Generality of Rotating Partial Cavitation in Two-Dimensional Cascades

Transcription

1 Proceedings of the 7 th Internationa Symposium on Cavitation CAV2009 Paper No. 90 August 17-22, 2009, Ann Arbor, Michigan, USA Generaity of Partia Cavitation in Two-Dimensiona Cascades Byung-Jin AN Osaka University Graduate Schoo Suita, Osaka, JAPAN Takeo KAJISHIMA Osaka University Suita, Osaka, JAPAN Kie OKABAYASHI Osaka University Graduate Schoo Suita, Osaka, JAPAN ABSTRACT Numerica simuations of 2-dimensiona (2D) unsteady cavitating fows were carried out under various conditions of the number of bades, incidence anges and cavitation numbers. When the incidence ange increased or the cavitation number decreased, the steady baanced cavitation transited to unsteady and non-uniform patterns. Typica patterns reported in the previous studies such as rotating, asymmetric and aternating for 3- and 4-bades were successfuy reproduced. In this study, cascades of the arger number of bades were deat with to consider the generaity of unsteadiness by reducing the infuence of periodicity. The cavitation is basicay triggered in the backward next section. However, the period of time for growing causes compexity in the discrimination of propagation. In most cases of rotating partia cavitation, except for 4-bades, the cavity deveops in the second passage of backward direction after the decay of argest cavity. In case of many bades, mutipe cavities rotate simutaneousy and the particuar patterns observed in cascades of sma even numbers of bades attenuate. Key Words : Cavitation, Inducer, Numerica Simuation, Turbomachinery, Unsteady Fow, Cascade, Gas-Liquid Two- Phase Fow INTRODUCTION A typica exampe of the high-speed hydro-machineries is the inducer of a iquid fue rocket engine, which is the target of this research. In case of a rocket engine, since weight and performance have an important effect on payoad, it is crucia to perform studies about high speed in order to improve performance and ower the weight. The generation of cavitation in high-speed hydro-machinery sometimes resuts in serious accidents. During the deveopment stage of the LE-7 rockets in Japan, it was pointed that cavitation is the cause of axia vibration [1]. However, it is actuay difficut to perform experimenta studies on rea working fuid. Tsuimoto, et a. deveoped the inear theory to derive the stabiity condition in cavitating fow through cascade. It is usefu for the prediction of the rotating cavitation. And they anayzed behavior of unsteady cavitation by a inear anaysis and singuar point method [2-4]. Numerica simuations in order to investigate the properties of the fows at the cavitation condition, based on the Navier- Stokes equation, started from 1990 s. Cavitation usuay invoves strong unsteadiness and can have various types ike bubbes, sheets, couds, vortices, etc. Aso, in the unsteady phenomena which occur in the cascade, various modes are observed, ike rotating cavitation, aternating cavitation, cavitation surge, etc. Therefore, research on numerica simuation technoogy, which can dea with various modes of unsteady cavitations, continues. The authors reproduced an unsteady cavitating fow fied in a 2D cascade [5-6] and in a 3D inducer [7], using their simuation method. On the other hand, by a barotropic mode, Iga, et a. [8] and Fortes-Patea, et a. [9] anayzed unsteady cavitating fows in 2D modes. Coutier-Degosha, et a. [10] conducted 3D simuation for a 3D inducer. Recenty, Yamanishi, et a. [11] appied LES to the 3D turbuent fows incuding a cascade. An inducer with 3- or 4-bades is necessary in order to raise the pressure in front of main engine. In case of the LE-7A engine of the H-IIA rocket in JAPAN, 3-bades inducer is used, the Vucain engine of Ariane rocket in Europe and the Space Shutte Main Engine in USA used an inducer of 4-bades. Therefore, conventiona experiments and anayses are based on 3- and 4-bades cascade. However, it is generay difficut to dea with behavior of the non-synchronous cavitation because the infuence of the periodicity in a few numbers of bades is strong. Namey, in Fig. 1, when cavitation propagates to the passage next, the fow passage is same 2nd to the opposite direction in case of 3-bades cascade. Aso, in case of 4-bades cascade, the propagation of cavitation to the ±2nd passage is a same and a rotating symmetry at the same time. For these 1

