THREE DIMENSIONAL WATER FLOW IN NOZZLES

Transcription

1 8 th GRACM Intnational Congss on Comtational Mchanics Volos, 1 Jly 15 Jly 015 THREE DIMENSIONAL WATER FLOW IN NOZZLES Johanns V. Solis 1, Modstos A. Lokas 1 Flid Mchanics/Hydalics Division, Datmnt of Civil Engining, Dmocition Univsity of Thac, Xanthi, GR-67100, Gc -mail: solis@civil.dth.g Flid Mchanics/Hydalics Division, Datmnt of Civil Engining, Dmocition Univsity of Thac, Xanthi, GR-67100, Gc -mail: modlokas@yahoo.com Kyods: El Eqations, Finit-volm Mthod, Tbomachiny Flos. Abstact: A tim-maching finit-volm nmical ocd is sntd fo th-dimnsional incomssibl tbomachiny flos. Th cod D3flo is alid to th consvativ fom of th El qations ittn in gnal cvilina co-odinats. A siml bt comtational fficint gid is constctd. Nmical soltion slts fo 3D nozzl, shon in Figs 1, 3 and 4, a sntd and comad ith th Flnt ANSYS 14.5 ogam lasd in 01, Figs 5-1. Pdictd slts sing ith mthod yild satisfactoy comaison. Th oosd nmical mthod is an accat and liabl tchniq fo solving inviscid, incomssibl flo qations in 3D tbomachiny gomtis. 1 INTRODUCTION Tbomachiny dvlomnt of ith ms o tbins has bn lagly conditiond by th imovmnts achivd in comonnt fficincis. Dsign and fomanc stimation of ms and tbins has bn basd and ill contin to b so, almost comltly, on th ndstanding of flid flo bhavio. Hov, flo assags thogh tbomachins a gomtically vy comlx and th intaction btn otos and stationay ats constitts a majo challng. Comtational flid dynamics mthodology alid to tbomachiny flos has bn achivd a considabl ogss in th ast sval yas. Althogh fficint algoithms a no availabl to intgat th Navi-Stoks qations, this aas to b still a fomidabl task fo oss of actical alications. In th comlx tbomachin nvionmnt th diction accacy of sch flo calclations is limitd by th limitations of tblnc modlling. Mixing lngth ddy viscosity modls a by fa th most commonly sd mthod 1. Th soltion of inviscid flos El qations thogh tbomachiny assags is a good fist aoximation fo analysis. It is of actical intst fo th dsign of tbomachiny comonnts. Inviscid flo qations a nmically tatd in to distinct catgois, namly El solvs and otntial flo iotational solvs. Potntial mthods do not aa to hav bn idly sd fo dsign oss. Noadays, 3D El solvs a ll dvlod and a availabl fo otin tbomachiny calclations. Sval of ths daling ith intnal and xtnal flos a dscibd by Hisch. El solvs fo 3D tbomachiny flos hav bn otd as aly as 1974 by Dnton 3, ho dvlod an xlicit tim-maching mthod. His idly acctd mthod mloys an oosd diffnc schm in od to solv th El qations. Th schm ss ind diffncing fo flxs of mass and momnta, bt donind diffncs fo sss in th stamis diction. In addition, coction factos fo ach of th hysical qantitis a alid in th stamis diction. Th mthod is of th finit-volm ty. Nmical soltion tchniqs sntd by Shih and Dlanay 4. Th ho-scotch schm as alid to th consvativ fom of th El qations ittn in gnal cvilina co-odinats sing an O-ty gid systm. Wb t al 5 sntd a 3D El analysis on a C-ty gid sing th ll-knon Bam-Waming imlicit algoithm. Rslts fo a cascad and oto flos sntd. Ats 6 sntd an inviscid flo soltion fo axial tbin stag. Holms and Tong 7 dscibd a 3D El solv fo tbomachiny blad os. Th algoithm as basd on th xlicit, fo-st, Rng-Ktta finit-volm mthod advocatd by Jamson. Th objctiv of this a is to otlin an accat and fficint nmical ocd fo simlating th tim-avagd, 3D, inviscid at flo fild ithin a tyical tbomachiny nozzl. Th main sco as to

