The cavitation behavior with short length blades in centrifugal pump

Transcription

1 Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~ DOI /s The cavitation behavior with short ength bades in centrifuga pump Quangnha Thai and Changjin Lee * Department of Aerospace Information Engineering, Konkuk University, 1 Hwayang-dong, Gwangjin-gu, Seou , Korea (Manuscript Received September 3, 2009; Revised May 31, 2010; Accepted June 23, 2010) Abstract A CFD code with 2-D cascade mode was deveoped to predict the cavitation behavior around the impeer bades of impeer in a centrifuga pump. The governing equations are the two-phase Reynods Averaged Navier-Stokes equations in a homogeneous form in which both iquid and vapor phases are treated as incompressibe fuid. To cose the mode, a standard k-ε turbuence mode is introduced. And the mass transfer rates between iquid and vapor phases are impemented as we. The vaidations are carried out by comparing with reference data in impeer of a centrifuga pump impeer. The cavitation characteristics of current centrifuga pumps is tested at an ondesign point (V=8 m/s) and two off-design points (V=20 m/s and V=30 m/s), respectivey. The criteria of cavitation and fow instabiity around bades are presented. The resuts show that the current centrifuga pump can safey operate without cavitation at on-design point. Aso, the simuation shows cavitation deveops inhomogeneousy among the bades at off-design points. Moreover, the effects of additiona bades in the impeer are studied as we. From the numerica resuts, it is expected that a haf-ength bade is the optimum configuration as additiona bades in cavitation point of view. Keywords: Cavitation; Haf-ength bade; 2-D cascade; Centrifuga pump; Impeer Introduction Cavitation is a phenomenon in which iquid evaporates and vapor bubbes occur in the region where the pressure of the iquid fas off under vapor pressure. It is usuay observed in high-speed fuid machinery such as propeers, pumps, and impeer bades where the fow acceerates and the pressure decreases. Cavitation can give rise to erosion damage, noise, vibration and hydrauic performance deterioration by periodic inception, growth, depetion of vapor bubbes. Speciay, cavitation in a centrifuga pump reduces efficiency and causes pressure head damage. The understanding of cavitating fow is one of the common subjects for designers of high-speed fuid machinery. The fow in a centrifuga pump is intrinsicay turbuent, 3-D and unsteady; sometimes cavitation appears. The design of centrifuga pump is mainy based on the steadystate theory, empirica correation, combination of mode testing, and engineering experiences. Over the ast few years, however, with the deveopment of the computer, there have been many researches on the centrifuga pump in numerica cacuation. Croba et a. [1] considered a more reaistic through 2-D, unsteady, incompressibe and turbuent fow. Anagnostopouos (2006) [2] simuated the 3-D turbuent fow This paper was recommended for pubication in revised form by Associate Editor Won-Gu Joo * Corresponding author. Te.: , Fax: E-mai address: cjee@konkuk.ac.kr KSME & Springer 2010 in a centrifuga pump impeer with Cartesian grid to represent an adequate accuracy for the compex geometry of the centrifuga pump impeer. Cheah et a. (2007) [3] simuated the compex interna fow in a centrifuga pump impeer with six twisted bades by using 3-D Navier-Stokes code with a standard k-ε two-equation turbuence mode. Different fow rates were specified at the inet boundary to predict the characteristics of the pump such as impeer passage fow, fow separation and pressure distribution. As aforementioned, cavitation in an impeer is naturay a 3-D phenomenon. However, due to the quite intrinsic compex nature with phase change, a mathematica mode and numerica method are consideraby difficut to estabish. Generay, the pump is designed to operate with non-zero incidence ange at a nomina fow rate. Nevertheess, in the range of operationa conditions, the ange of attack of impeer bades remains very sma, because of variabe fow thread and the associated curvature of bades. The back fow at the inet may happen sometimes but it s not strong. So, the approach of a 2- D cascade can be adopted. The cavitation in a 2-D cascade cacuation has been investigated by many researchers so far, instead of 3-D cacuation. For exampe, Joussein et a. [4] simuated rotating cavitation and aternate bade cavitation occurring four-bade cycic cascade by appying 2-D unsteady numerica method incorporated with cavitation mode by barotropic state aw. Iga et a. [5] simuated propagating phenomena of cavitation, which corresponds to the rotating cavi-