2 reasons, it is difficut to understand that reationship between the directions which cavitation affects and the passage which appears ceary next. The obective of this study is to find generaity in the rotating cavitation. And arger number of bades is considered to reduce the infuence of the periodicity which appears 3- and 4-bades cascade. First, the resuts of the prime numbers which are 5-, 7- and 11-bades cascades are expained. Then, the 6- and 8-bades cascades are treated to observe how aternating cavitation to be appeared. Finay, generaity about the propagation of unsteady cavitation is considered. 2. OUTLINE OF COMPUTATION 2.1 Computationa setup In this study, a fow through 2D cascades of fat bades is considered. The computationa domain shown in Fig. 1 is approximated to the tip region in an inducer of a iquid fue rocket engine. The chord ength and the pitch are C and h respectivey. The pitch chord ratio h/c is 0.5 and the stagger ange β = The infow boundary of the computationa region is at 4C upstream from the eading edge of the bades, the outfow boundary is at 5C from the traiing edge. The bades are numbered as: B.1, B.2,, B.N. The fow passages between bades are numbered as P.1-2, P.2-3, and the ast passage becomes P.N-1 due to periodicity. An H-type grid in a curviinear coordinate system is generated. The number of grid points is in each passage. The non-sip condition is appied at the bade surface, the non-uniform grid near the bade surface, especiay in the vicinity of eading and traiing edges, is generated with high resoution. The H-type grids of the passages next to the boundary are connected to circumferentia direction of cascade, and the periodic condition is set at both ends. The uniform inet veocity and the incidence ange are fixed in each simuation. The convective outfow boundary condition and the non-sip condition on the soid was are appied. Speciay, the non-refective boundary condition is used on the pressure equation [12]. At this time, the pressure p in a far downstream L wi be fixed. The saturated vapor pressure p v is given from the cavitation number using a reference pressure p. Eq. (1) is the cavitation number: p pv (1) 1 2 uin 2 where u in is inet veocity and ρ is the density of iquid. 2.2 Governing equations Hereafter, a variabes are non-dimensionaized by a characteristic ength C, veocity u in and the iquid density ρ. Aso, the fow fied is assumed to be isotherma. The tota density becomes ρ = f ρ when iquid is treated homogeneousy, because the vapor is assumed as a cavity of density 0. At this time, f is the iquid voume fraction. The iquid is considered as weaky compressibe and a ow-mach number assumption is appied to the mass conservation equation, which can be expressed as: Figure 1: Computationa domain and boundary conditions Df JU Dp f M Dt Dt J The momentum equation considering f, can be written as foows: k dui ui p U k dt f xi k 1 u u i v k T J J Re x x x i Where x i is the Cartesian coordinate, is the curviinear coordinate and J is the Jacobean of the coordinate transformation. u i is x i direction component of the veocity, U is the contra-variant veocity of direction. Reynods number is Re = Cu m / v (v is the kinematic viscosity of iquid) 6 and it is set to Re v T is the turbuent kinematic viscosity coefficient which is non-dimensionaized. The Mach number M is a