Nonuniform Flow in a Compressor Due to Asymmetric Tip Clearance

Transcription

1 Seung Jin Song e-mil: Seung Ho Cho Shool of Mehnil nd Aerospe Engineering, Seoul Ntionl University, Seoul , Kore Nonuniform Flow in Compressor Due to Asymmetri Tip Clerne This pper presents n nlytil study of flow redistriution in ompressor stge due to symmetri tip lerne distriution. The entire stge is modeled s n tutor disk nd it is ssumed tht upstrem nd downstrem flow fields re determined y the lol tip lerne. The flow is ssumed to e invisid nd inompressile. First, n xisymmetri flow model is used to onnet upstrem nd downstrem flows. Seond, liner perturtion pproximtion is used for nonxisymmetri nlysis in whih eh flow vrile is ssumed to onsist of men (xisymmetri vlue) plus smll perturtion (symmetri vlue). Thus, the perturtions in veloity nd pressure indued y the tip lerne symmetry re predited. Furthermore, rotordynmi effets of suh flow nonuniformity re exmined s well. S X Introdution Nonxisymmetri tip lerne degrdes oth erodynmi nd struturl performne of turomhinery, nd the tip lerne symmetry n hve mny uses, suh s rotor shft ending, whirling, sing symmetry, nd deformtion of omponents. The effets of rotor tip lerne symmetry on turine rotors were initilly suggested y Thoms 1 nd Alford. They suggested tht the vrition in effiieny with lol lerne would led to destilizing forwrd whirl-induing fore. This suggestion ws experimentlly verified y Urlihs 3, Wohlr 4, nd Mrtinez-Snhez et l. 5. Mrtinez-Snhez et l. lso identified nonxisymmetri pressure ting rdilly on the turine hu s seond soure of foring mehnism in ddition to the nonxisymmetri torque initilly hypothesized y Thoms nd Alford. Anlytilly, Song nd Mrtinez-Snhez 6,7 developed n tutor disk model tht ould urtely predit oth nonxisymmetri torque nd pressure effets in turines. To exmine the effets of ompressor tip lerne symmetry, Horlok nd Greitzer 8 nd Colding-Jorgensen 9 formulted tutor disk models. Also, Ehrih 10 nd Grf et l. 11 developed prllel ompressor models. All of these models require ompressor performne dt s inputs. Therefore, Prk 1 developed n nlytil model to predit the effets of nonxisymmetri rotor tip lerne in single-stge ompressor without empiriism. All the models predit kwrd whirl-induing fore due to torque symmetry t the design point. In ddition, Prk 1 predits forwrd whirl-induing ross fore due to pressure symmetry. Until now, ttention hs een foused only on the effets of rotor tip lerne symmetry. However, rel mhines operte with tip lernes in oth rotors nd sttors. Furthermore, the results from reent experiment onduted in the Low Speed Reserh Compressor LSRC t GE 13 strongly suggest ontriutions from the sttor tip lerne symmetry. Therefore, this investigtion ims to understnd the flow fields nd rotordynmi effets in ompressors used y nonxisymmetry in oth rotor nd sttor tip lernes. The sope of urrent investigtion is limited to the effets of stti tip lerne symmetry in single stge ompressor. An nlytil tutor disk pproh is used in this investigtion. Anlytil Model The modeling pproh is similr to the pproh of Song nd Mrtinez-Snhez 6,7. A two-step proess is used to solve for Contriuted y the Interntionl Gs Turine Institute nd presented t the 45th Interntionl Gs Turine nd Aeroengine Congress nd Exhiition, Munih, Germny, My 8 11, 000. Mnusript reeived y the Interntionl Gs Turine Institute Ferury 000. Pper No. 000-GT-416. Review Chir: D. Blll. the flow through ompressor stge. First step is n xisymmetri, two-dimensionl, meridionl plne nlysis, or the lde sle nlysis. This nlysis exmines rdil redistriution of flow due to xisymmetri rotor nd sttor tip lernes. The top prt of Fig. 1 shows the lde sle view of the stge. Due to tip lerne flows, the rdilly uniform upstrem flow is ssumed to split into three strems,, nd upon going through the inlet guide vne IGV, rotor, nd sttor. Strems nd re ssoited with the rotor nd sttor tip lernes, respetively. Strem is the rest of the pssge flow, whih hs pssed through the lded prt of oth rotor nd sttor rows. The seond step is nonxisymmetri, two-dimensionl, rdil plne