence such as vibration, noise, and performance decrement on these hydraulic machineries. Moreover, the cavitation brings about the oscillations which are called cavitation instabilities in the turbomachinery on the occasion. The cavitation instabilities are generally classified into two type, which is cavitation oscillation and rotating cavitation. The cavitation oscillation [1] leads to flow rate and pressure fluctuation, also, it causes an oscillation for entire of pump systems by a resonance phenomenon. The rotating cavitation [2] causes asynchronous axial vibration in turbomachinery. As the example that the accident caused by this phenomenon, launch failure in the Japanese liquid propellant rocket happened due to the rotating cavitation in 1999. Therefore, it is important to find the suppression and control method of these instabilities for designing of the highly performance and reliable turbomachinery. For the above reasons, the methods which aim to suppress the cavitation instabilities have been developed and adopted in the liquid rocket engine

such as the using pogo-suppresser [3] and extension of the casing diameter at inducer inlet [4]. However, these methods are not perfect owing to the complex characteristics of cavitation.

In these days, we can obtain commercial software and open source codes easily, and numerical simulation by the software is able to utilize in designing of the fluid machineries. But numerical simulation has not been utilized in designing of fluid machineries with cavitation and theoretical analysis and experiment are still predominant. It is because the accuracy of the CFD for cavitation is not enough. In benchmark simulation by 6 research groups and 4 software venders by

Turbomachinery society of Japan，it was reported that

even time averaged lift and drag can’t be predicted by the all commercial software and the in-house codes [5] . In such a situation, our group developed numerical method which is suitable for unsteady cavitation by using homogeneous model and compressible scheme, and has several experiences of the numerical simulation for cavitation instabilities, which is although arising in 2 dimensional cascade. Hence, it is considered that the numerical simulation for development of suppression technique of cavitation instabilities is possible as application of our numerical method.

In this paper, the suppression effect for cavitation instabilities by using slit on blade is investigated. A

Numerical Analysis of Suppression Effect of Asymmetric Slit on Cavitation Instabilities -2

numerical simulation of unsteady cavitation in the cascade with an asymmetric slit is performed, and results are compared with cascade without slit. The occurrence region of cavitation instabilities, cavity volume, mass flow rate, and head performance of the asymmetric slit are discussed and compared with results of cascade without slit. These results indicate the suppression and control effect of an asymmetric slit for cavitation instabilities in cascade without decline of performance. 2. NUMERICAL METHOD

2.1 Cavitation model In this study, a locally homogeneous model of a

compressible gas-liquid two-phase medium [6] was used for numerical simulation of cavitation. This model allows to consider of a gas liquid two-phase field as a pseudo-single phase medium. Therefore, the Navier-Stokes equations for continuum can be applied to a cavitating flow field in which there is discontinuity between the gas and liquid phases. As a governing equation, the compressible gas-liquid Navier-Stokes equations are used for the gas-liquid medium.

j j

t x

E EQS (1)

,iu

Y

Q j

j i j i j

j

u

u u p

u Y

E

0

,

0

j ij

E

0

0

S ,

where ρ, p, and u are the density, static pressure, and velocity of the mixture phase respectably. The mass fraction of the gas phase is denoted Y. u and μ , which is necessary for estimating the stress tensor τ in Eq. (1), is determined from the following equation:

i

j

j

iijij

x

u

x

uudiv

2

12

, (2)

1 1 2.5 l g , (3)

where 𝛼 is the volume fraction of the gas phase which means the void fraction, and subscript g and l mean the gas and liquid phase.

The government equations in Eq. (1) are closed by an equation of the state for the locally homogeneous compressible gas-liquid two-phase medium [6]. In this equation, the local equilibrium between gas and liquid phase at a pressure and temperature, and a linear combination of the masses of the gas and liquid phases are assumed. Additionally, the liquid is assumed to be

compressible, and the gas is assumed to obey the ideal-gas low:

1

c

l c g c

p p p

K Y p T T R Y p p T

, (4)

where 𝐾𝑙 and 𝑅𝑔 are the liquid constant and gas

constant. 𝑝𝑐 and 𝑇𝑐 are the pressure and temperature constant of the liquid, respectively, and are assumed to be constant. The speed of sound in the two-phase medium is determined from Eq. (4) is found to be in good agreement with that obtained in an experiment on variations in the void fraction [6]. Thus, this numerical method is able to reproduce pressure wave propagation in a gas-liquid flow field. This is important for numerical simulation of the mutual interface between cavitation and fluid machinery systems.