compressibiity parameter considered uniform (M = 0.1) in the computationa domain. The obective of this cacuation incudes unsteady phenomena; however, LES is not suitabe because it is a 2D cacuation. Therefore Reynods average mode is used. The Badwin-Lomax turbuence mode for a singe-phase fow is used to cacuate v T, because there is no turbuence mode estabished for cavitation at the present. This simuation does not account for the interaction between turbuent fuctuations and cavitation, which is our future work. 2.3 Cavitation mode Omitting the second-order derivatives, viscous and surface tension terms in the Rayeigh-Pesset equation derives: (2) (3) dr 2 pv p/3 (4) dt 2

3 where R is the bubbe radius. Eq. (4) gives the increase rate of bubbe radius for p < pv. An expansion of square-root gives: df 1 f p pv dt R (5) This equation simpy means that the cavitation region wi expand under the condition of p < pv, whereas it wi contract in the region of p > pv. It is simiar to the mode proposed by Chen and Heister [13]: Dρ /Dt = C ( p - pv ). An empirica constant C in their mode seems to be reated to the bubbe diameter as supposed from Eq. (5). However, R can not be determined because bubbes of varied size are incuded in the actua cavitating fow. Chen and Heister [13] tested a very wide range of vaues for C: namey, 500, 5000 and It may be reated to the variation of R. Another probem of Eq. (5) in the actua computation is that the right hand side can not become negative when f = 1. Considering these probems, Okita and Kaishima [12] proposed: Df Cg ( f ) C f p pv Dt (6) The mode constants are C = 1 and Cg = 1000 for p < pv (inception and growth of cavity) whereas Cg = 100 for p > pv (contraction and disappearance of cavity). These two parameters were determined by comparison of experimenta and numerica data of cavitating fow around a rectanguar prism. The same vaues were used by Okita and Kaishima [12], without making any particuar adust for the current cacuation. 2.4 Computationa method The time marching procedure is based on the fractiona step method. The 2nd order Adams-Bashforth method is used for the convective and viscous terms, and backward Euer method is used for the pressure term and continuity equation (Okita and Kaishima [12], Ohashi and Kaishima [5]). The viscous term is discretized with centra differences of 4th order accuracy. A modified upstream difference is appied for the convective term. 3. RESULTS First, a cavitation number σ = 0.2, is considered for incidence ange α = 3. And the infuence of incidence ange α = 3, 6, 9 is considered when cavitation number is fixed to σ = 0.6. Then, the cacuation conditions, for exampe σ = 0.6 and α = 3, are expressed as Caseσ.6-α3. The other parameters are fixed in common. A uniform fow is given as initia condition. Then, the fow deveops in a singe-phase for a designated α. After that, the cavitation condition is set by the vaue of σ and the unsteady fow fied deveops enough with the time marching. The resuts are summarized in the tabe 1. It is divided into the parts of odd and even number bades. Sma cavities occurred homogeneousy for a number of bades near the eading edge on suction side on the Caseσ.6-α3. In this case, cavitation is vibrates and rotates. However, it s possibe to Figure 2: Infuence of σ and α on time evoution of cavitation voume in each passage of a 3-bades cascade consider it quasi-steady and homogeneous because the ampitude of the vibration is very weak. Therefore, this case is excuded in tabe 1. Except for the N = 4 cases and Caseσ.2-α3, on the three remaining cases, the scae of cavity and the propagation speed varies depending on the fow parameters, but the quaitative behavior is amost simiar. 