nlysis. The tip lerne symmetry is nonuniformity with length sle on the order of the ompressor rdius. Therefore, this ltter nlysis is referred to s the rdius sle nlysis. It exmines the flow redistriution in the zimuthl diretion used y nonxisymmetri tip lerne distriution. This view is shown shemtilly in the ottom prt of Fig. 1. This nlysis is smll perturtion tip lerne symmetry nlysis out the men xisymmetri solution provided y the lde sle nlysis. Therefore, the results from the lde sle nlysis re pertured to provide onneting onditions ross the tutor disk. The tutor disk in this study onsists of n inlet guide vne IGV row, rotor lde row, nd sttor lde row. The IGV hs full-spn ldes while rotor nd sttor hve prtil spn ldes. Axil, tngentil, nd rdil diretions re denoted y x, y, nd z, respetively. On the lde sle, refers to lotion fr upstrem of the IGV. Ner upstrem of the IGV is referred to s Sttion 0. Inlet to the rotor is referred to s Sttion 1, nd the rotor exit is lled Sttion. Downstrem of the sttor row is lled Sttion 3. Fr downstrem is referred to s. On the rdius sle x0 nd x0 re equivlent to nd t the lde sle, respetively. The ompressor s rottionl speed, solute veloity, nd reltive veloity re U, C, nd W, respetively. is the solute flow ngle, nd is the reltive flow ngle. The model ssumes n invisid, inompressile flow. The ompressor geometry is ssumed to e two dimensionl t the men rdius vlues. Also, the flow is ssumed to follow the ldes perfetly. Thus, effets suh s lokge nd devition re not ounted for in this model..1 Tip Sle Anlysis. Mrtinez-Snhez 14 developed n invisid tip lerne flow model Fig. whose preditions greed with the theory nd dt of Chen 15. The tip lerne flow jet in Fig. is modeled s jet driven y the pressure differene etween the pressure nd sution sides. This jet then ollides with n equl mount of pssge pss in Fig. flow Journl of Turomhinery Copyright 000 y ASME OCTOBER 000, Vol. 1 Õ 751

2 Fig. 3 Geometry of ompressor tip vortex roll up Fig. 1 Blde nd rdius sle views of ompressor stge tn W G W ps C p C ps p ss (4) 1C pps efore rolling up into vortex. Finlly, this tip vortex forms lyer tht is underturned reltive to the pssge flow. For exmple, t the rotor exit, Strem is the pssge flow nd Strem is the underturned flow due to the rotor tip lerne. The turine tip lerne flow model hs een modified for ompressors 16, nd the ompressor tip lerne flow model is shown shemtilly in Fig. 3. The flow veloities on sution nd pressure sides re otined from the Bernoulli eqution W ps W 1 p psp 1 W ss W 1 p ssp 1 where the susript 1 refers to the rotor inlet ondition. Also, W G p psp ss (3) Sine the flow is ssumed to e invisid, the two strems jet nd ss in Fig. 3 tht ollide hve sme totl pressure, temperture, nd lso equl stti pressures long their ontt line. Therefore, these two strems must hve equl veloity mgnitudes, nd the line OP isets the ngle mde y W jet nd W ss Fig. 3. Then (1) () where C p (pp 1 )/(W 1 /). Notie tht C l C pps C pss, nd it n e shown tht the degree of underturning of the vortex reltive to the pssge flow n e otined s where os 1 4 4C l C l ZW os 1 os nd ZW refers to the Zweifel oeffiient ZW s/ H/ os tn 1 tn (7). Blde Sle Anlysis. The zimuthl momentum eqution for the flow is C C y 0 (8) where C ıc x kc z is the meridionl veloity deoupled from the zimuthl veloity. The zimuthl omponent of vortiity y (C x /z)(c z /x) n e desried in terms of the strem funtion s y x (9) z The upstrem flow is irrottionl ( y 0). Therefore, oeys the Lple eqution. Downstrem, y is onentrted t the interfes etween Strems,, nd Q R nd Q S in Fig. 1. Defining the Bernoulli onstnt s B P/(1/)C, (5) (6) y db (10) d From the definition of B, with the ontinuity onstrint nd the ssumption of spnwise uniform lding, Fig. Shemti of the tip lerne flow model of Mrtinez- Snhez 14 B 3 B 1 P 3P 1 (11) Upstrem of the stge, db 1 /d0 due to the flow s irrottionlity. Then, Eq. 10 eomes y3 db 3 d db 3B 1 d d d P 3P 1 (1) 75 Õ Vol. 1, OCTOBER 000 Trnstions of the ASME

