IOP Conference Seres: Earth and Envronmental Scence Desgn optmzaton of a hgh specfc speed Francs turbne runner To cte ths artcle: Y Enomoto et al 2012 IOP Conf. Ser.: Earth Envron. Sc. 15 032010 Vew the artcle onlne for updates and enhancements. Related content - Numercal smulaton of turbulence flow n a Kaplan turbne -Evaluaton on turbne performance predcton accuracy- P Ko and S Kurosawa - Desgn optmzaton method for Francs turbne H Kawar, Y Enomoto and S Kurosawa - Expermental and numercal nvestgaton of unsteady behavor of cavtatng vortces n draft tube of low specfc speed Francs turbne Y Tamura, K Tan and N Okamoto Recent ctatons - A revew of cavtaton n hydraulc machnery Xan-wu LUO et al Ths content was downloaded from IP address 37.44.207.9 on 24/12/2017 at 19:43
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 Desgn optmzaton of a hgh specfc speed Francs turbne runner Y Enomoto, S Kurosawa and H Kawar Toshba Corporaton, 20-1 Kanse-cho, Tsurum-ku, Yokohama, 230-0034, Japan E-mal: yasuyuk.enomoto@toshba.co.p Abstract. Francs turbne s used n many hydroelectrc power statons. Ths paper presents the development of hydraulc performance n a hgh specfc speed Francs turbne runner. In order to acheve the mprovements of turbne effcency throughout a wde operatng range, a new runner desgn method whch combnes the latest Computatonal Flud Dynamcs (CFD) and a mult obectve optmzaton method wth an exstng desgn system was appled n ths study. The valdty of the new desgn system was evaluated by model performance tests. As the results, t was confrmed that the optmzed runner presented hgher effcency compared wth an orgnally desgned runner. Besdes optmzaton of runner, nstablty vbraton whch occurred at hgh part load operatng condton was nvestgated by model test and gas-lqud two-phase flow analyss. As the results, t was confrmed that the nstablty vbraton was caused by oval cross secton whrl whch was caused by recrculaton flow near runner cone wall. 1. Introducton Hydroelectrc power generaton s one of the envronment-frendly power generaton systems compared wth other exstent electrc power generatng equpment and extremely outstandng renewable energy. In partcular, Francs turbnes are the most wdely used n varous hydraulc machnes. The demands for mprovement of turbne performance n wder operatng range have been ncreasng n recent years. Man characterstc of turbne performance are 1) effcency, 2) cavtaon and 3) pressure fluctuaton. In order to mprove the turbne performance such as effcency and cavtaton, runner shape optmzaton of varous optmzaton methods, for example a genetc algorthm and desgn of experments, were carred out for balancng the load dstrbuton on the runner blade surfaces [1-4]. On the other hand, recently the pressure fluctuaton whch occurred at hgh partal load operatng condton was reported[5-6]. Ths phenomenon occurs n a very small doman and s accompaned by very bg nstable vbraton. Therefore, t s mportant to nvestgate the mechansm of the nstablty vbraton phenomena and fnd out the countermeasures. Ths phenomenon s caused by runner outlet whrl whch has oval cross secton. However, the mechansm of outbreak of ths oval cross secton whrl was not solved. In ths study, we have two purposes. One s optmzaton of runner shape whch has hgh effcency, and another one s nvestgaton of ths nstablty vbraton phenomenon. As for the optmzaton of the runner shape, a new runner desgn method whch combned the latest CFD and a mult obectve optmzaton method wth an exstng desgn system was ntroduced. In ths study the optmzaton runner shape was conducted for hgh specfc speed about 0.21. As for the nvestgaton of nstablty Publshed under lcence by IOP Publshng Ltd 1