3.1 Cascades of 3- and 4-bades A resut for 3-bades cascade is shown in this section. A part of this case was treated in the past research [6]. Fig. 2, show tempora evoution of the cavity voume Vc in each fow passage. A uniform and quasi-steady state cavitation is observed in Caseσ.6-α3 (C), but a weak rotating cavitation is observed in Caseσ.3-α3 (B) which owered σ. Furthermore, cavitation turns into quasi-steady state in Caseσ.2-α3 (A). For this case, cavitation turns into the super cavitation state, namey when the area of a cavity reaches a traiing edge. On the other hand, rotating cavitation is observed in Caseσ.6-α6 (D), when the incidence ange α is increased and the cavitation number σ is fixed 0.6. Moreover, in Caseσ.6-α9 (E), the cavity voume changes drasticay, and the wave pattern is aso distorted. In 3-bades cascade, for a unsteady state, the fow passages that cavitation voume Vc becomes biggest are moving to the rotating direction. Therefore, these are distinguished as forward rotating cavitation. In tabe 1, this is expressed with +1 in rotating direction, but -2 aso is wrote in the same coumn because the 1st fow passage is same the 2nd it to opposite direction in case of 3-bades cascade. The propagation becomes sower for arger Vc. In addition, when rotating cavitation is observed, a of it is partia cavitation. In Caseσ.3-α3, σ.6-α6 and σ.6-α9, the cavity rotates. There are cavities in a passages, except in Caseσ.6-α9. The back dashed ines in Fig. 2 show the mean of the cavity voume between a fow passages. Except in Caseσ.6-α9, the mean cavity voume in terms of time is amost constant. This anticipates that the whoe fow passage system is unstabe due to the variation of incidence ange, which becomes arger. 3

4 Tabe 1: Summary of cavity transfer patterns Number of bades N Case σ.2 - α3 σ = 0.2, α = 3 Case σ.3 - α3 σ = 0.3, α = 3 Case σ.6 - α6 σ = 0.6, α = 6 Case σ.6 - α9 σ = 0.6, α = 9 Quasi-Steady Asymmetric (ong period) +1, -4 (2.95) (ong period) +4, -3 (3.73) (ong period) +3, -8 (4.27) +1, -2 (0.86) +3, -2 (0.66) +5, -2 (0.52) +3, -8 (0.26) +1, -2 (0.94) +3, -2 (0.78) +5, -2 (0.63) +3, -8 (0.29) +1, -2 (1.15) +3, -2 (0.96) +5, -2 (0.79) +3, -8 (0.36) Aternate (ong period) Quasi-Steady (ong period) +1, -3 (3.69) Even +3, -1 (3.58) 2 (1.53) Osciating Pair Pair Pair 6 Pair +4, -2 (0.43) +4, -2 (0.47) +4, -2 (0.59) (ong period) 8 +3, -5 (4.76) +3, -5 (0.38) +3, -5 (0.43) +3, -5 (0.52) Numbers, m and ( ), indicate the order of appearance and the numbers in parentheses denote T/N. 4 In Fig. 3, the propagation phenomenon is considered based on the vapor voume fraction and the variation of the speed vectors. The soid ine near the cavity indicates the difference between the veocity vector and the time-mean fow. It s assumed that a cavity deveoped on the suction side of the B.2 at some moment (eft side of Fig. 3). The bocking effect is caused when the cavity is shrinking, then a new cavity can be generated in B.3, the next passage (midde of Fig. 3), but first, the cavity deveoping in B.1 reaches its maximum size. Namey, the propagation of the phenomenon is observed in the backward direction (B.2 B.3), but the biggest cavity is observed in the forward direction (B.2 B.1). As a typica exampe, see Fig. 4, the cavity voume in each passage between the bades is expressed by a soid ine, and the time evoution of the oca fow ange at 0.1C upstream side from eading edge is expressed by dashed ine. The oca fow ange increases in opposite direction of rotation when a cavity deveops in a certain bade, shown with ova marks. In other words, the phenomenon propagates backward through the dotted ine. However, as expained by Fig. 3, the biggest cavity foows the direction of the dash-dot ine between the passages. It is corresponding to +1 in forward direction, but it becomes -2 in backward direction. In Fig. 4, the 2 dotted ines and the dashdot ine are passing in a certain time. The expression, two pieces of information are propagating by Iga et a. [8], is consistent to