3 nd y n e determined from the stti enthlpy ddition (P 3 P 1 )/ y the ompressor. From Euler s eqution, the stti enthlpy rise is given y P 3P 1 UC y C y1 1 C 3 C 1 (13) At the IGV exit, tngentil veloity is given s C y1 C x1 tn 1 (14) At the rotor exit, the flow hs split into two strems. For the lded strem Strem, the tngentil veloities t the rotor exit nd sttor exit 3 re C y UC x tn (15) C y3 C x3 tn 3 (16) Thus, the pressure rise for Strem is P 3P 1 UC y C y1 1 C x3c y3 C x1 C y1 (17) For Strem, the tip sle nlysis predits its rotor exit zimuthl veloity to e C y UC os Rsin m R x (18) os m where m is the men flow ngle through the rotor nd R is the underturning of the Strem reltive to the Strem. Also t the sttor exit, C y3 C x3 tn 3 (19) Similrly, for Strem, the tngentil veloities t the rotor exit nd sttor exit 3 re given s C y C y (0) C y3 C x3 os s sin m s os m (1) Thus, the pressure rises for Strems nd re P 3P 1 UC y C y1 1 C x3c sin y3 R C x os m nd P3P1 C x1 C y1 () UC y C y1 1 C x3 C y3 C x3 sin S os m C x1 C y1 (3) The third terms on the right-hnd side of Eqs. nd 3 re the kineti energy dissipted when the lekge through the tip gps ollides with the pssge flow efore rolling up into vorties. To fous on the tip lerne effets, the oordinte system is trnsformed to the stremline oordinte from the z oordinte. Then, the eqution for eomes Upstrem x0 0 (4) Downstremx0 Q R rotortip Q s sttortip (5) where Q j j i y db 3 i B 3 strems i nd j, nd is Dir s delt funtion. The oundry onditions re x, 00 is the strength of the y etween x,hc x0 H x0, zc x0 z 1 z 3 z x0, z0 (6) x 1 z z 3z z From the definition of B nd Eqs. 17,, nd 3, the strengths of vortiity, Q R nd Q S, in the sher lyers Fig. 1 re Q R UC y C y1 UC y C y1 1 C sin y3 R C x os m C y3 Q S UC y C y1 UC y C y1 1 C sin y3 S C x3 os m C y3 (7) (8) Susequently, the veloities t vrious xil lotions n e determined. At the rotor exit, xil veloities for Strems nd re C x C x1 1 q R R 1 (9) C x C x1 1 q R R (30) where q R Q R /C x1 nd R is the nondimentionl mss frtion of Strem. Tngentil veloities re given in Eqs. 15 nd 18. At the sttor exit, the xil veloities re C x3 C x1 1 q R 1 R q S S (31) C x3 C x1 1 q R R q S S (3) C x3 C x1 1 q s 1 S q R R (33) nd tngentil veloities re given in Eqs. 16, 19, nd 1. Fr downstrem of the sttor, the xil veloities re C x C x1 1q R 1 R q S S (34) C x C x1 1q R R q s s (35) C x C x1 1q s 1 s q R R (36) nd tngentil veloities re equivlent to those t the sttor exit given in Eqs. 16, 19, nd 1. Sustituting for veloities in Eqs. 7 nd 8 yields two qudrti equtions for Q R nd Q S s funtions of lde geometry nd mss frtions of Strems nd, R nd S. 1 R tn 3 q R tn 3 1 sq s q R 1 q R 1 R sin R os m 1 q R 1 R T 1 q R R tn 0 (37) nd for the sttor where Tos R sin m R /os m Journl of Turomhinery OCTOBER 000, Vol. 1 Õ 753

4 q S S tn 3 G S 1 s G S 1 q R R S tn 3 G S G S q S 1 q R R tn 3 G S 0 (38) where G S sin s os m. os Ssin m S os m Next, the mss frtions, R nd S, n e determined from the given tip lernes s nd R 4t R /H 1 q R q S S 1 q R q S S 4 q R t R H (39) Fig. 4 Coordinte system for the model S 4t S /H 1 q S q R R 1 q S q R R 4 q S t S H (40) Thus, from the presried tip lernes t R /H nd t S /H, Eqs n e solved for Q R, Q S, R, nd S. Fr downstrem, fter the ompletion of flow redjustment, the thiknesses of Strems nd n e determined s R H R 1 q R 1 R q S S S H S 1 q S 1 S q R R 1 q R 1 R q S S 1q R 1 R q S S(41) 1 q s 1 S q R R 1q s 1 s q R R(4) Also, the pressure rise ross the ompressor stge n e determined s P P UC y1 C y 1 C x 1 C x 1 C y3 (43).3 Rdius Sle Anlysis. The rotor offset, e, is ssumed to e muh smller thn the lde spn. Therefore, the tip lerne distriution is given y t t Reêe iy/r (44) where t is the men rotor tip gp, nd y is the distne from the mximum tip gp in the zimuthl diretion Fig Upstrem Flow. The irrottionl upstrem flow is given y 0 (45) where is the veloity potentil. Fr upstrem, C x () C x. Ner upstrem of the stge t x0, the xil nd tngentil veloities re C x 0,y,tReC x Kˆ 0e iy/r (46) C y x,y,treikˆ 0e x/riy/r (47) where Kˆ 0 is the omplex mplitude of xil veloity perturtion s the flow pprohes the disk. Therefore, the upstrem pressure is given y Px,y,tPReC x Kˆ 0e