2 2008 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~2016 tation through three-bade cycic cascade and discussed the difference of the resuts obtained in different conditions at the inet boundary. And the coaborative work of French researchers [6, 7] carried out the numerica and experimenta anaysis about cavitation behavior of four-bade inducer using a 2-D mode of unsteady cavitating fow in a bade cascade. For the reduction of computing time and anaysis in design, the 2-D cascade ooks ike a new technique to 3-D cacuation of cavitation anaysis. Based on the prediction of 2-D cascade, a cearer understanding can be expected for 3-D cavitation. This is the reason why the present paper woud ike to simuate cavitation around pump bades by using the 2-D cascade. We are interested in simuating the cavitation behavior around bades of impeer in centrifuga pump. The main objective is to estabish the criterion of cavitation inception, the infuence of the interaction among cavitations in fow passages. And, the effect of additiona bades in impeer of centrifuga pump is aso investigated by evauating the characteristics of cavitation incuding the area of cavitation region and the fuctuation of mass fow rate. The benefits and disadvantages when putting the additiona bades wi be discussed in this study as we. 2. Governing equations and numerica procedure 2.1 Governing equations The vapor-iquid fow is described by a singe-fuid mode which is treated as a homogeneous bubbe-iquid mixture. The set of governing equations under singe-fuid mode comprises the conservative form of Reynods-averaged Navier-Stokes equations, the k-ε two-equation turbuence cosure and a transport equation for the iquid voume fraction. The continuity, momentum and iquid voume fraction equations are written in Cartesian coordinate system as foows: rr ρmdv + ρmv. nds = 0 t V S (1) r rr r r r r ρ vdv vv nds T nds bdv t V m + ρ S m. =. + ρ S V m (2) r r α ( ) + = + V dv α S v. nds m& m& dv t V (3) where 2 u j ( ) ( ) u u i j T ij = p + µ + µ t δ ij µ µ t 3 x j x j xi The constitutive reations for the density and dynamic viscosity of the mixture are: ρ m = ρ α + ρ ( 1 α ) and µ = µ = µ α + µ ( 1 α ) v And the turbuent viscosity is defined as foows: m v µ t = 2 ρm C µ k ε 2.2 Cavitation mode Cavitation terms, based on Kunz et a. s mode [8], are used in this study. The evaporation and condensation rates are given as foows: m& m& C = dest ρvαmin 1 ρ ρu 2 [ 0, p p ] 2 t ( α ) 2 C + prod ρvα 1 = ρ t where C prod = 9x10 5, C dest = 3x10 4, t = Numerica procedure The discretization of the governing Eqs. (1), (2), (3) is done by using finite voume method. And the coocated grid system is used to aocate veocity components and dependent variabes. The convective and diffusive terms are differenced by upwind scheme and centra scheme, respectivey. The soution of pressure and veocity may show an unphysica osciation due to the use of coocated grid system and shoud be treated by using interpoation in the momentum equation to avoid osciation. Detais of the numerica description can be found in the reference [9] D cascade and boundary condition Anayzing the fow fied data wi provide deep insights into the fow mechanism. 2-D cascade fow was adopted as the methodoogy for fow anaysis in this study. The 2-D bade-tobade cascade was drawn by cutting the 3-D inducer geometry at constant radius equa to 70% of the tip radius as shown in Fig. 1. Detais are found in the reference [10]. The axia fow entrance combines the rotationa speed of the bades and wi resut in the reative veocity between fow and bades. In the present 2-D cacuation, the bades are considered stationary and the reative fow comes to bades with a certain ange of attack. The boundary conditions used in the present simuations incudes infow, outfow, non-sip, and periodic boundary condition. At the inet, the veocity and iquid fraction are imposed and the pressure is extrapoated from the interior points. At the downstream, pressure is imposed whie the other variabes are extrapoated. At the wa, the veocity is zero whie the other variabes are extrapoated from the interior points. Aong the ines which separate the bades, a variabes are extrapoated from the interior points. The periodic condition is appied between the 4th and 1st passages. Dummy ce must be imagined as simiar as possibe to the ces of the origina corresponding row to consider a periodic boundary v

