Romuald Rzadkowski. Vitally Gnesin, Luba Kolodyazhnaya. Ryszard Szczepanik

Transcription

1 Proceedngs of 10 th European Conference on Turbomachnery Flud dynamcs & Thermodynamcs ETC10, Aprl 15-19, 2013, Lappeenranta, Fnland THE UNSTEADY LOW-FREQUENCY AERODYNAMIC FORCES ACTING ON ROTOR BLADES IN THE FIRST TWO STAGES OF A JET ENGINE AXIAL COMPRESSOR IN THE CASE OF A BIRD STRIKE Romuald Rzadkowsk The R. Szewalsk Insttute of Flud Flow Machnery, Polsh Academy of Scences n Gdansk, Poland Fszera 14, Gdańsk z3@mp.gda.pl Vtally Gnesn, Luba Kolodyazhnaya Natonal Academy of Scences of Ukrane, 2/10 Pozharsky st., Kharkov, Ukrane, gnesn@pmach.kharkov.ua Ryszard Szczepank Ar Force Insttute of Technology, Kseca Bolesława 6, Warszawa, Poland, ryszard.szczepank@twl.pl ABSTRACT A brd strke can cause damage to statonary and rotatng arcraft engne parts, especally the engne fan. Ths paper presents a brd strke smulated by blockng four stator blade passages. It ncludes the numercal results of the unsteady low-frequency aerodynamc forces and the aeroelastc behavour caused by a non-symmetrc upstream flow affectng the frst two rotor blade stages n the axal-compressor of a jet engne. The appled approach s based on the soluton of the coupled aerodynamc-structure problem for a 3D flow through the compressor stages n whch flud and dynamc equatons are ntegrated smultaneously n tme. An deal gas flow through the mutually movng stators and rotor blades wth perodcty on the entre annulus s descrbed usng unsteady Euler conservaton equatons, ntegrated usng the Godunov-Kolgan explct monotonous fnte-volume dfference scheme and a movng hybrd H-H grd. The structure analyss uses the modal approach and a 3D fnte element model of a blade. The blade moton s assumed to be a lnear combnaton of the frst natural modes of blade oscllatons wth the modal coeffcents dependng on tme. The obtaned results show that dsturbances n the engne nlet strongly nfluence the level of unsteady forces actng on the rotor blades. Wth a partally blocked nlet the whole spectrum of low-frequency harmoncs s observed. Such harmoncs can lead to rotor blade damage. The low-frequency ampltudes are hgher n the frst stage rotor blades than n the second stage. OPEN ACCESS Downloaded from 1 Copyrght by the Authors