x/riy/r (48).3. Downstrem Flow. Downstrem of the stge, the flow onsists of three regions: Strems,, nd. The ontinuity eqution for eh strem n e written s R t S t C x R x C x S x C y R 0 (49) y C y S 0 (50) y H R S C x H R S C yh R S t x y 0 (51) where H is nnulus height nd R nd S re given y Eqs. 41 nd 4. The momentum equtions for the three strems n e written s C,, t C,, C,, 1 P0 (5) where C is two-dimensionl veloity with xil nd tngentil omponents. Now, eh flow prmeter n e expressed s nd CC C (53) CReĈe xiy/r (54) is smll perturtion out the men. A homogeneous set of equtions for eigenvlues is otined y sustituting for eh flow vrile nd linerizing. 754 Õ Vol. 1, OCTOBER 000 Trnstions of the ASME

5 R i R /R A S i S /R C D E B B 0 A A C C B B 0 0 i/r Ĉx Ĉ x Ĉ x ˆ R ˆ S Pˆ /0 (55) where AC x i(c y /R), BC x i(c y /R), CC x i(c y /R), D(H R S ), Ei(H R S)/R. Then, the nontrivil homogeneous solution is Ĉ x,ĉ y,ĉ x,ĉ y,ĉ x,ĉ y,ˆ R,ˆ Pˆ S, Kˆ ie i (56) 8 i1 where E i s re eigenvetors, nd the omplex onstnts Kˆ i s re to e determined from mthing..3.3 The Upstrem Downstrem Coupling. To onnet upstrem nd downstrem flows, the results from the lde sle nlysis re used. Aording to the lde sle nlysis, the flow vriles depend on the lol nondimensionl tip lernes, t R /H & t S /H, nd flow oeffiient,. For exmple, C x4 U C x4 U t R H, t S H, (57) The downstrem nd upstrem perturtion quntities re determined from the lde sle results s shown elow. The xisymmetri lde sle results on the right-hnd side of Eq. 58 re pertured to ount for the given geometri nonxisymmetry. The perturtion solutions then eome nonxisymmetri rdius sle results on the left-hnd side of Eq. 58. The lol flow oeffiient,, is lso determined from mthing upstrem nd downstrem flows. Rdius Sle Blde Sle Ĉx Ĉ x Ĉ x ˆ R ˆ S Pˆ /x0 ˆ Perturtion H H e R t R H H e S t S Cx C x C x C y 8 C y C y R S P/i1 K i E i (58).4 Clultion of Rotordynmi Coeffiients. From the perturtions in flow vriles, rotordynmi exittion fores n e predited. The tngentil fore exerted on the ompressor y the fluid per zimuthl length is defined s f y R qc y1 C y 1 R qc y1 C y (59) where q is the lol mss flux. The men nd the perturtion of f y re, respetively f y Rq C y1 C y 1 Rq C y1 C y (60) f Rq C y1 y C y R q R 1 Rq C y1 C y q C y1 C y C y1 C y R q 1 R q C y1 C y C y1 C y (61) The perturtion in f y is lmost like the torque vrition envisioned y Thoms nd Alford. However, they ssumed tht the flow remins xisymmetri upstrem nd downstrem of the ompressor, nd, thus, ignored the effets of mss flux perturtion, q/q. However, s Eq. 48 shows, rotor nd sttor tip lernes do indeed indue zimuthl flow redistriution, nd this flow redistriution results in nonxisymmetri pressure distriution. Journl of Turomhinery OCTOBER 000, Vol. 1 Õ 755

6 The pressure ting on the rotor hu is pproximted s the verge of pressures t the inlet nd the exit of the rotor, P. P P 1P P 1 P 1P P 1 1 C x tn UC y1 (6) RC model of Prk 1 shown in Figs. 7 nd 8. Underturning is due to the effets of the tip lekge flow. The xil momentum defet is used y flow migrtion wy from the tip lerne where the pressure rise ross the ompressor stge is sensed more. Suh effets inrese with inresing tip lerne, nd this PP 1 C x C x tn UC y1 C y1 (63) Nonuniform tngentil fore nd nonuniform pressure n thus e otined from Eqs. 61 nd 63. Upon projetion onto the X, Y xes, the totl exittion fore oeffiients, or the rotordynmi stiffness oeffiients, re Y i X totl fˆ yilpˆ (64) f ye/h The totl oeffiients totl, re omposed of ontriutions from tngentil fore symmetry (wd), nd pressure symmetry (p). Fores long the rotor offset re lled diret fores nd re denoted with susript X. Fores perpendiulr to the rotor offset re lled ross fores nd re denoted with susript Y. 3 Model Preditions This setion presents the model preditions for the seleted seline ompressor nd other ompressors. Initilly, the seline ompressor hosen for this study is explined, nd the preditions for this ompressor re given in the following order. First, the rdil flow redistriution indued y xisymmetri rotor nd sttor tip lernes is presented. Seond, the