Application of an Analytical Method to Locate a Mixing Plane in a Supersonic Compressor

Transcription

1 Journal of Energy and ower Engineering 9 (015) doi: / / D DAVID UBLISHING Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Emmanuel Benichou 1 and Isabelle Trébinjac 1. Turbomeca, Groupe SAFRAN, Bordes 64511, France. Laboraoire de Mécanique des Fluides e d Acousique, UMR NRS 5509, Ecole enrale de Lyon, Ecully edex 69134, France Received: Sepember 1, 014 / Acceped: November 03, 014 / ublished: January 31, 015. Absrac: In order o achieve greaer pressure raios, compressor designers have he opporuniy o use ransonic configuraions. In he supersonic par of he incoming flow, shock waves appear in he fron par of he blades and propagae in he upsream direcion. In case of muliple blade rows, seady simulaions have o impose an azimuhal averaging (mixing plane) which prevens hese shock waves o exend upsream. In he presen paper, several mixing plane locaions are numerically esed and compared in a supersonic configuraion. An analyical mehod is used o describe he shock paern. I enables o ake a criical look a he FD (compuaional fluid dynamics) seady resuls. Based on his mehod, he shock losses are also evaluaed. The good agreemen beween analyical and numerical values shows ha his mehod can be useful o wisely forecas he mixing plane locaion and o evaluae he shif in performances due o he presence of he mixing plane. Key words: Supersonic compressor, shock wave, pressure loss, RANS, mixing plane. Nomenclaure Symbol V V n V, V r a 1 Saic pressure Sagnaion pressure in he impeller frame ircumferenially averaged sagnaion pressure ircumferenial pich Densiy Velociy in he impeller frame Normal velociy componen Tangenial velociy componens Radius Speed of sound erfec gas consan Roaion speed osiion of he bow shock on he profile x 0 symmery axis x, y oordinaes of a poin in he profile frame M Mach number orresponding auhor: Emmanuel Benichou, h.d. suden, research fields: aerodynamic insabiliies in cenrifugal compressors, including roor-saor ineracions and flow conrol issues using boundary layer aspiraion. emmanuel.benichou@ec-lyon.fr. µ Mach angle Angle of he flow Angle of he bow shock e B Axial disance beween poins x 0 and B d Deachmen disance of he shock wave ichwise disance on which he flow is considered isenropic θ ircumferenial direcion K Toal pressure loss coefficien Subscrip B, Relaive o poins B and 1, alculaed in Secion 1 or Secion Value of he quaniy a infinie upsream Exponen Value of he quaniy a M = 1 1. Inroducion The need for compac, efficien high pressure raio compressors ofen resuls in high roaion speeds. In some cases, he enry flow may herefore be supersonic over he enire- or upper-span. The resuling physics of he flow field in he enry zone can be complex because