Γ in the source term of S denotes instantaneous equilibrium evaporation model [6]. In the evaporation model, an empirical constant is not included. Therefore, it is robust to rapid pressure jumps during pressure wave propagation. For the reason above, this model can be applied to flow field such as the cavitation oscillation and other phenomenon.

In this locally homogeneous model, the cavity surface which is expressed as the void gradient, can be treated as the contact discontinuity problem in a compressible fluid. Therefore, the thickness of the cavity surface depends on the resolution of the computational grids. However, it can reproduce various types of unsteady cavitation.

2.2. Numerical scheme In this study, the discretization of the governing

equations Eq. (1) is based on finite difference method. It is necessary to simulate stably the cavity flows, which has a large density jump between gas and liquid phase. Accordingly, the total variation diminishing (TVD) scheme is applied to preserve the monotonicity of the solution. Specifically, the explicit TVD-Mac Cormack scheme [7] with second-order accuracy in time and space is used. In this way, no turbulence model is applied because reliable turbulence model has not been found for simulate for unsteady cavitating flow.

2.3. Validation In order to develop a suppression technique for

cavitation instabilities by using numerical simulation, numerical method which can reproduce the cavitation instabilities is needed. Our numerical method by homogeneous model and compressible scheme has been validated for time averaged and unsteady characteristics in several cavitating flow field. The present numerical method was validated [8] through comparisons with experimental data about time-averaged lift and drag in non-cavitation and cavitation conditions around a Clark Y 11.7% single hydrofoil and the time-averaged pressure distribution on a Clark Y


11.7% cascade hydrofoil in non-cavitation conditions in several cascade arrangements. Additionally, the availability of an existing RANS turbulent model of single-phase flow for the CFD of a cavitating flow was discussed [9]. Although, as shown in Figure 1. which is for time averaged pressure distribution on a hydrofoil, the RANS simulation is effective in non-cavitation condition but ineffective in cavitation condition. Additionally, in break-off frequency of sheet cavitation on a hydrofoil, the frequency of developed sheet cavitation can be predicted by simulation without RANS model as shown in Figure 2., which is closer to experimental frequency than that with RANS model.

(a) Non-cavitation condition (σ = 3.0) [9]

(b) Cavitation condition (σ =1.4) [9]

(c) Cavitation condition (σ =1.0)

Figure 1. Grid convergence characteristics of the single wing

surface mean pressure distribution, Comparison between RANS turbulence model and laminar flow analysis

(NACA0015, αin = 8deg)

Figure 2. Inspect calculation of the sheet cavity break period,

Comparison between RANS turbulence model and laminar flow analysis (NACA0015, αin = 8.36 deg) [9]

Consequently, we decided not to use a turbulence model in the present study in order to reduce computational costs. However, the macroscopic disturbance in the flow field, which is caused by cavity volume variation, can be reproduced in the present numerical simulation.

Additionally, the authors have researched the cavitation instabilities arising in three-blade cyclic cascades by the present numerical method in the past. Seven kinds of cavitation instabilities, caused by several different mechanisms, have been reproduced without adding any additional models or boundary conditions [9], the occurrence map of cavitation instabilities is shown in Figure 3. The propagation velocity ratio of rotating cavitations and frequency of cavitation oscillation can be reproduced as shown in Figure 4. and

1.2

1.0

0.8

0.6

0.4

0.2

f C

/ U

1.81.61.41.21.00.80.6

Experiment by Kawanami [11]

:

U= 6.0m/s

U= 8.0m/s

U=10.0m/s

U=12.0m/s

Present:

U=10.0m/s Without turb. model

U=10.0m/s BL-DS


5. which corresponds well-known unsteady characteristics of cavitation instabilities in inducer of liquid propellant rocket engine. Furthermore, two-kinds of break-off mechanism of cavitation in cascade was reproduced by the present numerical method, one is re-entrant jet which is in the case of low cascade interference and the other is cavity surface instability acconpanied by pressure wave propagation inside the cavity which is in a case of high cascade interference [8].

Therefore, the present numerical method is available for numerical simulation of unsteady cavitation and cavitation instabilites, hence, it is considered to be able to utilize for development of supression techeneque of caviation insatabilities with some extent of reliability.