3 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~ Fig D cascade modeing [10]. Fig. 2. Variabes transfer between segment 1 and segment 2 in periodic boundary condition. Fig. 3. Cascade domain and boundary condition. Fig. 4. Comparison of instantaneous fow fied through contour of iquid voume fraction. condition. The variabes must be transferred according to the arrows shown in the Fig. 2. Two ines on the segment 1 must be transferred to two corresponding ines on segment Code verification 3.1 Cavitating fow in a pump inducer with 2-D cascade mode A 2-D four-bade cascade is considered to check the vaidity of our in-house code. The computationa domain, boundary condition in the cacuation is described in Fig. 3. The bade cross section is Cark Y-6%; pitch chord ratio and stagger ange of hydrofoi are 1.62 and , respectivey. The working fuid is pure iquid water (α = 1) which comes to the bade cross section with the veocity of m/s and ange of attack of 4 0. Reynods number is fixed as 5x10 5 [11]. The pump inducer is operated under standard pressure and the cavitation number is 0.5. Fig. 4 shows instantaneous fow fied with cavity indicated by voumetric fraction and a fow passage fied with iquid is modified in gray coor instead of red coor in the present cacuation to highight the cavitation region. Since the bade rotation was captured by 2-D cascade modeing, the sequence of cavitation behavior can be examined by the variation of size on consecutive bades from the first. The cavitation on bade A 1 coapses on bade A 2 and then continues coapsing on

4 2010 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~2016 Fig. 5. Comparison of spectrum of Strouha number of ift coefficient fuctuation. bade A 3 or the cavitation on bade B 1 deveops on bade B 2, and then continues deveoping on bade B 3. The growth and depetion of cavitation causes fow instabiity in the fow. The good agreement of spectrum of Strouha number of ift coefficient fuctuation can be found by the comparison with reference resut [11] as shown in Fig. 5. The intensity of fuctuation is very arge at S t = 0.25 and then becomes smaer as the increase in frequency. This means that the sheet cavitations among the passages are inhomogeneous at the beginning of the cacuation, but graduay become more homogeneous due to the gradua decrease of the fuctuation. If the fow condition reaches to steady, the intensity of the fuctuation wi diminish to a ower imitation. In other words, the smaest fuctuation is obtained in steady. In this case, the fuctuation sti remains but the intensity is smaest. 3.2 Unsteady fow in impeer using 2-D cascade mode Recenty, Kitano Majidi soved the unsteady 3-D viscous fow in entire impeer and voute casing of a centrifuga pump and showed the resut of mass fow rate fuctuation through the bade passages [12]. The present paper woud ike, for verification purpose, to make sure that the in-house code can catch the characteristics of cavitation not ony on the hydrofoi in a pump inducer but aso on the bades in a centrifuga pump. The origina mode of a commercia impeer in reference [12] is shown in Fig. 6. It is shrouded and has five backswept Fig. 6. Bade passage and casing [12]. bades. The bade profie varies between the hub and the shroud. The bade ange at the inet varies from at the shroud to 30 0 at the hub; the bade ange at the outet is The outet diameters, outet passage width of impeer are 508 mm, 72.5 mm, respectivey. And the base circe diameters, voute width at the base circe of voute casing are 523 mm, 94.3mm, respectivey. At the design point, the mass fow rate, tota head, rotationa speed and specific speed are kg/s, m, 1482 rpm, 68 min -1 respectivey. Detais are found in Tabe 1 of reference [12]. A 1780x30 structured mesh is generated; the boundary condition and the numerica method of the cacuation are simiar to the previous one. The water fow comes to the inet of the passages with a veocity of 6 m/s. The centrifuga pump is