the resut indicated by dotted ines of Fig. 4. Next, the resut of 4-bades cascade is discussed. Fig. 5 is the time change of cavity voume between each passage. Caseσ.3-α3 is quasi-steady and uniform as we as σ.6-α3. On the other hand, unsteady aternate cavitation occurs in Case σ.6-α6. The above-mentioned tendency accords with an observed resut by Fortes-Patea [9]. In Caseσ.6-α9 with increased α, and Caseσ.2-α3 with owered σ, the fuctuation is so ong that vapor-phase appears and disappears in some of the passage. Particuary in the atter, the mean of cavity voume at a passages aso fuctuates remarkaby, and unstabeness as a system is suggested. Figure 3: Typica pattern of cavity propagation (soid ines: fow; dashed ines: trigger; dash-dot ines: transfer) Figure 4: Time evoutions of oca fow ange (dashed ines) and cavity voume (soid ines) in each passage of N = 3; Case σ.6- α6 4

5 Caseσ.2-α3 Caseσ.6-α6 Caseσ.6-α9 Figure 5: Infuence of σ and α on time evoution of cavitation voume in each passage of 4-bades cascade (red ines: P.1-2; back ines: P.2-3; bue ines: P.3-4; sky-bue ines: P.4-1; back dashed ines: average) Figure 6: Chart for propagations of maximum cavitation ( ): trigger ( -1, dashed ines), basic mode ( -2, soid ines) and forward rotation ( +1, dash-dot ines) However, aternate cavitation is hard to distinguish whether meaning every 1 (±2) or rotating symmetry site (±180 ). The above-mentioned resuts were consistent to the previous studies [8] [9] for 3- and 4-bades cascade. However, it is difficut to concude a universa tendency because the infuence of the periodicity was strong in sma number bades. 3.2 Cascades of a prime number bades To reduce the infuence of the periodicity and find the generaity of unsteady cavitation, the resuts are compared for arger prime number of bades. In the tabe 1, there is a common characteristic about propagation of the rotating cavitation for Caseσ.3-α3, σ.6-α6, σ.6-α9 with N = 3, 5, 7. Namey, reative positions of the propagation of maximum cavity is +1 (N = 3), +3 (N = 5), +5 (N = 7) to the forward, respectivey, but it is aways `-2' to the backward. From Tabe 1, Fig. 3 and Fig. 4, in cases of N = 3, 5, 7, the cavity has an infuence on the incidence ange of passage -1, which is next to the backward side, but, the passage where the cavity carifies is -2 reativey. The forward rotating +1 observed in case of 3-bades is incuded in this universaity. In tabe 1, the correation caed -2 is not seen in case of arger prime numbers such as N = 11. This is because more than one cavitation is rotating at the same time, due to the increased number of bades. The method of discrimination is discussed in Sec Cavitation fuctuates for a very ong period in Caseσ.2-α3 for N = 5, 7, 11. The wave pattern resembes the condition of N = 4 (eft side of Fig. 5), but it becomes more compicated with the increase of N. On the other hand, the non-uniform and quasisteady state (Caseσ.2-α3 (A) of Fig. 2) seems to be imited to a cascade with a sma number of bades ike N = Cascades of an even number bades The probem if whether aternating cavitation mode exists staby or not is eft in case of even number bades. Here the resuts of 6- and 8-bades are considered. In case of 6-bades, a strong correation is observed in the two passages of rotating symmetry, namey, B.1 and B.4, B.2 and B.5, B.3 and B.6. The size of three sets of cavities becomes non-uniform in Caseσ.2-α3, and the cavity becomes big in a particuar set of passages facing each bade, but it doesn't rotate. On the other hand, three pairs rotate in Caseσ.3-α3, σ.6-α6 and σ.6-α9. From the viewpoint of rotating cavitation that the correation is strong as -2, the tendency is the same for N = 3, 5, 7. On the other hand, as for the pairs of cavity in the rotating symmetry, the infuence of N = 4 remains. In case of N = 6, not "the aternation" seen in N = 4, but the rotating pair manifests. In case of the 8-bades cascade, cavitation is not observed 5