2 calclat flos thogh all tys of tbomachins axial, mixd, adial no matt ho comlx thi gomty may b. Th oosd schm has sval advantags: a. Th gid sd is th simlst ossibl fomation fo nmical calclations. b. Th consvativ fom of th qations is ittn in gnal cvilina co-odinats, ths nabling comlx gomty to b fficintly analyzd. c. Calclatd at mass flos into and ot of th nozzl a matchd. d. Bonday conditions a asily and accatly satisfid in a staight foad mann.. Th tim intgation nmical ocd is a staightfoad mthod qiing minimal algoithm coding. f. Atificial viscosity is ovidd via a siml ss coction fomla. Fo th tim bing alications a stictd to at flo thogh nozzls ith 3D comlx gomty. Th comt cod dvlomnt as fomd at th Comtation Laboatoy of th Flid Mchanics/Hydalics Division, Civil Engining Datmnt, Dmocition Univsity of Thac. GOVERNING FLOW EQUATIONS Th basic qations govning th flo insid of any tbomachin a divd fom th incils of consvation of mass, momntm and ngy. It is convnint to it th 3D El qations in a cylindical ola co-odinat systm z, θ,. Sinc th algoithm as dvlod fo comssibl flid analysis, it is convnint to it don th qations in comssibl flo taking into accont th otation and to tat th incomssibl flo as at of it fo at flo analysis. Ths qations a xssd in consvation fom as, ] z t 1, ] ] z t, ] z t 3, ] ] z t 4, ] ] ] z t 5 z, θ and a th axial, tangntial and adial dictions sctivly, t is th tim, ρ is th dnsity,, υ and a th absolt vlocity comonnts along th z-, θ- and - dictions sctivly, is th ss of th flid, Ω is th otational sd of th imll and is th total intnal ngy givn by th qation blo,, γ is th atio of th scific hats. Th nknons of th oblm a th six hysical qantitis, υ,, ρ, and. Th oblm mst b closd ith a comlt scification of th bonday conditions. Ustam conditions Th stagnation at th inlt flo is assmd to b constant thoghot th stamis diction. Donstam conditions Th static ss is st on th hb sfac and th adial ss distibtion is dtmind by th siml adial qilibim qation,

3 7 3 MESH GENERATION A comlx gid systm as dvlod and imlmntd on th main nmical mthod to allo flos insid comlx gomty to b calclatd. An xaml of a comlx gomty hich as sd fo analysis is shon in Fig 1a. Hb and casing adial co-odinats along th machin axis a ovidd as an int to th comt cod. Int coss-sctions a intolatd to calclat th axial, sction sfac and blad thicknss. Midional gid oints a nifomly sacd in th -diction. Axial oints nd not b qally sacd. Fig 1b shos th 3D gid sd fo nozzl analysis Flnt. Us of itchis lins gatly simlifis th alication of th iodic bonday otis btn th bonding qasi-stamlins of th assag. Hov, th nmical schm can b sd ith any gid fomation, hich nd not b nifomly sacd in any co-odinat diction. Th gid is not stictd to th on dscibd. Any msh gnation tchniq can b adotd. Edgs qi scial nmical tatmnt. Th oblm can b ovcom by fitting mo gid oints h aoiat. Th D3flo ogam tilizs 800 nods hil th Flnt nods. a b Fig 1. a Gomty of th 3D nozzl, b msh sd fo analysis Flnt 4 TRANSFORMATION EQUATIONS Th disct aoximation to th govning flo qation has bn dvlod by dividing th hysical domain into cboid clls hich can b dfind abitaily to odc sfac-fittd gids, th stct of hich follos th tbomachiny intnal configation. Onc this has bn achivd, a tansfomation is intodcd thogh hich cboids of th hysical domain a mad into comtational domain cbs, Fig, Solis 7. Fig. Distotd cbs of th hysical domain a mad into cbs of th comtational domain Th tansfomation fom global z, θ, to local ξ, η, ζ co-odinats can b xssd as,

4 8 Johanns V. Solis, Modstos A. Lokas 8 8 z N, i 1 i z i N, i 1 ii N, i 1 i i 8 N i a th fist-od, lina sha fnctions associatd ith th cboid nods, Solis t al. 8. Th s of fistod sha fnctions has bn dtmind by th ncssity to stict th comlxity of th nmical cod, hich is inhnt to almost all 3D comtational mthods. Ths, in od to nmically solv th systm of govning flo qations 1 6 on a body-fittd gid systm, th qations a tansfomd to an abitay cvilina systm: J J U J V J W, t 9 1 J J t 1 U z ] J 1 V z ] J 1 W z ], J J U / ] J V / ] t 1 J W / ] 1 J W, 11 J U ] J V ] t J 1 J W ] 1 J, J {J U / ]} {J V n / ]} t 1 {J W / ]}, 13 U, V and W a th contavaiant vlocity comonnts in th ξ-, η- and ζ- dictions, sctivly. Th invs Jacobian J -1 of th tansfomation fom th hysical to th local co-odinat systm is dfind, Solis 9, as, Th mtics ξ z, η z and ζ z, of q. 10 a, z z z 1 J z / J, z / J, z / J. 15 Similaly th mtics ξ θ, η θ and ζ θ, of q. 11 a, 1 z z / J, z z / J, z z / J, 16 th mtics ξ, η and ζ, of q. 1 a, 1 1 z z / J, z z / J, z z / J Th contavaiant vlocitis a latd to th hysical vlocitis by th qations, 1 1 1