Figure 3. Occurrence map of cavitation instabilities of the

present three-blade cyclic cascade [9]

Figure. 4 Propagation velocity ratio of uneven cavity

volume in each rotating phenomenon [9]

Figure 5. Comparison with empirical frequency and actual

frequencies of cavitation oscillation [10]

3. Results and Discussion

3.1. Computation conditions The flow field simulated in present study is the flat

plate three blade cascade, which has a slit on the each blade randomly, it is named asymmetric slit. The solidity C/h, stagger angle 𝛾 , chord length C, and blade thickness are 2.0, 75°, 0.1 m, and 0, respectably. In this study, three types cascade are analyzed. The detail of the three types of cascade are shown in Figure 1. The three blades in each cascade are denoted by blade1, blade2, blade3; the direction of numbering is opposite to that of the rotation of the cascade, as shown as in Figure 6. The first one is single-stage cascade which is denoted by Cascade 1. The cascades which is denoted

1.5

1.0

0.5

PV

R

0.50.40.30.20.1

Marks

Even Blade Cascade:

:in=3.0 deg

: =5.0 deg

: =7.0 deg

: =9.0 deg

: =9.5 deg

: =1 1 deg

Uneven Blade Cascade:

:in=3.0deg

Colors

:Super-S R.C

:Sync.R.C

:Sub-S R.C

:R-stall C

:C.S Type 3

Figure 6. Schematic diagram of the single-stage

cascade and two cascades with an asymmetric slit

h


Cascade 2 and Cascade 3 have an asymmetric slit on each blade which are a kind of tandem cascade.

In the Cascade 2, the asymmetric slit is arranged in each blade. The position of the asymmetric slit in blade 1, 2 and 3 are 0.5C, 0.6C, and 0.7C, in the case of Cascade 3, the position of the asymmetric slit are 0.3C, 0.5C, and 0.55C from leading edge respectably. The width of slit is constantly 0.05C each cascades. In present cascade arrangement, the cascade throat is located at the point of 0.48C. Consequently, the slit of blade1 is located slightly downstream of the throat in Cascade 2. On the other hand, the slit of blade 1 is located upstream of the throat in Cascade 3.

As a boundary condition, constant total pressure, flow angle, and void fraction are applied at inlet. At the outlet boundary, constant static pressure is applied. Under this condition, the static pressure at the inlet is extrapolated and the inlet velocity is calculated by the inlet total pressure. Besides, the density and velocity at the outlet is extrapolated. Although the flow angle is constant in the boundary condition, the flow rate can change according to the change of prewhirl flow. Also, a non-slip condition is imposed at the wall boundary of the blade. In addition, the cyclic boundary condition is imposed on the third flow channel in the cascade in order to reproduce the circumferential instabilities in the cascade, such as rotating cavitation.

The inlet flow angle is αin fixed at a value of 9.5° and the time averaged inflow velocity Uin is approximately 13.5 m/s. The calculation is carried out with changing cavitation number σ, which is a dimensionless quantity that indicate the ease of cavitation. σ can be changed by changing the outlet static pressure. A mesh per

blade-to-blade channel is 361 71 points. The distance

between the inlet boundary and leading edge is two chord lengths, and the distance between the trailing edge and outlet boundary is three chord lengths. The cavitation number σ and static pressure coefficient ψ are estimated using the following equations:

25.0 inin

outin

U

pp

(5)

2,out in

in t

p p

U

(6)

here pv is the saturated vapor pressure. σ and ψ are calculated from the time-average flow field of the

numerical result. The flow rate coefficient / ,a tU U is

uniquely determined by the flow angle αin. In this study,

is corresponding to 0.0963, where Ua and Ut are the axial inlet flow velocity and the circumference inlet flow velocity respectably.

3.2 Suppression of cavitation instabilities by

asymmetric slit The head performance of present cascades are

drown in Figure 7. At high σ value region, the head performance of the Cascade 2 and 3 is better than that observed in the Cascade 1. Around head drop region, the head performance of Cascade 2 and 3 are almost same or slightly better than that of Cascade 1 although the head drop point is almost same. As the reason why the head increases in the cascade with slit is as following: The local angle of attack of rear blade takes the positive value by the pressure gradient between the suction side and pressure side at the slit. Therefore, the flow decelerates and the pressure rises twice at front and rear blades in the cascade, then head performance is improved in Cascade 2 and 3.