zimuthl flow redistriution due to nonxisymmetri tip lernes is shown. For oth, differenes nd similrities etween the preditions of the new model with rotor nd sttor lernes RSC model nd those of the model with only the rotor lerne RC model 1 re rought out. Third, rotordynmi oeffiients t the design point nd off-design points re disussed. Finlly, the effets of vrious ompressor designs on rotordynmi stiffness oeffiients re presented The hrteristis of the seline ompressor re given in Tle 1. The design flow oeffiient, retion, nd work oeffiient hve ll een set to 0.5 euse they re representtive of modern ompressors. The tip lerne vlues of perent of the nnulus height hve een seleted euse suh vlue is ommon in reserh experiments. 3.1 Blde Sle Preditions. Figure 5 shows the rdil profiles of xil veloity, solute tngentil veloity, nd reltive flow ngle t the rotor exit fter the flow hs split into two strems. The hu nd endwll re t z/h0.0 nd z/h1.0, respetively. Strem is retrded in the xil diretion nd underturned in the tngentil diretion reltive to Strem. Figure 6 shows the solute veloity nd flow ngle profiles t the sttor exit. Now, the flow hs split into three strems. Reltive to Strem, whih goes through oth rotor nd sttor ldes, Strem shows hrteristis similr to those of Strem. Suh results gree with the orresponding preditions from the Fig. 5 Rdil distriutions of xil veloity, solute tngentil veloity, nd reltive flow ngle t rotor exit predited y the new RSC model Fig. 6 Rdil distriutions of xil veloity, solute tngentil veloity, nd solute flow ngle t sttor exit predited y the new RSC model Tle 1 point Bseline ompressor speifitions t the design Prmeter Vlue D 0.50 D 0.50 R D 0.50 t R /H 0.0 t S /H 0.0 Fig. 7 Rdil distriutions of xil veloity, solute tngentil veloity, nd reltive flow ngle t rotor exit predited y the RC model of Prk Õ Vol. 1, OCTOBER 000 Trnstions of the ASME

7 Fig. 8 Rdil distriutions of xil veloity, solute tngentil veloity, nd solute flow ngle t sttor exit predited y the RC model of Prk 1 Fig. 10 ngle Upstrem pressure perturtion versus zimuthl trend grees with the experimentl findings of Hunter nd Cumpsty 17. Compring Figs. 6 nd 8, the ovious differene etween the new RSC model nd the RC model is Strem, whih does not exist in Fig. 8. Thus, the new model n inorporte the effets of sttor gp on the flow field. Fousing on Strem, the strem s mss frtion, degree of xil momentum defet, nd underturning in Figs. 6 nd 8 re virtully identil. This is euse the downstrem sttor tip lerne effet ours over length sle on the order of the tip lerne. However, the xil lde sping is on the order of the lde hord, whih is t lest ouple of orders of mgnitude lrger thn the tip lerne. Thus, Strems nd re prtilly deoupled from eh other. 3. Rdius Sle Preditions 3..1 Azimuthl Flow Redistriution. First, the upstrem zimuthl flow redistriution indued y tip lerne symmetry is disussed. Nondimensionl veloity nd pressure perturtions upstrem of the ompressor predited y the RC model nd the new RSC model re plotted versus zimuthl lotion in Figs. 9 nd 10, respetively. The minimum gp is t 0 deg nd mximum gp is t 180 deg. Roughly, the mss flux is higher ner the minimum gp in oth ses. Agin, the higher downstrem pressure is felt more ner the mximum gp (180 deg). The result is tngentil flow migrtion wy from the lrger gp towrd the smller gp. Then from the Bernoulli reltion, the pressure dereses s flow elertes. The mgnitudes of oth perturtions inrese signifintly when the sttor lerne is introdued. As Eq. 58 shows, the lerne symmetry ts s the foring term, whih indues zimuthl flow redistriution. Therefore, imposing sttor tip lerne symmetry in ddition to the rotor tip lerne symmetry strengthens the foring effet. Thus, the flow eomes more nonuniform with rotor nd sttor tip lernes. Next, the rotordynmi onsequenes of suh flow redistriution re presented. Tngentil fore perturtion lso referred to s the torque symmetry or lde loding vrition is plotted versus in Fig. 11. Sine the fore on the ompressor y the fluid ts in diretion opposite to the diretion of rottion, the men vlue of tngentil