5 U z, 18 V z, 19 W z. 0 In all th abov qations th sbscits z, θ and f to atial divativs. Thn, a nmical algoithm is bing sd to solv th govning flo qs. 9-13, Solis 10. Convgnc is achivd hn all hysical qantitis ls th mass balanc achs a 10-7 accacy. In od to achiv th dsid convgnc, th D3flo ogam ss 5 factos: a. FT =1.0 is an itation st facto dtmining th itation st. b. SF =0.01 is a smoothing facto acting on changs of th vlocitis and ss. c. OMIN =0.15 is a laxation facto acting on ss changs dating flo conditions at inlt. d. FACT1 =0.1 is a mltilication facto alid ov continity qation dating ss.. FACT =0.05 is a mltilication facto alid ov axial, tangntial and adial momntm qations dating th axial, tangntial and adial vlocitis. 5 GEOMETRY Th gomty of th 3D nozzl is shon in figs 3 sid vi and 4 to vi, Lokas 11. To constct th nozzl gomty th adis in qation 7 attains high val so as th ss vaiation acoss th adis at th otlt to b ngligibl. 60 mm z 50 mm 40 mm 30 mm 0 mm 10 mm Fig 3. Gomty of th sid vi fo th 3D nozzl 50 mm y 0 mm -50 mm -100 mm Fig 4. Gomty of th to vi fo th 3D nozzl

6 6 BOUNDARY INITIAL CONDITIONS To achiv th most favoabl comaisons btn alid mthods, th otlt D3flo bonday conditions a accodingly adjstd. Th bonday conditions fo th 3D nozzl oblm sing D3flo and Flnt tchniq a shon in tabl 1. Flnt D3flo Static ss at inlt Pa Pa Static ss at otlt Pa Pa Total ss at inlt Pa Pa Bondais Solid ndicla vlocitis a st zo Solid ndicla vlocitis a st zo Flid dnsity at ρ kg/m kg/m 3 Gavity acclation g 9.81 m/s 9.81 m/s Dynamic sha viscosity μ 0.0 kg/m-s 0.0 kg/m-s Tabl 1 : Bonday conditions sing D3flo and Flnt nmical tchniqs NOTES: a. Gavity acclation g has ngligibl ffcts on th soltion in ith ogam. b. Th flo is assmd to b inviscid μ=0.0 kg/m-s. c. Th otational sd Ω is st qal to zo. 7 COMPUTATIONAL RESULTS Th cnt sach ok snts th slts of solving th 3D nozzl oblm sing Flnt and D3flo. Th comtational static ss distibtion comaison btn th mthods D3flo and Flnt a shon in figs 5,6,7 and 8 fo bottom-ight sid, bottom-lft sid, -ight sid and -lft sid, sctivly Pa Pa Pa Pa Pa Pa Fig 5. Static ss Pa distibtion comaing D3flo Flnt slts fo bottom ight sid

7 Pa Pa Pa Pa Pa Pa Fig 6. Static ss Pa distibtion comaing D3flo Flnt slts fo bottom lft sid Pa Pa Pa Pa Pa Pa Pa Fig 7. Static ss Pa distibtion comaing D3flo Flnt slts fo ight sid Pa Pa Pa Pa Pa Pa Pa Fig 8. Static ss Pa distibtion comaing D3flo Flnt slts fo lft sid Th comtational total vlocity distibtion comaison btn th mthods D3flo and Flnt a shon in figs 9, 10, 11 and 1 fo bottom ight-sid, bottom-lft sid, -ight sid and -lft sid, sctivly.