6 aternatey, but the rotating cavitation becomes dominant. Namey, rotating cavitation is observed in the cascade of even numbers of bades. In cases of N = 4 and 6, they are ikey to form a pair. However, the simutaneous appearance of two cavities in the rotating-symmetry position can not be observed in N = 8. Thus the pair mode is thought to be a particuar phenomenon observed in sma even numbers. 3.4 Generaity of rotation cavitation A genera characteristic about Caseσ.3-α3, Caseσ.6-α6 and Caseσ.6-α9 at cases of the N = 3, 5, 7, 8, 11 is summarized in Tabe 2. The Fig. 6 diagrams of the propagation in cases of N = 3, 5, 8. The bades (marked ) in which the cavity reaches it maximum size on the suction side are tied, using dashed ines, soid ines and dash-dot ines. The trigger indicated by dashed ines means that the infuence is transmitted to a sequence of -1, the soid ine shows the basic mode -2. And the forward mode denoted by dash-dot ines shows the forward rotation +1. The three kinds of ines ike in Fig. 6, tie up the bades in which the biggest cavity occurs. The numbers of ines that go through the diagram is summarized in Tabe 2. Then, we can estabish a aw as foows: Tabe 2: Summary of cavity transfer patterns Number of Bades N Trigger M -1 Basic M -2 Forward M (i) M 1 M 2 M 1 (ii) After the biggest cavity occurs in bade I, the sequence that the biggest cavity appears in the backward 2nd is corresponding to the M 2 -th counting form I bade. In other words, we can find the number of simutaneousy rotating cavitations is M 2. The number of propagating information in 3-bades cascade is M 1. It is seen in Fig. 6 that the forward rotating +1 and the basic mode -2 tie the same bade to the opposite direction in N = Period of rotation cavitation Fig. 7 shows the period of rotating cavitation for various numbers of bades for some conditions. In the three cases of an odd number, T/N decreases for N ineary. Then, the traverse speed of the cavity is not constant in case of partia rotating cavitation. The period T becomes onger for arger N, which means the reduction of the infuence of periodicity, and for onger time-scae of each cavitation. In cases of an even number, when the number of bades increases, the cavitation is shifted from aternate mode to rotate mode. The resut for N = 8 gets coser to the characteristic of the odd number bades. Therefore, it coud be stated again that pair mode is a characteristic in case of sma number of bades based on this anaysis of periodicity. 4. CONCLUSION The numerica simuations of the cavitating fow in 2D cascades, which modeed a cross section of a iquid fue rocket engine inducer, were carried out. To consider the generaity in the rotating cavitation, we treated two series of bade numbers; namey prime numbers and even numbers in addition to the numbers, 3 and 4, in conventiona inducers. Figure 7: Period of rotating cavitation At first, the characteristic patterns of steady and unsteady fows observed in 3- and 4-bades cascade coud be reproduced. We therefore think our method can be used for quaitative discrimination of unsteady cavitation patterns by a 2D mode. The foowing generaities were observed in case of rotating patterns for the a bades numbers. (1) The enhanced cavitation causes next cavitation by increasing the oca fow ange at the eading edge of next bade to the backward direction (-1). (2) But it takes time for growing up, another cavity which has been deveoping in a different passage grows in advance. (3) The basic pattern of rotation is every-second-passage of backward direction. These characteristics incude the speciaty of sma number of bades and the cases of mutipe rotating ces in arge number of bades. On the other hand, the period of rotation is very ong in case of super cavitation and it is transformed to aternate or quasi-steady mode in case of sma numbers of bades. A generaity was found about rotating partia cavitation by making a chart that shows the trigger sequence of -1, the basic mode of -2 and the forward mode of +1. Besides we coud find by counting the number that each ines go through at a certain moment. As for the time scae of rotating cavitation, T/N decreased for number of bades N ineary. 6