2 NOMENCLATURE C dampng matrx F unsteady aerodynamc forces vector F 1 Fourer coeffcent F 2 Fourer coeffcent F y crcumferental force F z axal force h = h y crcumferental blade dsplacement h z axal blade dsplacement I dagonal matrx K stffness matrx L blade length M mass matrx n normal vector q modal coeffcent of -th mode R 1 frst rotor blade R 2 second rotor blade U dsplacement vector correspondng to -th mode shape v normal velocty w blade moton vector z o number of nlet stator cascade z 1 number of frst stator cascade z 2 number of second stator cascade torson blade angle about cross-secton gravty center modal dampng coeffcent modal force 1 st harmonc frequency densty dagonal matrx -th natural blade frequency INTRODUCTION Foregn objects enterng arcraft engnes are a perennal problem. Mathematcal analyss and computer smulatons can provde nformaton on component stresses already n the desgn phase. Foregn objects (e.g. a brd) were characterzed as a water-lke hydrodynamc response by Wlbeck and Rand [1] n ther tests. Storace et al. [2] developed a computer program to predct structural response due to soft body mpact. Hedar, Carlson and Yants [3] developed rotor dynamcs as a nonlnear transent analyss for a propulson system durng brd strke nduced fan blade loss. Experments were carred out on SO-3 compressor frst stage rotor blades [4] to ntate a crack by placng rectangular blocks on the stator blades, smulatng brds engulfed n the engne. The Fluent code was used to analyze the low frequency exctaton caused by nlet blockage of 1.5 stages of an SO-3 engne, but t dd not take nto account rotor blade vbraton Rzadkowsk et al. [5], Solńsk et al.[6]. Aeroeleastc behavour of rotor blade 1.5 frst compressor stage was analyzed by Rzadkowsk et al. [7]. In ths paper the unsteady forces actng on rotor blades wll be calculated for a 3D non-vscous deal gas flow through 2.5 compressor stages usng an n-house code and takng nto account rotor blade vbraton. AERODYNAMICAL MODEL Blade falures n the frst stage of SO-3 compressors were reported n the years [4]. Experments were carred out on a frst stage rotor blade n an SO-3 engne compressor at the Ar Force Insttute of Technology n Warsaw to ntate a crack by placng rectangular blocks on the stator blades, whch n real lfe could be caused by brds engulfed n the engne (see Fg. 1a). The 3D transonc flow of an nvscd non-heat conductve gas through the frst compressor stages of an SO-3 arcraft engne was smulated (Fg. 1a). A 3D whole-annulus model of the frst stage of a SO-3 jet engne compressor s shown n Fg. 1b. The model conssts of 44 blades n the Inlet Stator Cascade (S0), 28 blades n the Rotor Cascade (R1), 33 blades n the Stator Cascade of the frst stage (S1), 42 (R2) rotor blades and 44 (S2) stator blades. The numercal calculatons were carred out usng computatonal H-grds (R-Theta-Z): 2

3 10*40*84 grd ponts for the S0 passage, 10*60*78 for the R1 passage, 10*48*68 for the S1 passage, 10*44*64 for the R2 passage and 10*42*62 for the S2 passage. Fgure 1a: Test rg of SO-3 engne In order to model the engne nlet brd strke, four stator blade passages were blocked. Each of the passages s meshed, usng an H-type grd for the stators doman and a hybrd H-H grd for the rotor doman ([10], [11]). Here the outer H-grd of the rotor remans statonary durng the calculaton, whle the nner H-grd s rebult n each teraton by a gven algorthm, so that the external ponts of the nner grd reman unmoved, but the nternal ponts (on the blade surface) move accordng to the blade moton. Mesh dependency concernng the flud- flow and mesh sze were analysng n [9], [10]. Fgure 1b: Computatonal model of the 2.5 stage (S0-R1-S1-R2-S2) The spatal transonc flow, ncludng n the general case strong dscontnutes n the form of shock waves and wakes behnd the ext edges of blades s wrtten n the relatve Cartesan coordnate system rotatng wth constant angular velocty accordng to full non-statonary Euler equatons, presented n the form of ntegral conservaton laws of mass, mpulse and energy [9]. Wakes were not smulated. 3