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 vbraton phenomenon whch occurred at hgh partal load operatng condton, model test and CFD analyss was conducted. CFD analyss were conducted usng compressble cavtaton analyss based on LES turbulence model. By usng ths model, CFD analyss can successfully catch the oval cross secton whrl and be able to help nvestgatng the mechansm of outbreak of oval cross secton whrl. 2. Optmzaton runner shape 2.1. Methodology Conventonal hydro turbne runner desgn s tuned by usng follow method. At frst, the runner shape s optmzed at desgn pont (maxmum effcency pont). Next, ths runner performance at other operatng ponts s confrmed and tuned runner shape based on the results. In ths case, CFD s performed, but ths method depends on the skll of desgner too much. Whle n ths study, turbne effcency on whole operatng range were chosen n order to evaluate turbne performance wth changng varous runner desgn parameters. Ths whole optmzaton method s shown n fgure 1. In ths method, turbne effcency at 12 operatng ponts from partal load to over load and from hgh head to low head were defned as mult obectve functons and total weghted effcency was evaluated. It s possble to explore optmzed runner shape effectvely and systematcally by ths method. The concept of desgn system s shown n fgure 2. At frst, whole flow passage analyss from spral case nlet to draft tube outlet was conducted for conventonal turbne. Next some runner desgn parameters to defne a runner shape were chosen by usng Desgn of Experments (DOE) and all matchng of runner desgn parameters were executed on runner desgn and CFD. And then turbne performance predcton was carred out based on CFD results and the optmzed runner shape was decded by senstvty analyss of desgn parameters. In the search of optmzed runner shape, optmzed desgn parameters were sought n order that mult obectve turbne effcency values were maxmzed and restrcted condtons were suffcent. The ponts of parameter desgn method, whch manly nvolves evaluated functons and restrcted condtons, are as follows. Contour lne of effcency Dscharge factor QED Dscharge lne Speed factor n ED Operatng range Fgure 1. Whole optmzaton for hydro turbne Whole flow passage analyss Runner desgn parameter settng by DOE CFD analyss Runner + Draft tube Senstvty analyss Optmzaton of desgn parameters Desgn and CFD of NEW runner Evaluaton of CFD results Fgure 2. Optmzaton desgn system 2.2. Parameter desgn ponts Desgn parameters were defned based on consderng followng ponts. 1) As for whole runner blade shape optmzaton, pressure dstrbuton on the blade surface was gradual from nlet and outlet n order to restrct ncreasng hydraulc loss. 2
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 2) In the optmzaton of the blade nlet shape, the occurrence of flow separaton and nlet cavtaton was reduced and good turbne performance was acheved n wder operatng range. 3) In the optmzaton of the blade outlet shape, outlet angle and outlet wdth dstrbutons, secondary flow was decreased n wder operatng range to acheve hgh turbne effcency. 4) In the optmzaton of outlet wdth dstrbutons and runner band shape, mprovements of turbne effcency and ant-cavtaton characterstcs had been consstent. 2.3. Evaluated functon and restrcted condton Evaluated functons were defned as hydraulc effcency that was calculated based on hydraulc losses at 12 operatng ponts. Restrcted condton was defned as the pressure coeffcent on the blade surface (mnmum value). 