fore. f y /ṁu, is negtive. Therefore, ording to Fig. 11, the ompressor rotor lde is loded less ner the mximum gp. The unloding ner the mximum gp ours minly euse the tip lekge flow rte is higher there. Suh predition hs een verified y the experimentl dt from the GE LSRC 18. Also, introduing sttor lerne symmetry hrdly hnges the perturtion in lde loding euse the rotor tip lekge flow is prtilly deoupled from the sttor tip lerne. The perturtion in the rotor region stti pressure is plotted versus in Fig. 1. The pressure hs its mximum ner the mximum gp (180 deg). Although similr trend is suggested y the GE s LSRC dt, the orresponding experimentl dt do not exist yet to onfirm this effet in ompressors. Unlike the lde loding perturtion, the pressure perturtion is more sensitive to Fig. 9 ngle Upstrem xil veloity perturtion versus zimuthl Fig. 11 ngle Rotor lde loding perturtion versus zimuthl Journl of Turomhinery OCTOBER 000, Vol. 1 Õ 757

8 Fig. 1 Perturtion in the verge pressure on rotor hu versus zimuthl ngle Fig. 13 Predited totl nd pressure rotordynmi oeffiients versus operting flow oeffiient the ddition of sttor lerne symmetry Fig. 10. In turines, the pressure symmetry, predited y similr tutor disk model, mthed well with experimentl dt Design Point Rotordynmi Coeffiients. The predited exittion oeffiients from the new RSC model re listed in Tle nd those from the RC model re given in Tle 3. The oeffiients due to lde loding vrition re (wd) s nd those due to pressure vrition re (p) s. Both models predit the following: The lde loding vrition indues negtive ross fore, whih promotes kwrd whirl, nd negligile diret fore. The pressure effet leds to positive ross fore, whih indues forwrd whirl, nd positive diret fore. However, the models predit different totl oeffiients. The new model predits tht the Y totl, is positive euse Y(p) is igger thn Y(wd). Thus, net positive ross fore is predited. However, the RC model predits negligile ross fore euse the lde loding nd pressure effets nel eh other out in Y diretion. Both models predit positive Xtotl etween 0.4 nd 0.6. In omprison, the prllel ompressor model of Ehrih 10 n predit only Y(wd). The model uses the differene etween ompressor hrteristis t different xisymmetri tip lernes to predit torque symmetry, whih is ssumed to e in phse with the lerne distriution. However, no pressure informtion is ville for the prllel ompressor model to predit pressure symmetry. Nevertheless, like the new RSC model, the prllel ompressor model predits negtive ross fore due torque symmetry Off-Design Point Rotordynmi Coeffiients. Figure 13 shows grph of exittion fore oeffiients versus the operting flow oeffiient. In this se, to model n emedded stge, the IGV hs een repled with sttor row. If ompressor opertes elow its design flow oeffiient, exittion fore oeffiients inrese in mgnitude euse the mplitudes of flow perturtions re mgnified t low. These trends re similr to those predited nd mesured in turines 7. Figure 14 shows grph of the ross fore exittion oeffiient due to lde loding vrition, Y(wd), plotted versus. Y(wd) remins negtive ut dereses slightly in mgnitude s inreses. This trend hs een verified experimentlly in the LSRC t Generl Eletri Prmetri Anlysis Preditions. This setion presents the predited effets of ompressor design prmeters on the exittion fore oeffiients. The seleted prmeters re the design flow oeffiient, D, the design work oeffiient, D, nd the design retion, R D. They determine ompressor lde ngles s shown elow. R D 1 tn 1tn D (65) D 1 D tn 1 tn (66) Thus, hnge in the vlue of one of the prmeters hnges oth rotor nd sttor lde shpes i.e., ngles, nd the effets of vri- Tle Exittion fore oeffiients for the seline ompressor D Ä0.50, D Ä0.50, R D Ä0.50 predited from the new model RSC Diretion (wd) (p) totl X Y Tle 3 Exittion fore oeffiients for the seline ompressor D Ä0.50, D Ä0.50, R D Ä0.50 predited from the rotor lerne-only RC model of Prk 1 Diretion (wd) (p) totl X Y Fig. 14 Predited ross rotordynmi oeffiients due to lde loding perturtion versus operting flow oeffiient 758 Õ Vol. 1, OCTOBER 000 Trnstions of the ASME