5 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~ Tabe 1. Operationa range of centrifuga pump. Temperature ( 0 C) Density (kg/m 3 ) Dynamic viscosity (kg/m.s) Veocity (m/s) Fig. 7. Unsteady mass fow rate through each passage. operated in standard pressure condition. The mass fow rate at right after bades in term of time is investigated to compare the present resut with reference. As seen in Fig. 7, the same tendency of unsteady fow rate in each passage can be observed athough the resut of the present cacuation is not so cose to the reference. The main reason for the difference between two resuts here is due to different methods adopted in each cacuation (3-D cacuation in reference paper versus 2-D cascade cacuation in the present paper). However, it can be seen that the 2-D cascade mode can catch a fairy good prediction of cavitation compared to the 3-D approach. 4. Resuts and discussion 4.1 Cavitation around the bades of impeer using 2-D cascade Fig D geometry and projections of impeer. Fig. 9. Configuration of pump impeer without and with haf-ength bades. The centrifuga pump in this study is for feeding fue JP-7; the configuration of impeer is shown as Fig. 8. The bades have two circuar arcs shape, 0.5mm thickness (s) and 18mm chord ength (c). The outer (d 2 ) and inner diameters (d 1 ) are 60 mm and 21 mm, respectivey. And the inet (β s1 ) and outet (β s2 ) ange of bades are 15 0 and There are 3 bades and 3 haf-ength bades with the passage width (h) of mm. The rotationa speed (n) is 5660rpm. This centrifuga pump usuay operates at the on-design point at which the inet veocity and standard temperature are 8 m/s and 20 0 C, respectivey. The operationa temperature range of this centrifuga pump varies from C to 43 0 C and corresponding vaue of density, dynamics viscosity and veocity tabuated as shown in Tabe 1. The pump impeers considered in the present paper are shown as Fig. 9. The cavitation behavior around bades of pump impeer without haf-ength bades is investigated at first and then, the pump impeer with haf-ength bades is aso considered to study the roe of haf-ength bades. The method and techniques appied in this cacuation are described in the previous section 2.4. Steady cavitation is investigated and the resut can be observed in Fig. 10. At the ondesign point, the resut shows there is no cavitation. However, symmetrica and stabe cavitation occurs at the suction side of each bade at the off design point. From the on-design point, increasing the veocity unti around 20 m/s, the inception of

6 2012 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~2016 Fig. 10. Cavitation shape of steady cacuation at on and off design point. Fig. 11. Criterion curve of cavitation inception. Fig. 12. Instantaneous fow fied represented by contour of iquid voume fraction at off-design point. cavitation was found. And steady cacuation is carried out again by graduay increasing veocity. Finay, we obtained the criterion curve of cavitation as shown in Fig. 11. The big back point and the rectanguar box depict the condition at the on-design point and the operationa range of this centrifuga pump, respectivey. It s obvious that this centrifuga pump operates in a safe region without appearance of cavitation. Sometimes the operating conditions of a centrifuga pump may not be at the on-design point due to rotating speed or temperature variation. So, the unsteady cavitating fow at certain off design point is studied as we and an arbitrary veocity of 30 m/s is picked up in the cacuation. The instantaneous fow fied represented by the contour of iquid voume fraction is investigated and the resut is obtained as Fig. 12. The sheet cavitation among bades at the suction side is not symmetrica anymore as in the steady cacuation. It deveops inhomogeneousy due to fow rate variation through the passages. The growth and depetion of cavitation are created and move from one bade to next bade according to the arrow in Fig. 12. And it causes the instabiity of the bade cascade. 4.2 The roe of haf-ength bades Another interesting issue has been studied to define the roe of the haf-ength bades in a centrifuga pump impeer. As aforementioned, the cavitation occurred at off-design points at which veocity is arge enough. The present paper is interested in defining the cavitation inception in terms of veocity and ength of additiona bades. So, the area of cavitation region is investigated in the cacuation. And Fig. 13 on the top shows the reduction of cavitation region due to the imposition of additiona bades. With haf-ength bades, we can expect the minimum cavitation area compared to the other cases. The bottom graphs gives a cear visuaization of cavitation shape in the case with haf-ength bades, and the back region was zoomed in to have a better observation as shown in