Figure 7 is also colored according to predicted cavitation instabilities. The occurrence of cavitation instabilities is judged by the visualization of aspects of the cavity and waveforms of variations in cavity volume, upstream pressure and upstream flow rate. In Figure 8 and 9, typical aspect of cavitation oscillation (C.O) and super synchronous rotating cavitation (Super-S R.C) are shown respectively [11] In C.O, three sheet cavities develop and shed, and three cloud cavities collapse in

Figure 7. The head performance and occurrence region of cavitation instabilities


Figure 9. Time evolution of void fraction distribution under condition of Type 3 C.O. (= 0.0699, = 0.296, =

0.142) [11]

same phase in each blade. In Super-S R.C, cavity develops in term Blade 3, 2, 1, which is rotating direction of cascade in stationery frame. In the Cascade 1, C.O, sub-synchronous rotating cavitation (Sub-S R.C) and rotating-stall cavitation (R-Stall C) are observed according to decrease of σ. In the rotating cavitation, the asymmetrical sheet cavity pattern rotates in the same or opposite direction as the cascade rotation. Besides, in the C.O, the sheet cavity break off and collapse simultaneously in the all blade. As indicated in the Figure 7, the multiple cases of C.O are observed around head drop region in Cascade 1. In comparison with the result of Cascade 1, C.O is not observed in the both case of Cascade 2 and 3. Also, in case of Cascade 3,

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

t = T0

t = T0 + 2 ms

t = T0 +4 ms

t = T0 + 6 ms

t = T0 + 8 ms

t = T0 + 10 ms

t = T0 + 12 ms

t = T0 + 14 ms

t = T0 + 16 ms

t = T0 + 18 ms

t = T0 + 20 ms

t = T0 + 22 ms

t = T0 + 24 ms

Figure 8. Time evolution of the void fraction distribution

under the condition of Type 2 C.O. (= 0.105, = 0.162,

and = 0.186; time interval = 2.0 ms) [11]

t = T1

t = T1 + 14.4

ms

t = T1 + 26.8

ms

t = T1 + 30.4

ms

t = T1 + 46.4

ms

t = T1 + 56.4

ms

t = T1 + 62.4

ms

t = T1 + 76.4

ms

t = T1 + 84.4

ms

t = T1 + 86.4

ms

t = T0 + 2.0 ms

t = T0 + 2.2 ms

t = T0 + 2.4 ms

t = T1

t = T1 + 14.4 ms

t = T1 + 26.8 ms

t = T1 + 30.4 ms

t = T1 + 46.4 ms

t = T1 + 56.4 ms

t = T1 + 62.4 ms

t = T1 + 76.4 ms

t = T1 + 84.4 ms

t = T1 + 86.4 ms


Sub-S R.C and R-Stall C is not detected in all σ region. In Cascade 2, the Super-S R.C, which is not detected in Cascade 1, is observed. These results indicate that the cavitation instabilities are suppressed or the type is controlled at certain σ region by arranging the asymmetric slit on the blade, and changing the slit position.

3.3 Flow field around the asymmetric slit Figure 10. shows the flow field around the slit by void

fraction and velocity vector distribution (upper) and pressure distribution (lower) of the Cascade 2 (Blade 2). As shown in Figure 10, the sheet cavity length is limited by a jet flow which flows from pressure side to suction side through the slit. This slit jet is caused by pressure gradient between the slit, which is increased by collapse of the cloud cavity in the channel of pressure side and also increased by the cavity growth in the channel of the suction side. When a re-entrant jet flows into the sheet cavity from upper side of the slit jet, a vortex cavity is generated in the vicinity of trailing edge of the front blade. Also, the slit jet induces increasing of the local angle of attack of the rear blade, which promotes the growth of a small sheet cavity at leading edge of the rear blade. Then the cavity on the blade with a slit will be composed by three vortex structures as shown in upper side of Figure 3. After that, these three cavities separate from the blade, shed to downstream and collapse with interfering with each other.

3.4 Time averaged characteristics Table. 1 shows the time average of the cavity

volume in each blade and total cavity volume in cascade 2 and cascade 3 with an asymmetric slit at almost same σ condition about σ = 0.10. The cavitation instabilities are suppressed in the both cases. From Table 1, the total cavity volume values of Cascade 3 are less than the case of Cascade 2 in all blades. Also, the cavity volume of each blade is getting larger with increasing front blade length in each cascade. As mentioned in previous section, the slit has an effect to limit the cavity length by jet flow.

Therefore, in case of front blade length is shorter, the sheet cavity becomes shorter.