9 Fig. 15 Predited totl nd pressure rotordynmi oeffiients versus design flow oeffiient Fig. 17 Predited totl nd pressure rotordynmi oeffiients versus design retion ous ompressor designs n e exmined. For prmetri nlysis, one of the three vriles is hnged while the other two re held onstnt t the seline vlues. Figure 15 shows vrition of totl nd (p) s the design flow oeffiient is inresed. (wd) is the differene etween totl nd (p). For the ross fore, Ytotl dereses with inresing D primrily euse Y(p) dereses s the mgnitude of zimuthl flow nonuniformity is deresed. Also, Y(wd) remins negtive nd its mgnitude inreses with D. For the diret fore, X(p) domintes over X(wd) nd hnges sign s the phse of pressure non-uniformity reltive to tip lerne distriution shifts. These trends re similr to those predited for turines in Song nd Mrtinez-Snhez 7. Figure 16 shows vrition of exittion fore oeffiients s the design work oeffiient is inresed. In the Y diretion, Ytotl inreses primrily euse Y(p) inreses in mgnitude. In the X diretion, Xtotl does not hnge muh while X(wd) inreses. Overll, inresing the work oeffiient is equivlent to strengthening the intensity of disontinuity ross the tutor disk. Thus, for given imposed tip lerne symmetry, the perturtions in the flow field inrese. In ddition, the phses of the flow perturtions reltive to the lerne distriution lso shift. Figure 17 shows the vrition of exittion fore oeffiients versus the design retion. As R D inreses, X(p) does not hnge muh, ut Y(p) is redued signifintly. This hnge is due to the derese in the mgnitude of zimuthl flow nonxisymmetry. (wd) is reltively insensitive to R D. Thus, t low design retions, Ytotl is inresed. 4 Conlusions The new onlusions of this study n e summrized s follows: 1 A new nlytil model hs een developed to exmine the effets of nonxisymmetry in rotor nd sttor tip lernes on the ompressor flow field. The new model hs reonfirmed the following previously found trends: rdil flow migrtion wy from the tip lerne; zimuthl flow migrtion towrds smller gp re; nd diretion of rotordynmi fores tht rise due to pressure nd torque i.e., lde loding symmetry. In ddition, for the seline ompressor, the following onlusions n e drwn. 3 Diret fore is mostly due to the pressure symmetry nd is positive. 4 Torque symmetry results in negtive ross fore tht, without dmping, would promote kwrd whirl. However, pressure symmetry results in positive ross fore, whih would promote forwrd whirl. The net result is positive ross fore. 5 The dominne of pressure symmetry effets over those of lde loding symmetry is due to the introdution of sttor lerne symmetry. 6 The flow ssoited with the rotor tip lerne is hrdly ffeted y the existene of the downstrem sttor tip lerne. 7 The pressure symmetry indued y zimuthl flow redistriution inreses signifintly in mgnitude with the ddition of sttor tip lerne symmetry. 8 Operting t elow the design flow oeffiient inreses the mgnitude of exittion fore oeffiients. Finlly, from the results of prmetri vrition out the seline ompressor, the following onlusions n e drwn. 9 High design flow oeffiient nd high design retion derese the mgnitudes of exittion fore oeffiients. 10 High design work oeffiient inrese the exittion fore oeffiients mgnitudes. Fig. 16 Predited totl nd pressure rotordynmi oeffiients versus design work oeffiient Aknowledgments The finnil support for this study hs een provided y the Seoul Ntionl University Reserh Fund nd the Institute of Advned Mhinery nd Design. Also, the uthors hve enefitted from onstrutive disussions with Professor Mrtinez-Snhez of MIT. Journl of Turomhinery OCTOBER 000, Vol. 1 Õ 759