2 9 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor of he ineracion beween compression and expansion waves [1-3]. Besides, shock waves propagaing upsream he blades are dissipaive and mus necessarily be aken ino accoun in he predicion of he sage performances [4]. urren FD (compuaional fluid dynamics) offers hree main caegories for simulaions: RANS (Reynolds-averaged Navier-Sokes), LES (large eddy simulaion) and DNS (direc numerical simulaion). RANS simulaions approximae he mean effec of urbulence, while DNS enables a full resoluion of he Navier-Sokes equaions, from he smalles urbulence scale (Kolmogorov scale) up o he inegral scale. LES corresponds o a filered DNS: only he larges urbulence scales are resolved, he smalles ones being modeled. The more urbulence scales are resolved, he finer he mesh mus be, and hus he more expensive he simulaion becomes. In a curren engine design process, only RANS simulaions can be carried ou. This sor of simulaion is based on he Reynolds decomposiion and urbulence models are added o close he se of equaions. U-RANS (unseady RANS) simulaions are generally no affordable in a concepion approach, because of heir high U cos. Tha is why he only ool ypically available for designers oday consiss in seady RANS simulaions. In he case of muli-row urbomachinery, hese simulaions rely mos of he ime on he use of mixing planes, which average he daa in he circumferenial direcion, and hus do no le he non-uniformiies in he flow field ransmi in he upsream or downsream direcion. This aricle focuses on he enry zone of a supersonic compressor. The filraion of he shock paern upsream he blades by he mixing plane raises he issue of he mixing plane locaion. The presen paper compares numerically differen locaions o poin ou he influence of he mixing plane on he flow field, noably in erms of sagnaion pressure change. In a firs par, he numerical resuls are qualiaively analyzed and he role of he mixing plane is highlighed. An analyical model is hen used o reproduce he shape of he shock waves a he leading edge of he blades. Finally, a mehod is given ha enables o judge he reliabiliy of seady simulaions on supersonic configuraions.. Tes ase and Numerical rocedure The es case is a cenrifugal unshrouded impeller designed and buil by Turbomeca. In he presen sudy, only he fron par of i is concerned. There is herefore no need giving he compressor geomery and performance in his paper. Only one operaing poin is examined, and i corresponds o he sonic blockage region. The flow, supersonic over 60% of he span, is examined a several secion heighs beween h 1 and h in Fig. 1. A simulaion performed wihou any mixing plane is used as reference (Fig. a). In his one, he inle block is roaing wih he impeller, and here is no paricular inerface. In order o evaluae he change in performance induced by he mixing plane approach, hree differen mixing plane locaions are numerically esed (Fig. b). They are labeled a, b and c in Fig. 1. In hose hree simulaions, he inle block is fixed, like a saor row and he periodiciy enables o use a smaller domain since he flow upsream of he mixing plane is circumferenially uniform. ompuaions were performed wih he elsa sofware developed a ONERA [5]. The code is based on a cell-cenered finie volume mehod and solves he h 1 h Air inle Fig. 1 Meridional skech of he compressor inle par.

3 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 93 Inle Oule Splier blade Main blade eriodic boundaries (a) for convecive fluxes and a nd-order cenered scheme for viscous fluxes. An LU implici phase (lower upper decomposiion) is associaed o he backward-euler scheme for ime inegraion. The near-wall region is described wih y + < 13. The inle condiion imposes he velociy angles and he sandard sagnaion pressure and emperaure. The urbulen values are deermined from a free-sream urbulence rae of 5% (resuling from previous measuremens). The oule condiion imposes a uniform value of saic pressure. The walls are described wih non-slip and adiabaic condiions. The seady sae enables o simulae only one blade passage, he azimuhal boundaries being periodic. Blade-o-blade surfaces are hen exraced from all simulaions a he same secion heighs beween h 1 and h. A he mixing plane inerface, a circumferenial average using Riemann invarians is compued a boh upsream and downsream faces. The resuling values (m 1 -m 5 ) are hen applied o he adjacen face wih a non-reflecive boundary condiion: Mixing plane = 0 wih 1 m 1 ( av )d S n S 1 m ( avn )d S S 1 m 3 ( a )d S S 1 m V ds 4 1 S 1 m V ds 5 S S ds he oal surface of he inerface. (b) Fig. (a) 3D view of he domain wihou mixing plane; and (b) 3D view of he domain wih mixing plane. compressible RANS equaions on muli-block srucured meshes. The k-l model of Smih [6] (chosen according previous work [7]) is used for urbulence modeling. The se of equaions are resolved in he relaive frame of each row, using he Roe space scheme 3. Numerical Resuls Fig. 3 shows he relaive Mach number in he reference case (wihou any mixing plane), in a blade-o-blade surface. The whie lines indicae he hree mixing plane locaions and he purple line shows he iso-conour M = 1. Wih he mixing plane locaed in (c), he subsonic zone beween he leading edge and he shock wave is clearly cu.