4 In the general case, when axal velocty s subsonc, at the nlet boundary ntal values for total pressure, total temperature and flow angles are used, whle at the outlet boundary only the statc pressure has to be mposed. Non-reflectng boundary condtons must be used,.e., ncomng waves (three at nlet, one at the outlet) have to be suppressed, whch s accomplshed by settng ther tme dervatve to zero [9, 10]. On the blade's surface, because the grd moves wth the blade, the normal relatve velocty s set to zero (v - w) n=0, (1) where: v normal velocty vector, w s blade moton vector, n s normal vector. STRUCTURAL MODEL The blade vbraton formulaton s based on a modal approach of the coupled problem ([11], [12]). The dynamc model of the oscllatng blade n lnearzed formulaton s governed by matrx equaton: M u C u K u F, (2) where M, C, K are the mass, mechancal dampng and stffness matrces of the blade respectvely; u(t) s the blade dsplacement; F s the unsteady aerodynamc forces vector, whch s a functon of blade dsplacement. The frst step of the modal approach conssts of solvng the problem of the natural mode shapes and egenvalues wthout dampng and n a vacuum. Then the dsplacement of each blade can be wrtten as a lnear combnaton of the frst N modes shapes wth the modal coeffcents dependng on tme: N Uq U q (3) u 1 where U s the dsplacement vector correspondng to -th mode shape; q (t) s the modal coeffcent of -th mode. Takng nto account the equaton (3) and the orthogonalty property of the mode shapes the equaton (2) can be wrtten n form of: Iq Hq q (t), (4) where I =dag (1, 1,... 1), H =dag (2 h 1, 2 h 2,... 2 h n ), =dag ( 2 1, 2 2,..., 2 n ) are dagonal matrces; s -th natural blade frequency; { (t)}s the modal forces vector correspondng to the mode shapes, h =, where s the -th modal dampng coeffcent ([11]). Thus the dynamc problem (2) reduces to the set of ndependent dfferental equatons relatvely to modal coeffcents of natural modes ([9], [10]): 2 q 2h q q, (5) The equatons of moton (5) can be solved usng any standard ntegraton method. The modal forces are calculated at each teraton wth the use of the nstantaneous pressure feld n the followng way ([10]): pu n d. (6) 2 U dv where p s the pressure along the blade surface. v 4

5 NUMERICAL RESULTS The numercal calculatons presented below were carred out for the two and half stages of the SO-3 engne compressor. The blade vbraton was defned by takng nto account the frst fve natural mode shapes of the rotatng R1 and R2 blades. The values of natural frequences are gven t Table 1. Mechancal dampng was not taken nto consderaton, becuse we dd not have expermental values. Aerodynamcal dampng was part of the calculaton process [10]. Table 1: Natural frequences of the rotatng compressor rotor blades R1, R2 Freq.(Hz) Rotor Rotor The boundary condtons at the nlet and outlet were as follows: the total pressure n absolute system p o = Pa, total temperature n absolute system T o =288 K. In case of the unblocked nlet mass flow was equal to 3.5 kg/s and pressure rato was 1.67, for the blocked nlet mass flow 19.4 kg/s and pressure rato The calculaton ncluded two regmes. In the frst regme the calculatons of the unsteady flow through the turbne stage were made by frst takng nto account the rotor rotaton wthout blade vbraton. Next the blades vbraton began. The tme step was constant for all the calculated domans and defned by the stablty condton of the dfference scheme of the lnearzed equatons system [11]. A Fourer analyss was used for the tme -dependant numercal results: ( t 1 F t) F F cos(2 t) F sn (2 ), where F(t) s a physcal unsteady load; F o s the averaged value of load; F 1, F 2 are the Fourer coeffcents; s the harmonc number; s the 1 st harmonc frequency. The rotor rotaton tme was sec and the rotaton frequency was 256 Hz. Unsteady Aerodynamcs Forces and Moment The averaged values of unsteady aerodynamc loads (crcumferental, axal forces and moment about the blade cross-secton centre of gravty) actng on rotor blades R1, R2 are presented n Tab 2, 3. Table 2: The averaged values of unsteady loads actng along the R1 blade, blocked nlet, S0- R1-S1-R2-S2 L Fy (N) Fz ( N) M (N*m) The average values of crcumferental forces were hghest n the root, whereas the axal forces and moments were hghest n the perpheral blade cross-secton. In the case of the blocked nlet, the 4 th low-frequency harmonc of crcumferental R1 force was the hghest n the root cross-secton, the 6 th harmonc n the 0.55 L cross-secton and the 6 th harmonc n the 0.96 L cross-secton (Fg. 2a). 5