7 ACKNOWLEDGMENTS This study was supported by the Grant-in-Aid for Scientific Research of the Ministry of Education, Cuture, Sports, Science and Technoogy Japan. It was aso party supported by a coaborative research proect with IHI Co. The authors express their gratitude to the contribution of Dr. Takashi Ohta and Dr. Kohei Okita. And the first author woud ike to acknowedge the Japanese Ministry of Education, Cuture, Sports, Science and Technoogy, MEXT, for financia support through schoarship program. NOMENCLATURE C Chord ength C, C Constants for cavitation mode f h g Voumetric fraction of iquid phase Pitch of cascade J x / Jacobian of coordinate transformation i M u / c Mach number (c: speed of sound) p pv p in Static pressure Vapor pressure Static pressure at far upstream Re Cu / v Reynods number ui u in Vc, U in Cartesian and contra-variant components of veocity Inet veocity Voume of cavity x, Cartesian and curviinear coordinates i v Incident ange Stagger ange of cascade Kinetic veocity of iquid Cavitation number [4] Watanabe, S., Sato, K., Tsuimoto, Y., and Kamio, K. 1999, Anaysis of rotating cavitation in a finit pitch cascade using a cosed cavity mode and a singuarity method, Trans. ASME, 121, [5] Ohashi, Y. and Kaishima, T. 2005, Numerica simuation of rotating cavitation in a 2-D cascade, Proc. of the 6th KSME-JSME Therma & Fuids Engineering Conf., Jeu, Korea, DG.01(CD). [6] Ohta, T. and Kaishima, T. 2008, Transition of different un-steady cavitating fows in 2D cascade with fat bades, Proc. 12th Int. Symp. on Transport Phenomena and Dynamics of Machinery, Honouu, Hawaii, USA, ISROMAC (CD). [7] Ugain, H., Kawai, M., Okita, K., Matsumoto, Y., Kaishima, T., Kawasaki, S. and Tomaru, H. 2006, Numerica simuation of unsteady cavitating fow in a turbo pump inducer, Proceedings of the 42 nd AIAA/ASME/SAE/ASEE Joint Propusion Conference, Sacramento, Caifornia, USA, AIAA (CD). [8] Iga, Y., Nohmi, M., Goto, A. and Ikohagi, T. 2004, Numerica anaysis of cavitation instabiities arising in the three-bade cascade, Trans. ASME, J of Fuids Engineering, 126, [9] Fortes-Patea, R., Coutier-Degosha, O., Perrin, J. and Reboud, J. L. 2007, Numerica mode to predict unsteady cavitating fow behavior in inducer bade cascades, Trans. ASME, J. of Fuids Engineering, 129, [10] Coutier-Degosha, O., Fortes-Patea, R., Reboud, J. L., Hakimi, N., and Hirsch, C. 2005, Numerica simuation of cavitating fow in 2D and 3D inducer geometries, Int. J. of Numerica Methods in Fuids, 48, [11] Yamanishi, N., Fukao, S., Qiao, X. G., Kato, C. and Tsuimoto, Y. 2007, LES simuation of backfow vortex structure at the inet of an inducer, Trans. ASME, J. of Fuids Engineering, 129, [12] Okita. K. and Kaishima, T. 2000, Numerica investigation of unsteady cavitating fow around a rectanguar prism, Proceedings of the 4 th JSME-KSME Therma Engineering Conference, Kobe, Japan, 2, [13] Chen, Y.-L. and Heister, S. 1995, Two-phase modeing of cavitated fows, Computers & Fuids, 27, No.7, Liquid density REFERENCES [1] Kamio, K. and Yoshida, M. 1991, Experimenta study of LE-7 LOX pump inducer, Trans. JSME, 57, Series B, 544 (in Japanese), [2] Tsuimoto, Y., Kamio, K. and Yoshida, Y. 1993, A theoretica anaysis of rotating cavitation in inducers, Trans. ASME, J. of Fuids Engineering, 115, [3] Tsuimoto, T., Watanabe, S., Kamio, K. and Yoshida,Y. 1996, A noninear cacuation of rotating cavitation in inducers, Trans. ASME, J. of Fuids Engineering, 118,