2.4. CFD analyss method When the optmzaton of runner shape s carred out by usng CFD, calculaton tme and precson of the solutons become mportant. In order to mprove CFD accuracy, t s desrable to carry out analyss of the whole flow passage, but t nvolves great deal of tme. Therefore when the runners shape optmzaton s carred out, the analyss doman s lmted to the regon from runner to draft tube. The boundary condton of the runner nlet was set applyng the velocty dstrbuton obtaned from CFD results of a whole flow passage from the spral casng to draft tube. 2.4.1. Whole passage flow calculaton. As for the whole flow passage analyss, turbulence flow smulaton based on Reynolds Averaged Naver-Stokes (RANS) equaton was adapted. As for RANS smulaton, t s mportant to employ the ansotropy of the Reynolds stresses n the turbulence model snce the secondary flow and rotatonal force effect are strong n runner and draft tube flow. Therefore Reynolds-Stress model (RSM) s adopted as the turbulence model n the numercal model test. RSM model approxmates RANS equatons by solvng the transport equatons for Reynolds stresses, together wth an equaton of the dsspateton rate. As for the wall modelng, the non-equlbrum wall functon s used. Ths wall functon s more sutable n regons, where the mean flow and turbulence are subected to severe pressure gradents and change rapdly, and t s effectve for the off-desgn flow smulaton, whch s nvolvng the separaton and reattachment. The non-equlbrum wall functon s based on two layers concept n computng the budget of turbulent knetc energy at the wall adacent cells, whch s needed to solve the k equaton at wall-neghborng cells. The computatonal models are shown n fgure 3(a). The number of grd ponts s about 14 mllon. The computatonal boundary condtons were appled at the nlet surface and at the outlet surface of the computatonal doman. About the nlet boundary condton, t was assumed the unform velocty dstrbuton. As for the outlet boundary condton, the average pressure was set fxed. Furthermore, about the surface of the wall, the non-slp boundary condton was prescrbed,.e. the velocty components were set to zero. 2.4.2. Runner shape optmzaton. Fgure 3(b) shows computatonal doman whch s used n runner shape optmzaton. The three-dmensonal Reynolds-averaged Naver-Stokes code was used to calculate the flow. The dscretzaton of the governng equaton was done by fnte volume method, and the convectve terms were approxmated by Self Fltered Central Dfferencng scheme. RNG k-ε turbulence model, whch had been confrmed accurate n predcton of many hydro turbne and pumpturbne, was appled to calculate the Reynolds stress (Ref 7). As mentoned above, the nlet boundary condton at runner nlet was set applyng the velocty dstrbutons obtaned from whole flow passage analyss. Outlet boundary condton was defned as the fxed average pressure. The number of grd ponts s about 5,500,000. 3
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 (a) for whole passage flow calculaton (b) for runner shape optmzaton Fgure 3. Overvew of computatonal models 2.5. Runner shape optmzaton In order to carry out runner shape optmzaton by DOE, flow-passage shape must be defned by parameter values, so geometry defnton programs were used to desgn the runner shape. As for the runner shape defnton program, ts 3D shape s defned by desgn parameters, and runner merdan passage and blade shape can be generated accordngly by changng the parameters. In ths study, 8 desgn parameters were used to optmzng the varables such as blade nlet dameter, blade nlet angle, runner crown shape, runner band shape and so on. Fgure 4 shows runner shape. Where the conventonal runner was optmzed based on conventonal optmzaton method. In ths method runner shape was optmzed by usng CFD. However, boundary condton of runner nlet was used obtaned by sngle component flow analyss of stay vane and gude vane. The velocty dstrbuton was dfferent from the velocty dstrbuton that a whole flow passage analyss result provded. Thus, the optmzed runner shape s dfferent from conventonal runner n the shape of the blade nlet. Fgure 4 shows runner shape and surface flow near the desgn pont of conventonal and optmzed runner. Fgure 4 also shows surface flow near desgn pont of conventonal and optmzed runner. As for the conventonal runner, a secondary flow occurs from runner crown sde to runner band sde near the leadng edge. However a secondary flow s restraned n the optmzed runners because the runner shape was optmzed based on nlet boundary condton obtaned from whole flow passage analyss. (a) Conventonal runner (b) Optmzed runner Fgure 4. Runner shape and surface flow Fgure 5 shows velocty contour n draft tube. There was not the large dfference n both velocty contours, and the loss n draft tube was almost same. 4