6 Table 3: The averaged values of unsteady loads actng along the R2 blade, blocked nlet, S0- R1-S1-R2-S2 L Fy (N) Fz (N) M (N*m) The 5 th low-frequency harmonc of axal unsteady force has the hghest value n the root crosssecton, the 6 th harmonc n the 0.55 L cross-secton, and the 5 th harmonc n the 0.96 L crosssecton (Fg. 2b). a) b) Fgure 2: Ampltude frequency spectrum crcumferental F y and axal force F z for the perpheral layer of R1 blade The 5 th low-frequency harmonc of crcumferental R2 force was the hghest n the root crosssecton, the 5 th harmonc n the 0.55 L cross-secton, and the 4 th harmonc n the 0.96 L crosssecton (Fg. 3a). The 4 th low-frequency harmonc of axal R2 force was the hghest n the root cross-secton, the 4 th harmonc n the 0.55 L cross-secton, and the 3 th harmonc n the 0.96 L cross-secton (Fg. 3b). Therefore the low frequency unsteady force dstrbutons for R1 and R2 blades are dfferent. The low frequency harmoncs of the R2 blade were smaller (10% of steady part) than those of the R1 blades (22%), but the spectrum of frequences was smlar. The hgh-frequency R1 harmoncs ( 1 =256 33=8448 Hz), 2% of the steady part and ( z o =256 44=11264 Hz) 1%, where was the rotaton frequency had smaller values than the lowfrequency harmoncs. The hgh-frequency R2 harmoncs ( 1 =256 33=8448 Hz) (5%), ( o=256 44=11264 Hz and ( 2=256 44=11264 Hz) (1%), where was the rotaton frequency had smaller values than the low-frequency harmoncs, but the dfference was smaller than n the case of the R1 blades. 6

7 a) b) Fgure 3: Ampltude frequency spectrum crcumferental F y and axal force F z for the perpheral layer of R2 blade General dsplacements of rotor blades In the numercal model R1 and R2 rotor blade vbratons were taken nto consderaton, hence an aeroelastc analyss was possble. A comparson between the vbraton of rotor blades wth and wthout a blocked nlet s dscussed below. The relatve harmonc ampltudes of the R1 blade perpheral layer wth an unblocked nlet are presented n Table 4. The frst natural frequency of R1 blades was 540 Hz (Tab. 1), so the frequency n the flow decreased to 500 Hz. The unsteady bendng of hy components was 6.5% of the average dsplacements. The ampltudes of torson vbraton (about the blade cross-secton gravty center) were 21% of the average twst angle at 450 Hz and 5% at 1800 Hz. Table 4: Ampltude-frequency spectrum of generalzed dsplacements (hy, hz, φ) of the perpheral blade layer of the R1 blade, unblocked nlet, (S0-R1-S1-R2-S2) L hy (%) 450 Hz hz (%) 450 Hz φ(%) 450 Hz φ(%) 1800 Hz The relatve harmonc ampltudes of the R1 blade perpheral layer wth blocked nlet are presented n Table 5. The frequency n the flow s 450 Hz. The unsteady bendng of hy components was 9% of the average dsplacements. The ampltudes of torson vbraton were 300% of the average twst angle at 750 Hz and 2200% at 1500 Hz. Therefore blockng the nlet causes a greater decrease of rotor blade vbraton frequency n the flow than when the nlet s unblocked. In both cases the form of vbraton s bendng-torson and dfferent to the 1 st mode shape. Table 5: Ampltude-frequency spectrum of generalzed dsplacements (hy, hz, φ) of the perpheral blade layer of R1 blade, blocked nlet, (S0-R1-S1-R2-S2) 7

8 L hy (%) hz (%) φ (%) φ (%) 450 Hz 450 Hz 750 Hz 1500 Hz The bendng dsplacement of R1 blades wth an unblocked nlet s presented n Fgure 4a. Ths shows that R1 blades vbrate wth constant ampltude, whch means that they are self-excted. Fgure 4b shows bendng dsplacement of the R1 blades wth a blocked nlet. Here the rotor blade vbraton ampltude s consderably hgher than n the case of the unblocked nlet, but blade ampltude decreases. Fgure 4a: R1 blade dsplacement (perpheral layer, axal drecton hz), unblocked nlet hz (mm) t (sec) Fgure 4b: R1 blade dsplacement (perpheral layer, axal drecton hz), blocked nlet 8