8 14.00 m s m s m s m s m s m s 1 Fig 9. Vlocity magnitd m/s distibtion comaing D3flo Flnt slts fo bottom ight sid m s m s m s m s m s m s 1.00 m s m s m 0.05 m 0.10 m 0.15 m 0.0 m 0.5 m D3FLOW FLUENT Fig 10. Vlocity magnitd m/s distibtion comaing D3flo Flnt slts fo bottom lft sid m s m s m s m s m s m s m s 1 Fig 11. Vlocity magnitd m/s distibtion comaing D3flo Flnt slts fo ight sid

9 1.00 m s m s m s m s m s 1.00 m s m s m 0.05 m 0.10 m 0.15 m 0.0 m 0.5 m D3FLOW FLUENT Fig 1. Vlocity magnitd m/s distibtion comaing D3flo Flnt slts fo lft sid 8 DISCUSSION Fom th gomty of th 3D nozzl, fig 3, it is cla that th z axis attains th hight of 50.0 mm at inlt x=0.0 mm and ks it to x=19.0 mm, thn th gomty convgnc stats and nds at x=75.0 mm, h thoat is fomd. Thaft, divgnc stats nding at x=150.0 mm. Fom th distanc of x=150.0 mm to th xit of th nozzl th hight attains 60.0 mm. a. Rdction in th static ss occs fom inlt to th distanc of x=10.0 mm thoat at th bottom-ight sid of th nozzl. Thn, th static ss attains small changs fo a shot axial distanc. Thaft, an incas in th static ss occs to th otlt of th nozzl, fig 5. b. Rdction in th static ss occs fom th inlt to th distanc of x=130.0 mm and thn stats incasing to th nd of th nozzl at th bottom-lft sid, fig 6. Contos of static ss Pa at th bottom sid sing Flnt a shon in fig 13. c. Do of static ss occs aching its lost val at x=75.0 mm at th -ight sid, fig 7. Thaft, th static ss is covd. d. Do of static ss also occs aching its lost val at x=75.0 mm at th -lft sid, fig 8. Hov, th lo static val fo this sid of th nozzl is considabl mild to th on aaing in -ight sid, fig 7. Th is ss coving aft x=75.0 mm. Th ss ks this val hich as attaind at th thoat of th nozzl. Contos of static ss at th sid sing Flnt a shon in fig 14. Th lo static ss is claly dictd by bl colo.. Th vlocity distibtion is almost invsly ootional to th static ss 11 changs, figs 5-1. Fig 13. Contos of static ss Pa at th bottom sid sing Flnt

10 Fig 14. Contos of static ss Pa at th sid in Flnt 9 CONCLUSIONS Th comtational slts sing th to ogams a vy clos to ach oth so it can b assmd that th dictd slts of th ogam D3flo a clos to ality. Only xintial slts can sho th accacy of th dictions alays nd th assmtion mad abot th flo. Th cod is basd on a siml bodyconfoming gid systm. It tilizs a siml tim intgation tchniq. Th ogam has bn tstd on a nozzl xhibiting a 3D flo stct. Comaisons ith Flnt dictions dmonstat th accacy and comtational fficincy of th mthod. Main fats of th flo a asonably ll dictd, vn sing comaativly coas gids. Hov, mch fin gids old b ndd to solv dtails of th comlx 3D nozzl gomty. Whnv convgnc oblms hav occd, thy hav alays bn tacd to bad gids and hav vanishd hn th gid as find. REFERENCES 1] Dnton, J. D. 1990, Th calclation of th-dimnsional viscos flo thogh mltistag tbomachins, ASME Pa, 90-GT-19. ] Hisch, C. 1990, Nmical Comtation of Intnal and Extnal Flos, Vol., Wily, Chichst, Cha. 16, ] Dnton, J. D. 1974, A tim-maching mthod fo to-dimnsional and th-dimnsional blad to blad flos, ARC R&D ] Shih, C. F., Dlany, R. A. 1986, An accat and fficint El solv fo th-dimnsional tbomachiny flos, ASME Pa, 86-GT-00. 5] Wb, K. F., Tho D. W., Dlany, R. A. 1990, Analysis of th-dimnsional tbomachiny flos on C-ty gids sing an imlicit El solv, Jonal Tbomachiny, 11, ] Ats, T. 1985, Calclation of th th-dimnsional stady inviscid flo in a tansonic axial tbin stag, ASME Pa, 84- GT-76. 7] Solis, J. V. 1983, Finit-volm mthod fo th-dimnsional tansonic otntial flo thogh tbomachiny blad os, Int. J. Hat Flid Flo, 4. 8] Solis, J. V., Bllos, K. V. 1988, Consvation fom of flid dynamics qations in cvilina coodinat systms, Pat I, Mathmatical analysis, Tch. Chon. B, 8, 4, ] Solis, J. V. 1986, Comtational Flid Mchanics, Dmocition Univsity of Thac, Civil Engining Datmnt, Thssaloniki, Aivazis Zobolis. 10] Solis, J. V. 1995, An El solv fo th-dimnsional tbomachiny flos, Intnational Jonal fo Nmical Mthods in Flids, vol.0, ] Lokas, M. A., 015, 3D flo in tbomachiny nozzls, Dmocition Univsity of Thac, Civil Engining Datmnt, Xanthi.