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 Hgh Low (a) Conventonal runner (b) Optmzed runner Fgure 5. Velocty contours n draft tube 2.6. Model test In order to confrm the optmzed runner performance, model test was conducted. The model runner was made based on CFD analyss results and tested turbne effcency characterstcs. Fgure 6 shows model turbne vew of the model test. Turbne dscharge characterstcs on desgn speed factor are shown n Fgure 7. In ths fgure, the broken lne ndcates model test result of conventonal runner and the sold lne ndcates the optmzed runner s one and vertcal axs and horzontal axs ndcate non-dmensonal value based on the maxmum effcency pont of conventonal runner. The effcency of the optmzed runner was hgher than the conventonal one n whole operatng range. It was confrmed that ths optmzng method was a useful engneerng tool of a Francs turbne development. 1.02 Turbne effcency η/η0 (-) 1.00 0.98 0.96 0.94 0.92 0.90 Model Test Results (Optmzed runner) Model Test Results (Conventonal runner) CFD (Optmzed runner) CFD (Conventonal runner) 0.60 0.70 0.80 0.90 1.00 1.10 1.20 Dscharge Q ED /Q ED0 (-) Fgure 6. Vew of model turbne Fgure 7. Model turbne performance 3. Investgaton of nstablty vbraton of Francs turbne Recently the case whch pressure fluctuaton enlarges wth the hgh partal load of the Francs turbne s reported. In ths study, n order to nvestgate the nstablty vbratons whch occur at hgh partal load operatng condton n Francs turbne, a model test and a flow analyss were conducted. The runner whch causes especally large nstablty vbraton was selected. The speed factor of maxmum effcency pont was 0.4. 3.1. Investgaton of nstablty vbraton by model test The model test was conducted to nvestgate the area and phenomenon of the nstablty vbratons. In ths test measurement of pressure fluctuaton wth 8 locatons ((1)casng nlet, (2)(3) between runner and gude vane, (4)(5)upper draft tube, (6)draft elbow, (7)(8) draft elbow ext) and the observaton of whrl by the hgh-speed vdeo camera were conducted. Fgure 8 shows schematc vew of model test equpment. The test was conducted wth 3 gude vane openngs (a M =100%, 88%, 78%), whch are normalzed by the gude vane openng correspondng to the optmum operatng pont. By changng speed factor ( n ED ) and Thoma number (σ M ), the area that nstablty vbratons occurred was pnponted. Fgure 9 shows the area that nstablty vbraton occurred. As an example, fgure 10 shows the pressure fluctuaton under the condton of a M =88%, n ED =0.44. From ths fgure, the pressure 5
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 fluctuaton at casng, and between runner and gude vane ncreases wthnσ M condton from 0.15 to 1.5 0.27., :Pressure transducer for pressure Fgure 8. Schematc vew of model test equpment Pressure Fluctuaton H/H (%) 10.0 8.0 6.0 4.0 2.0 0.0 (2),(3) (4) (5) (7),(8) (6) (1) Hgh speed vdeo camera Relatve Dscharge Q ED /Q ED0 1 0.5 0.3 0.4 0.5 0.6 Speed factor n ED a M =100% a M =88% a M =77% Fgure 9. Instablty vbraton area 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Thoma number σ M Maxmum effcency pont Casng nlet Between runner and gude vane 1 Between runner and gude vane 2 Upper draft tube 1 Upper draft tube 2 Elbow Elbow ext 1 Elbow ext 2 Instablty vbraton area Fgure 10. Pressure fluctuaton at a M =88% and n ED =0.44 Casng nlet Between runner and gude vane 1 Between runner and gude vane 2 Upper draft tube 1 Upper draft tube 2 Elbow Elbow ext 1 Elbow ext 2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Tme (s) Fgure 11. Pressure fluctuaton wave form Fgure 11 and fgure 12 show the pressure fluctuaton wave form and the FFT analyss results at the condton of a M =88%, n ED =0.44,σ M =0.24. And observaton results of runner outlet are also shown n Fgure 13. From FFT analyss results, t can be seen that, there are about 5 Hz domnant frequency at upper draft tube and 39.6 Hz domnant frequency at casng nlet and between runner and gude vane. The rotatonal speed of runner (n M ) s 17.6 s -1. The 5 Hz domnant frequency at upper draft tube s about one thrd of rotatonal speed of runner, therefore the frequency was caused by whrl rotaton. However the 39.6Hz domnant frequency at casng nlet has no relaton to these runner rotatonal 6
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 speed and whrl rotatonal speed. From fgure 13 t can be found that the cross-secton of the whrl s an oval and the oval cross secton whrl rotates 1 round n the whrl core crcumference n about 0.05 seconds (about 20 Hz). So the oval cross secton whrl knocks the wall surface at 1 round twce and the frequency s about 40 Hz. Ths frequency s near the domnant frequency at casng nlet and between runner and gude vane. It s thought that the nstablty vbraton s caused by ths oval cross secton whrl. Ampltude(V) Ampltude (V) Ampltude (V) 5.E-02 0.E+00 5.E-02 0.E+00 5.E-02 0.E+00 n M =17.6Hz Fgure 12. FFT analyss results Casng nlet Between runner and gude vane 1 Between runner and gude vane 2 0 10 20 30 40 50 60 70 80 ( Upper draft tube 1 Upper draft tube 2 0 10 20 30 40 50 60 70 80 Elbow Elbow ext 1 Elbow ext 2 0 10 20 30 40 50 60 70 80 Frequency f (Hz) T=0.0s T=0.016s T=0.032s T=0.051s Fgure 13. Observaton results at runner outlet 3.2. CFD analyss In order to nvestgate the mechansm of outbreak of oval cross secton whrl, CFD was conducted. 3.2.1. Governng equaton. Governng equaton was used compressble Naver-Stokes Equatons whose term was transformed by Gausan Flter. u t + u = 1 p (u u + τ) = + ρ 0 u ν u + (1) (2) where u s averaged velocty, p s averaged pressure, ρ s flud densty, ν s knematc vscosty and τ s subgrd-scale s stress tensor. 7
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 τ = u u u u (3) 3.2.2. Turbulence model. Subgrd-scale Reynolds stress was formulated by Smagornsky model. 1 τ δ τ kk = 2ν SGS S (4) 3 where ν SGS = ( C Δ) s 2 1 u = 2 Cs s Smagornsky constant and Δ s a bandwdth flter. S 2 u + 2S S (5) 3.2.3. Cavtaton modellng. It s known that when cavtaton occurs, volume of the gas phase ncreases, and speed of sound suddenly decrease. Therefore t s mportant to consder the compressble condton and the speed of sound change Cavtaton model was used locally homogeneous compressble medum model whch use follow equatons (8). Fgure 14 shows the relaton between vod of fracton and speed of sound. 1) Transport equaton of vapor mass fracton Y Y t + u Y = s (6) s = < = (1 ) (7) 2 = (1 ) 2 where Pv* s saturated vapor pressure, Ts s condensng temperature and A s nterfacal area concentraton. Subscrpt l means lqud and v means gas. 2) State equaton (soluton of speed of sound) = ( + ) (1 )(+ ) + ( + ) (8) where, = 1944.61, =( + +)/, K =2aT + b, a = 3.353 10 p, b = 1.723 10 p, c = 1.222 10 p, and R = 461.6J/kg/K. 8
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 By usng these parameter and state equaton of lqud (9) and state equaton of vapor (10), the common pressure of lqud and gas was defned (11). 3) Calculaton of speed of sound + = ( + ) (9) = (10) = +(1 ) ( + ) (11) (12) (13) (14) (15) Fgure 14. The relaton between vod of fracton and speed of sound (8) The dscretzaton of the basc equatons was done by fnte volume method. Convectve terms were approxmated by the two order central dfferences, tme was approxmated by second order mplct method. As for numercal algorthm to solve the algebrac fnte volume equatons, the SIMPLE method was used. The analyss doman was modeled from the stay vane to draft tube ext. Fgure 15 shows computatonal doman. The total number of grd s about 700 mllon and boundary between statonary part and rotatng part was connected usng sldng mesh boundary condton. 9