9 a) b) Fgure 5: Ampltude-frequency spectrum of R1 blade torsonal oscllatons (perpheral layer), unblocked nlet 0.40 (deg) t (sec) a) b) Fgure 6: R1 blade angle (perpheral layer), blocked nlet The torson angle of R1 blades wth an unblocked nlet s presented n Fg. 5. Ths shows that R1 blades vbrate wth constant ampltude, whch means that they are self-excted. 9

10 Fg. 6 shows the torson angle of R1 blades wth a blocked nlet. Here the rotor blade vbraton ampltude s consderably hgher than n the case of the unblocked nlet. Ths shows that R1 blades vbrate wth constant torsonal ampltude, whch means that they are self-excted. R1 blades wth a blocked nlet vbrated wth a complex bendng twstng form, bendng. Ths form of vbraton s dfferent to the 1 st mode shape of the R1 blade. The relatve harmonc ampltudes of the R2 blade perpheral layer wth an unblocked nlet are presented n Table 6. The frst natural frequency of R2 blades was 670 Hz (Tab. 1), and the frequency n the flow decreased to 450 Hz. The unsteady bendng of hy components was 22% of the average dsplacements. The ampltudes of torson vbraton were 35% of the average twst angle at 450 Hz and 14% at 1500 Hz. Table 6: Ampltude-frequency spectrum of generalzed dsplacements (hy, hz, φ) of the R2 blade perpheral layer, unblocked nlet, (S0-R1-S1-R2-S2) L hy (%) hz (%) φ(%) φ(%) 450 Hz 450 Hz 450 Hz 1500 Hz R2 blades wth an unblocked nlet vbrated wth a complex bendng twstng form, bendng components at 450 Hz and twstng components at 450 Hz and 1500Hz. Ths form of vbraton s dfferent to the 1 st mode shape of the R2 blade. The relatve harmonc ampltudes of the R2 blade perpheral layer wth blocked nlet are presented n Table 7. The frequency of bendng vbraton was 450 Hz, the same as for unblocked nlet, although the unsteady component ncreased to 30%. The frequency of torson vbraton was 450 Hz and the unsteady part was 470%,.e. hgher than the 35% n the case of the unblocked nlet. The second torsonal component frequency was 750 Hz and the unsteady part was 270%. Table 7: Ampltude-frequency spectrum of generalzed dsplacements (hy, hz, φ) of the R2 blade perpheral layer, blocked nlet, (S0-R1-S1-R2-S2) L hy (%) hz (%) φ(%) φ(%) 450 Hz 450 Hz 450 Hz 750 Hz Fgure 7a shows bendng dsplacement of R2 blades wth a blocked and Fg. 7b unblocked nlet. Here the rotor blade vbraton ampltude was hgher than n the case of the unblocked nlet, n the case we had dampng vbraton. 10

11 0.00 hy (mm) t (sec) Fgure 7a: R2 blade dsplacement (perpheral layer, crcumferental drecton hy, blocked nlet) Fgure 7b: R2 blade dsplacement (perpheral layer, crcumferental drecton hy, unblocked nlet) 11

12 a) b) Fgure 8: R2 blade angle (perpheral layer), unblocked nlet 0.10 (deg) t (sec) a) b) Fgure 9: The ampltude-frequency spectrum of R2 blade torsonal oscllatons (perpheral layer), blocked nlet The torson angle of R2 blades wth an unblocked nlet s presented n Fgure 8. Ths shows that R2 blades have dampng vbraton. Fg. 9 shows the torson angle of R2 blades wth a blocked nlet. Here the rotor blade vbraton ampltude was hgher than n the case of the unblocked nlet. Ths means that nlet blockage causes dampng vbraton wth hgher ampltude and ncludes a greater number of torsonal components. 12