10 Nomenlture B Bernoulli onstnt, m /s C solute flow veloity, m/s xil lde hord, m C l lift oeffiient per unit spn C p pressure oeffiient e mgnitude of rotor offset, m eigenvetor for downstrem perturtions E i F X F Y lterl fore in the diretion of the offset, N lterl fore perpendiulr to the diretion of the offset, N H nnulus height, m H rotor lde spn, m Kˆ j omplex mplitude of flow perturtions L xil rotor hu thikness, m ṁ mss flux, kg/s P pressure, P Q strength of sher lyer, m /s q nondimensionl vortiity strength; lol mss flux R men ompressor rdius, m s lde pith, m t rdil tip lerne, m U R ompressor rottionl speed t the men rdius, m/s W reltive veloity, m/s X diretion long the rotor offset x xil diretion Y diretion perpendiulr to rotor offset y tngentil diretion z rdil diretion ZW Zweifel oeffiient solute flow ngle, deg; eigenvlue X Y diret exittion fore oeffiient ross exittion fore oeffiient reltive flow ngle, deg thikness of underturned lyer downstrem of tutor disk, m upstrem veloity potentil C x /U flow oeffiient nondimensionl mss frtion of underturned flow zimuthl ngle mesured in the diretion of rottion from the minimum gp lotion, deg; ngle of underturning reltive to pssge flow, deg density, kg/m 3 ngulr veloity of rotor shft rottion, s 1 meridionl strem funtion; work oeffiient Susripts D design vlue G gp m men p indites effet due to nonuniform pressure ps pressure side R rotor S sttor ss sution side t stgntion ondition wd indites effet due to vrition of torque or tngentil fore 0 ner upstrem of the tutor disk on the rdius sle 0 ner downstrem of the tutor disk on the rdius sle fr upstrem on the lde sle 0 IGV inlet on the lde sle 1 rotor inlet on the lde sle sttor inlet on the lde sle 3 sttor outlet on the lde sle fr downstrem on the lde sle meridionl omponent Supersripts the prt of downstrem flow ssoited with the rotor tip gp the prt of downstrem flow tht hs rossed the lded prt of ompressor the prt of downstrem flow ssoited with the sttor tip gp nonxisymmetri perturtion - zimuthl men, or xisymmetri vlue ^ omplex mplitude Referenes 1 Thoms, H. J., 1958, Unstle Nturl Virtion of Turine Rotors Indued y the Clerne Flow in Glnds nd Blding, Bull. de I A.I.M., 71, No. 11/1, pp Alford, J., 1965, Proteting Turomhinery From Self-Exited Rotor Whirl, ASME J. Eng. Power, 87, pp Urlihs, K., 1983, Clerne Flow Generted Trnsverse Fores t the Rotors of Therml Turomhines, NASA TM Wohlr, R., 1983, Experimentl Determintion of Gp-Flow Conditioned Fores t Turine Stges, nd Their Effet on the Running Stility of Simple Rotors, NASA TM Mrtinez-Snhez, M., Jroux, B., Song, S. J., nd Yoo, S., 1995, Mesurement of Turine Blde-Tip Rotordynmi Exittion Fores, ASME J. Turomh., 117, pp Song, S. J., nd Mrtinez-Snhez, M., 1997, Rotordynmi Fores Due to Turine Tip Lekge: Prt 1 Blde Sle Effets, ASME J. Turomh., 119, pp Song, S. J., nd Mrtinez-Snhez, M., 1997, Rotordynmi Fores Due to Turine Tip Lekge: Prt Rdius Sle Effets nd Experimentl Verifition, ASME J. Turomh., 119, pp Horlok, J. H., nd Greitzer, E. M., 1983, Non-Uniform Flows in Axil Compressors Due to tip Clerne Vrition, Pro. Inst. Meh. Eng., 197C, pp Colding-Jorgensen, J., 199, Predition of Rotordynmi Destilizing Fores in Axil Flow Compressors, ASME J. Fluids Eng., 114, pp Ehrih, F. F., 1993, Rotor Whirl Fores Indued y the Tip Clerne Effet in Axil Flow Compressor, ASME J. Vir. Aoust., 115, pp Grf, M. B., Wong, T. S., Greitzer, E. M., Mrle, F. E., Tn, E. S., Shin, H. W., Wisler, D. C., 1998, Effets of Nonxisymmetri Tip Clerne on Axil Compressor Performne nd Stility, ASME J. Turomh., 10, pp Prk, K. Y., 1998, Non-uniform Compressor Flow Fields Indued y Nonxisymmetri Tip Clerne, M.S. Thesis, Deprtment of Aerospe Engineering, Inh Univ. Kore. 13 Store, A., et l., 000, Unstedy Flow nd Whirl-Induing Fores in Axil-Flow Compressors; Prt 1 Experiment, ASME Pper No. 000-GT Mrtinez-Snhez, M., nd Guthier, R. P., 1990, Blde Sle Effets of Tip Lekge, Gs Turine Lortory Report #0, M.I.T. 15 Chen, G. T., 1991, Vortil Strutures in Turomhinery Tip Clerne Flows, Ph.D. thesis, Deprtment of Aeronutis nd Astronutis, M.I.T. 16 Roh, H. Y., 1997, Blde Sle Effets of Tip Lekge Flow in Axil Compressors, B.S. Thesis, Deprtment of Aerospe Engineering, Inh Univ., Kore. 17 Hunter, I. H., nd Cumpsty, N. A., 198, Csing Wll Boundry-Lyer Development Through n Isolted Compressor Rotor, ASME J. Eng. Power, 104, pp Ehrih, F. F., et l., 000, Unstedy Flow nd Whirl-Induing Fores in Axil-Flow Compressors; Prt Anlysis, ASME Pper No. 000-GT Õ Vol. 1, OCTOBER 000 Trnstions of the ASME