the two bottom figures. It is obviousy seen that cavitation size at the veocity of 20 m/s is smaer than that at 30 m/s corresponding to the smaer vaue in the Fig. 13 on the top. In reaity, when cavitation occurs, there is back fow on the pressure side of the fu bades. The occurrence of cavitation is the main reason creating the backfow. The arger the cavitation region is, the stronger back fow is going to be. When some appropriate additiona bades are ocated between fu bades, the back fow from the pressure side of fu bades wi touch the suction side of the additiona bades. It seems to be that the back fow has something to catch and moves aong the curve of additiona bades suction side instead of moving back

7 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~ Fig. 14. Comparison of cavitation criterion curve. Fig. 13. Comparison of area of cavitation region and visuaization of cavitation at two off-design points. to the eading edge of fu bades. That s the reason why the back fow near the midde of fu bade pressure side can t be moved back to the eading of the bade and is pushed downstream to the additiona bade suction side. Then, the back fow between pressure side of fu bade and suction side of additiona bade is going to be weaker. Moreover, the fow has a tendency to move toward the traiing edge of the additiona bade by passing through its pressure side. These additiona bades indirecty reduce cavitation by reducing backfow or fow instabiity. The resut shows that the optimum ength of the additiona bade shoud be a haf of a fu bade. Another benefit of the haf-ength bades is found as shown in Fig. 14. The non-cavitation region is improved and a itte bit wider than that in the case without haf-ength bades. This centrifuga pump more safey operates. For exampe, in the case without the addition of three haf-ength bades into impeer and at standard operationa temperature condition (20 ), the inception of cavitation happens if veocity is increased up to 20 m/s. However, it doesn t appear yet if three haf-ength bades are added into this pump impeer. The pump sti operates freey from cavitation. In this operationa condition, the pump is sti safe from cavitation. This is one of the benefits of adding three haf-ength bades into impeer. The veocity which makes centrifuga pump susceptibe to cavitation can be deayed up to 23 m/s, instead of 20 m/s in case without haf-ength bades. The fuctuation of mass fow rate downstream of the bades is aso important in the safe operation of a centrifuga pump. That s why FFT anaysis of the fuctuation of mass fow rate is investigated as we in this paper. In Fig. 15, resuts at the on-design point are shown with various engths of additiona bades. Generay with additiona bades, the fuctuations are weaker than that without additiona bades. However, it is a itte higher at some certain frequency. At ow and high frequency, the comparison can be easiy observed in Fig. 16 and Fig. 17, respectivey. With haf-ength bades, the ampitude of the fuctuation around 2000 Hz is much smaer compared to the other cases. For a centrifuga pump, arge fuctuations at high frequency affect destructivey to the safe operation and structure. So, the addition of haf-ength bades contributes the fow to be stabe by eiminating high frequency osciations of mass fow rate.

8 2014 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~2016 Fig. 15. Comparison of mass fow rate fuctuation at on-design point. Fig. 16. Comparison of mass fow rate fuctuation at ow frequency at on-design point. Fig. 18. Comparison of mass fow rate fuctuation at ow frequency at off-design point V = 20 m/s. Fig. 17. Comparison of mass fow rate fuctuation at high frequency at on-design point. Fig. 19 and Fig. 21 aso show neary the same tendency of characteristics of ampitude at high frequency as observed with haf-ength bades at two different off-design points. However, higher fuctuation is observed at ow frequency around 500 Hz in Fig. 18 and Fig. 20. This centrifuga pump is sti fine even though it operates under unsteady conditions. By adding three haf-ength bades into the impeer, the centrifuga pump can be expected to have more stabe operationa characteristics for the entire operationa range. Within the operationa range, additiona bades contribute the fow to be more stabe by suppressing the high frequency fuctuations. Even for out of range, additiona bades reduce cavitation size, high frequency osciation and improve non-cavitation region.