13 CONCLUSIONS Ths paper has presented the calculaton results for unsteady low-frequency aerodynamc forces actng on SO-3 jet engne compressor R1 and R2 blades, whch took nto account blade vbraton. The forces were analyzed n two operatng condtons: wth an unblocked and partally blocked engne nlet. The obtaned results show that dsturbances n the engne nlet strongly nfluence the level of unsteady forces actng on the rotor blades. Wth a partally blocked nlet the whole spectrum of low-frequency harmoncs was observed, ncludng the natural frequency of blades. The vbraton ampltudes of rotor blades wth a blocked nlet are consderably hgher than n the case of an unblocked nlet. In ths case R1 and R2 blade vbratons dffer from ther frst mode shape and nclude torsonal components. ACKNOWLEDGMENTS The authors wsh to acknowledge NCBR (project PBS1/B4/5/2012) for the fnancal support of ths work. All numercal calculatons were made at the Academc Computer Centre TASK (Gdansk, Poland). REFERENCES [1] Wlbeck, J. S. and Rand, J. L., The Development of a Substtute Brd Model, Journal of Engneerng for Power, ASME, vol. 103, p. 725, [2] Storace, A. F., Nmmer R. P. and Ravenhall, R., Analytcal and Expermental Investgaton of Brd Impact on Fan and Compressor Bladng. Journal of Arcraft, Vol. 21, July [3] Hedar, M. A., Carlson, D. L. and Yants, T., Rotor-dynamcs Analyss Process, p.3601, Worldwde Aerospace Conference & Technology Showcase, Aprl 8-10, Toulouse, [4] Szczepank R., Expermental Analyss of Rotor Blades of Arcraft Engnes n Varous Operatng Condtons, DSc Thess, ITWL Warsaw, (n Polsh) [5] Rządkowsk R., Solńsk M., Szczepank R., The Unsteady Low-Frequency Aerodynamc Forces Actng on the Rotor Blade n the Frst Stage of an Jet Engne Axal Compressor, Advances n Vbraton Engneerng, 11(2), , [6] Solńsk M., Rzadkowsk R., Pająk A., Szczepank R., The Unsteady Low-Frequency Forces Actng on the Rotor Blade n the Frst Stage on an Axal Compressor of SO-3 Jet Engne, Proceedngs of the 8th Internatonal Conference on Vbraton Engneerng and Technology of Machnery, VETOMAC VIII, p , Gdansk, Sept 3-6, 2012 [7] Rządkowsk R., Gnesn V., Kolodyazhnaya L., Szczepank R., Aeroelastc Behavour of Frst Stage Compressor Rotor Blades wth Foregn Object n Engne Inlet, ISUAAAT 2012, 13 TH Internatonal Symposum on Unsteady Aerodynamcs, Aeroacustcs and Aeroelastcty of Turbomachnes, Tokyo, Sept, [8] Godunov, S., K. et al., Numercal Soluton of Multdmensonal Problems n Gasdynamcse, Nauka, Moskva, 1976, n Russan. [9] Rządkowsk R., Gnesn V., 3D Unsteady Forces of the Transonc Flow Through a Turbne Stage wth Vbratng Blades, ASME Paper GT , ASME TURBOEXPO 2002, Amsterdam, (June 3-6, 2002) [10] Gnesn V., Kolodyazhnaya L., Rzadkowsk R., A Numercal Model of Stator-Rotor Interacton n a Turbne Stage wth Oscllatng Blades, Journal of Fluds and Structures, 19(8), , [11] Bathe K., Wlson E, Numercal Methods n Fnte Element Analyss, Prentce-Hall, Inc Englewood Clffs, New Jersey, [12] Rządkowsk, R., Dynamcs of Steam Turbne Bladng, Part Two Bladed Dscs, Ossolneum, Wrocław-Warszawa,