Table. 2 shows the time average of mass flow rate of each slit in cascade 2 and cascade 3 in same case in Table 1. In the Cascade 2, the time averaged mass flow rate increase incrementally with front blade length getting longer. However, the mass flow rate of Blade 1 in Cascade 3, is larger than the Blade 2, 3 of Cascade 3 and Blade 1 of Cascade 2. In case of front blade length is long such as Blade 2 and 3 in the Cascade 2, the effect of pressure gradient increasing is large due to the high pressure rising which is caused by cavity collapse at downstream. Also, the slit which arranged on the Blade 1 of the Cascade 3, is covered with sheet cavity long time than other slit. Then the pressure gradient is comparatively large. Because of the above reason, time averaged flow rate in each slit also varies by the random arrangement of the slit. When the intensity of the slit jet is different, each volume of the three cavities which was shown in Figure 3. Is considered to change. Then, the cavity volume is decided not only by the slit position but also by the flow rate of slit jet as shown in Table 1 in which cavity volume in Blade 1 in Cascade 2 is different from that in Blade 2 in Cascade 3 in spite of the slit positions are same in the blades.

3.5 Unsteady characteristics Figure 11. shows the waveform of total cavity volume

(upper) and that of upstream pressure(lower) in Cascade 1 and Cascade 3 at the same cavitation number (σ = 0.2). In the Cascade 1, the total cavity volume oscillates with regular cycle and the pressure oscillates as pulsation which is the characteristic of C.O [12]. In Cascade 3, the regular cycle is not observed in total cavity volume and upstream pressure. It means C.O is suppressed and the break-off cycle of sheet cavity in each blade is not in the same phase in Cascade 3. By the way, it is known that Strouhal number of cavity break-off cycle based on cavity length is known to remain constant [13], it means break-off cycle depends on the cavity length. Then it can be considered that the cavity break-off cycle in each blade is different each other in a cascade with asymmetric slit because the cavity volume is different as shown in Table 1. Therefore, C.O, in which cavity oscillation with constant cycle repeats in same phase in all blade, is completely suppressed in Cascade 2 and 3.

The waveform of cavity volume in the each blade is shown in upper figure in Figure 11. in cascade 3 in σ = 0.2. The wave form indicates that the non-periodic waveform of total cavity volume in Cascade 3 shown in Figure 11 (upper) is yielded not by a difference of a cycle in each blade but by non-periodicity of each cycle in each blade. Also the waveform of mass flow rate of each slit jet has irregularity as shown in lower figure in Figure5. The result indicates that pressure gradient at the slit is different in each cycle in each blade because

Figure 10. The void and vector distribution (upper), and

pressure distribution (lower) of Blade 2 in Cascade 2 (σ =0.13)


Blade 1

(slit position) Blade 2


(slit position) total

Cascade 2 0.000196 m3

(0.5C)

0.000206 m3

(0.6C) 0.000217 m3

(0.7C) 0.000618 m3

Cascade 3 0.000129 m3

(0.3C) 0.000169 m3

(0.5C) 0.000179 m3

(0.55C) 0.000477 m3

Blade 1



(slit position)

Cascade 2 6.78 kg/s

(0.5C) 9.44 kg/s

(0.6C) 9.56 kg/s

(0.7C)

Cascade 3 8.17 kg/s

(0.3C) 6.93 kg/s

(0.5C) 6.53 kg/s (0.55C)

of the relative situation between pressure side and suction side at the slit is different in each cycle, which can be inferred from cavity volume in upper figure in Figure 12. Therefore, strong non-periodicity of cavity oscillation can be realized and C.O is completely suppressed by arranging asymmetric slit on cascade. Besides, as shown in Figure 7, cavitation instabilities are completely suppressed in Cascade 3, on the other hand, rotating cavitations are observed in Cascade 2 although C.O is completely suppressed. Judging from the result, the non-periodicity of cavity oscillation in Cascade 3 is considered to be stronger than that in Cascade 2, which relate to the slit position.

4. Conclusion

In the present study, the numerical simulation of the cavitating flow around the cascade with an asymmetric slit is carried out, and examine the suppression and control effect of the slit for cavitation instabilities in the cascade, from the view point of head performance, occurrence region of cavitation instabilities, mass flow rate of slit and cavity volume. The result are summarized as follows.