9 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~ Fig. 19. Comparison of mass fow rate fuctuation of high frequency at V = 20 m/s. 5. Summary and concusion Based on a 2-D Kunz cavitation mode and standard k-e turbuent mode for cosure, a conventiona finite voume method has been impemented with a SIMPLE agorithm on the coocated body-fitted grid in the present study. A vaidation check has been carried out by comparing with reference data of the impeer in a centrifuga pump. The resuts show a good agreement with reference vaues and can successfuy provide the characteristics of cavitation on the bades of the impeer. A study of cavitation around the bades of target impeer suppying the fue JP-7 is carried out. The roe of the haf-ength bades in centrifuga pump impeer is aso investigated. Cacuation resuts confirm that the centrifuga pump design in this study was made free from cavitation if it operated within the design range. If the centrifuga pump operates due to temperature variation, the growth and depetion of cavitation are created among the bades and cause the instabiity of bade cascade. Moreover, the addition of three haf-ength bades pays an important roe to get better operationa characteristics of the impeer by reducing cavitation size, improving the cavitation free region and contributing the fow to be more stabe through the eimination of high frequency fow osciations. Acknowedgment This study was financiay supported by Hanwha with KHP program. Authors greaty appreciate the financia grant for this work. Nomencature Fig. 20. Comparison of mass fow rate fuctuation of ow frequency at V = 30 m/s. Fig. 21. Comparison of mass fow rate fuctuation of high frequency at V =30 m/s. α : Loca void fraction p, p v : Static pressure, vapor pressure µ : Kinetic viscosity µ t : Turbuent viscosity u i, u j : Cartesian veocity components x i, x j : Cartesian coordinates δ ij : Kronecker deta function C µ : Turbuent coefficient S : Contro surface V : Contro voume t : Time t : Time scae k : Turbuent kinetic energy ε : Dissipation rate C dest : Empirica constant C prod : Empirica constant U : Free stream veocity α : Ange of attack S t : Strouha number f : Frequency C L : Lift coefficient c : Chord ength of hydrofoi d 1, d 2 : Bade inet, outet diameter β s1, β s2 : Bade inet, outet ange h : Passage width s : Bade thickness

10 2016 Q. Thai and C. Lee / Journa of Mechanica Science and Technoogy 24 (10) (2010) 2007~2016 z : Number of bades n : Rotationa speed n r : Norma vector v r : Veocity vector b r : Body force vector ρ m, ρ, ρ v : Density of mixture, iquid and vapor, respectivey m & +, m& : Voume conversion rate of the condensation and of the evaporation References [1] D. Croba and J. L. Kueny, Numerica Cacuation of 2D, Unsteady Fow in Centrifuga Pumps: Impeer and Voute Interaction, Internationa Journa for Numerica Methods in Fuids, 22 (6) (1996) [2] J. S. Anagnostopouos, Numerica Cacuation of the Fow in a Centrifuga Pump Impeer Using Cartesian Grid, Proceedings of the 2nd WSEAS Int. Conference on Appied and Theoretica Mechanics, Venice, Itay (2006) [3] K. W. Cheah, T. S. Lee, S. H. Winoto and Z. M. Zhao, Numerica Fow Simuation in a Centrifuga Pump at Design and Off-Design Conditions, Internationa Journa of Rotating Machinery, 2007 (2007) 1-8. [4] F. Joussein, Y. Courtot, O. Coutier-Degosha and J. L. Reboud, Cavitating Inducer Instabiities: Experimenta Anaysis and 2D Numerica Simuation of Unsteady Fow in Bade Cascade, 4 th Internationa Symposium on Cavitation, Pasadena, CA, USA (2001). [5] Y. Iga, M. Nohmi, A. Goto, B. R. Shin