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 Fgure 15. Computatonal doman Fgure 16 shows Iso-surface of vod fracton and fgure 17 shows velocty vectors on merdan plane. It can be seen from these fgures that the oval cross secton whrl occurred near the runner cone wall surface. And the oval cross secton whrl s formed by the centrfugal force by the leanng of the runner corn shape and the dynamc pressure of the manstream drecton. Probably the reason why the nstablty vbraton occurs at hgh partal load condton s that, f the dscharge decreases less than ths operatng condton, the runner outlet flow tends to outer sde and the boundary layer of the runner corn wall surface developed, so t cannot mantan the recrculaton of oval cross secton near the runner cone where the startng pont of the whrl. Therefore, t s thought that the control a recrculaton flow near a runner corn wall surface s effectve to prevent the form of oval cross secton whrl whch causes nstablty vbraton. Dstrbuton of vod fracton Iso-surface of vod fracton(vod fracton =0.80) Fgure 16. Iso-surface of vod fracton Fgure 17. Velocty vectors on merdan plane 4. Conclusons In order to develop hgh performance turbne for hgh specfc speed Francs turbne, the optmzaton of runner shape and the nvestgaton of mechansm of nstablty vbraton phenomena was conducted. The results are obtaned as follows: 1) Ths optmzaton system was a useful engneerng tool of a Francs turbne development and the optmzed runner had hgh turbne effcency n whole operatng range. 2) When the nstablty vbraton at hgh partal load condton s caused, an oval cross secton whrl occurred. Ths whrl may be caused by the recrculaton flow near the runner cone wall. As for the nstablty vbraton, we wll conduct forward examnaton more and am gong to report measures. Nomenclature D M Model runner dameter H M Model test head n M Rotatonal speed of runner [m] [m] [m/s] Δ H ΔH / H a M Root-mean-square of Pressure fluctuaton [m] Pressure fluctuaton (= 2 2 Δ H / H ) [%] Gude vane openng [%] 10
26th IAHR Symposum on Hydraulc Machnery and Systems IOP Conf. Seres: Earth and Envronmental Scence 15 (2012) 032010 IOP Publshng do:10.1088/1755-1315/15/3/032010 g n ED σ M Gravty acceleraton [m/s 2 ] Speed factor ( =n M D M /(H g) 0.5 ) Thoma number ( =NPSH/H ) T Q M Q ED Tme [sec] Dscharge [m 3 /s] Dscharge factor ( =Q M /(D 2 M (H g) 0.5 ) ) References [1] Mazzou F et al. 2004 Multcrtera optmzaton: vscous flud analyss mechancal analyss 22nd IAHR Symp. on Hydraulc Machnery and Systems (Stockholm, Sweden, 2004) [2] Enomoto Y et al. 2006 Desgn Optmzaton of a Hgh Specfc Speed Francs Turbne Runner usng Mult-Obectve Genetc Algorthm 23rd IAHR Symp. (Yokohama, Japan, 2006) [3] Manfred S 2000 The Desgn of Francs Turbne Runners by 3D Euler Smulatons Coupled to a Breeder Genetc Algorthm 20th IAHR Symp.( Charlotte,North Carolna, August 2000) [4] Laurent T et al. 2002 Automated Desgn of a Francs Turbne Runner Usng Global Optmzaton Algorthms 21st IAHR Symp. (Lausanne,Sweden, September 2002) [5] Sh Q H Expermental Investgaton of Upper part Load pressure Pulsatons for Three Gorges Model Turbne Proc.24 th IAHR Symp. (Parana,Braz, 27 October 2008) [6] Koutnk J, Fagle P, Moser W, Pressure Fluctuatons n Francs Turbnes Theoretcal Predcton and Impact on Turbne Proc.24 th IAHR Symp. (Parana,Braz, 27 October 2008) [7] Sugshta K et al. 2001 Rehabltaton of 57.4MW Francs Turbne by Usng CFD Analyss J. Waterpower & Dams September 2001 [8] Sato Y, Takam R, Nakamor I, Ikohag T, Numercal analyss of unsteady behavor of cloud cavtaton around a NACA0015 fol, Computatonal Mechancs 2007 11