- The all cavitation instabilities can be completely suppressed in a cascade in which one slit locates upstream of the cascade throat. On the other hand, in the case of a cascade in which all slits locate inside the cascade throat, cavitation

Table 1. Time average of cavity volume in each blade and total volume in cascade 2 at σ = 0.10 and in cascade 3 at σ =

0.11

Table 2. Time average of mass flow rate of each slit in cascade 2 at σ = 0.10 and in cascade 3 at σ = 0.11

Figure 11. Comparison between Cascade 1 and 3 about

waveform of total cavity volume (upper) and upstream pressure (lower) (σ = 0.2)

Figure 12. The waveform of cavity volume on

each blade (lower) and flow rate of slit jet (upper) in Cascade 3 (σ = 0.2)


oscillation is perfectly suppressed, but some rotating cavitation is observed.

- By arranging the slit as random in each blade, the cavity volume and the slit jet fluctuates non-periodically in each cycle in each blade. It is because that pressure gradient at the slit is different in each cycle in each blade because of the relative situation between pressure side and suction side at the slit is different in each cycle in the asymmetric slit cascade

- The head performance of the cascade is not declined and slightly improved in some case by arranging the asymmetric slit in each blade.

References

[1] Tsujimoto, Y., Yoshida, Y., Maekawa, Y., Watanabe, S. and Hashimoto, T.: Observations of Oscillating Cavitation of an Inducer, J. Fluids Eng. 199 ,1997, pp. 775-781.

[2] Hashimoto, T., Yoshida, M. and Watanabe, M.: Experimental Study on Rotating Cavitation of Rocket Propellant Pump Inducer, J. Propul. Power, 13, 1997, pp. 488-494.

[3] Morino,M. and Kohsetsu, Y.: (Vision and Description) POGO Fluctuation in Liquid Rocket, Turbomachinery, 17, 1989, pp. 516-525 (in Japanese)

[4] Shimagaki, M., Haga, H., Hashimoto, T., Wataname, M., Hasegawa, S., Shimura, T. and Kamijo, K.: Effect of the Casing Configurations on the Internal Flow and unsteady Pressure Fluctuation in Rocket Pump Inducer, J. Jpan. Soc. Aeronaut. Space Sci., 53, 2005, pp. 266-273

[5] Kato, C., 2011, Industry-University Collaborative Project on “Numerical Prediction of Cavitating Flows in Hydraulic Machinery – Part1: Benchmark Test on Cavitating Hydrofoils,” Proc. AJK2011-FED, Hamamatsu, AJK2011-06084.

[6] Iga, Y., Nohmi, M., Goto, A., and Ikohagi, T., “Numerical Analysis of Cavitation Instabilities Arising in The Three-Blade Cascade,” J. Fluids Engineering, Trans. ASME, Vol. 126, No. 3, 2004, pp. 419–429.

[7] Yee, H. C., "Upwind and Symmetric Shock-Capturing Schemes," NASA-TM, 8946, 1987.

[8] Iga, Y., Nohmi, M., Goto, A., Shin, B. R., and Ikohagi, T., “Numerical Study of Sheet Cavitation Break-off Phenomenon on a Cascade Hydrofoil,” J. Fluids Eng., Trans. ASME, Vol. 125-4 , 2003, pp. 643–651.

[9] Iga, Y. and Yoshida, Y., “Mechanism of Propagation Direction of Rotating Cavitations in a Cascade,” Journal of Propulsion and Power, AIAA, Vol. 27, No. 3, 2011, pp. 675–683.

[10] Yuka IGA, Yoshiki YOSHIDA, “Influence of Pipe Length on Cavitation Surge Frequency in a Cascade”, Journal of Propulsion and Power, AIAA, Vol. 30, No. 6, 2014, pp. 1520-1527.

[11] Yuka IGA, Kei HASHIZUME, Yoshiki YOSHIDA "Numerical Analysis of Three Types of Cavitation Surge in Cascade" Journal of Fluids Engineering,

Trans. ASME, Vol. 133, No. 7 (2011), 071102-1- 071102-13.

[12] Yamamoto, K.: Instabilities in an Cavitating Centrifugal Pump, JSME Int. J. Ser. II, 34, 1991, pp. 9-17.

[13] Pham, T. M., Larrarte, F. and Fruman, D. H.: Investigation of Unsteady Sheet Cavitation and Cloud Cavitation Mechanisms, J. Fluids Eng., Trans. ASME, 121, 1999, pp. 289-296.

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