and T. Ikohagi, Numerica Anaysis of Unstabe Phenomena of Cavitation in Cascade with Finite Bade Numbers, Proc. 9 th Internationa Symp. on Transport Phenomena and Dynamics of Rotating Machinery, Honouu, Hawaii, USA (2002) 1-6. [6] R. Fortes-Patea et a., A Numerica Mode to Predict Unsteady Cavitating Fow Behaviour in Inducer Bade Cascades, Journa of Fuid Engineering, 129 (2) (2007) [7] O. Coutier-Degosha, R. Fortes-Patea, J. L. Reboud, M. Hofmann and B. Stoffe, Experimenta and Numerica Studies in a Centrifuga Pump With Two-Dimensiona Curved Bades in Cavitating Condition, Journa of Fuids Engineering, 125 (6) (2003) [8] I. Senocak and W. Shyy, A pressure-based method for turbuent cavitating fow computations, J. Comput. Phys., 176 (2) (2002) [9] C. Lee and D. Byun, Cavitation Fow Anaysis of Axisymmetric Bodies Moving in the Water, Internationa Conference of Computationa Science and Its Appications ICCSA, Gasgow, UK (2006) [10] O. Coutier-Degosha, J. L. Reboud and G. Abano, Numerica Simuation of the Unsteady Cavitating Behaviour of an Inducer Bade Cascade, Proc. ASME Fuids Engineering Summer Conference, Boston Massachusetts, USA (2000). [11] K. Okita, Y. Matsumoto and K. Kamijo, Numerica Anaysis for Unsteady Cavitating Fow in a Pump Inducer, 5 th Internationa Symposium on Cavitation, Osaka, Japan (2003). [12] K. Majidi, Numerica Study of Unsteady Fow in a Centrifuga Pump, Journa of Turbomachinery, 127 (2) (2005) [13] Hiroki Ugajin, Masafumi Kawai, Kohei Okita, Takashi Ohta, Takeo Kajishima, Masataka Nakano and Hiroshi Tomaru, Numerica Anaysis of the Unsteady Cavitating Fow in a 2D Cascade and a 3D Inducer, 43 rd AIAA/ASME/SAE/ASEE Joint Propusion Conference and Exhibit, Cincinnati, Ohio, USA (2007) [14] F. C. Visser, J. J. H. Brouwers and J. B. Jonker, Fuid Fow in Rotating Low-Specific-Speed Centrifuga Pump Impeer Passages, Fuid Dynamics Research, 24 (1999) [15] O. Coutier-Degosha, J. L. Reboud, N. Hakimi and C. Hirsch, Numerica simuation of cavitating fow in 2D and 3D inducer geometries, Internationa Journa for Numerica Methods in Fuids, 48 (2) (2005) [16] R. F. Kunz et a., A preconditioned Navier-Stokes method for two-phase fows with appication to cavitation prediction, Computers & Fuids, 29 (8) (2000) [17] I. Senocak, Computationa methodoogy for the simuation of turbuent cavitating fows, Ph.D. Dissertation, University of Forida (2002). [18] R. Fortes-Patea et a., Numerica Mode to Predict Unsteady Cavitating Fow Behavior in Inducer Bade Cascades, Journa of Fuids Engineering, 129 (2) (2007) [19] D. Wicox, Turbuence modeing for CFD, DCW Industries, Inc, La Canada, CA (1993). [20] S. Patankar, Numerica heat transfer and fuid fow, Tayor & Francis, Kentucky, USA (1980). Quangnha Thai received his B.S. in Aeronautica Engineering from Ho Chi Minh City University of Technoogy (HCMUT, Vietnam) and Ecoe nationae superieure de mecanique et d aerotechnique (ENSMA, France) in He is pursuing his M.S. in Aerospace Information Engineering from Konkuk University in Seou, Korea. His research interests are in the area of computationa fuid dynamics of twophase fow and fow instabiity of rocket iquids. Changjin Lee received his B.S. and M.S. in Aeronautica Engineering from Seou Nationa University in 1983 and He then went on to receive his Ph.D. from University of Iinois at Urbana-Champaign in Dr. Lee is currenty a Professor at the department of Aerospace Engineering at Konkuk University in Seou, Korea. His research interests are in the area of combustion instabiities of hybrid, iquid rocket and jet propusions.