4 94 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Secion 1 (a) (b) (c) Secion Wihou mixing plane ase (a) Fig. 3 Relaive Mach number wihou mixing plane. Fig. 4 shows ha in he four configuraions, he deachmen disance of he shock remains consan, which ends o prove ha he mixing plane gives a correc value of he averaged Mach number. Indeed, as explained in he following, he deachmen disance only depends on he blade geomery and inle Mach number. The whie lines represen Mach iso-conours from 0.7 o 1.4. However, he shape of he subsonic zone is seriously affeced. The more he mixing plane is locaed downsream, he less he shock waves can exend upsream. Thereby, according o he posiion of he mixing plane, he oal pressure loss is under-esimaed. The change in sagnaion pressure can be quanified wih he value of he loss coefficien K calculaed as: K 1 wih, he momenum-averaged relaive sagnaion pressure inegraed on he whole surface a Secion (locaed a he blade leading edge) and on he whole surface a Secion 1 (locaed upsream) as: S 1 V ds S n n V ds ase (b) ase (c) Fig. 4 Relaive Mach number in he four es cases. As is expeced, he more he mixing plane is locaed downsream, he lower he losses are, and consequenly, he more he massflow is over-esimaed. Table 1 gives he difference beween overall inle massflow in cases

5 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 95 Table 1 erformance shif o he reference configuraion. Massflow K/Kref Mixing plane locaed in (a) +0.04% -1.% Mixing plane locaed in (b) +0.35% -8.3% Mixing plane locaed in (c) +0.80% -1.% (a), (b), (c) compared o he reference configuraion wihou mixing plane. The effec of he mixing plane can also be shown by he evoluion of he enropy along a consan span heigh in he supersonic region (black arrow in he meridional view in Fig. 5). By prevening he shock wave o exend upsream, he mixing plane inroduces a shif in he global level of enropy. These daa show ha he resul of he seady sae simulaion significanly depends on he mixing plane locaion boh in erms of performance (massflow and loss) and flow opology. The objecive of he following par is o propose a mehod which can be used o: (1) forecas he locaion of he mixing plane minimizing he shif in performance; () forecas he change in performance for a given mixing plane locaion. 4. Analyical Descripion Le us consider a supersonic incoming flow compressor wih subsonic axial velociy componen. Depending on he inle Mach number and on he back pressure level, wo differen regimes can exis: he unsared regime, characerized by a deached, quasi-normal shock across he passage; he sared regime, characerized by an aached oblique shock. In case of a blun leading edge, he shock canno be sricly aached and a small subsonic area exiss upsream he blade. The deachmen disance of he shock is obviously smaller in he case of sared regime han ha of unsared regime. The model presened hereafer is only valid for a sared regime. This is he firs reason why he sudy akes place near he sonic blockage: he shock wave has o remain aached o he leading edge of he blades. Fig. 5 Evoluion of enropy along he roaion axis z, a a consan span heigh. Moreover, he presence of splier blades here is likely o influence he shock sysem a he leading edge of he main blades hrough poenial effecs. Thus, he back pressure has o be imposed very low. The wo inpus of his model are he upsream Mach number and he geomery of he blade leading edge. The deachmen disance is calculaed wih Moeckel s mehod [8], which assumes ha he deached shock has a hyperbolic shape (Fig. 6). The equaion of he hyperbola is specified by is asympoe (of angle ) and he posiion of poin locaed on he sonic line [B], which is supposed o be sraigh. y M > 1 O Wihou mixing plane µ x 0 (a) M < 1 e B Fig. 6 Skech of a deached shock. d A (b) B (c) B LE x

6 96 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor In order o apply Moeckel s model, he upsream flow is supposed o be wo-dimensional, uniform and he profile is approximaed by a symmeric shape. The hermodynamic shock relaions for ideal gas are used. The maximum enropy rise, corresponding o a normal shock, is locaed on he sreamline passing by he leading edge. This line and he sonic line are assumed o be sraigh. The available equaions are: The equaion of he hyperbola: y ( x x ) g c c The equaion of he angen a poin : 0 (1) x g g () y which, combined wih Eq. (1), gives: x 0 y g g g (3) The equaion of he disance e B : g eb y ( y y ) g B g y g g g (4) In order o calculae he ordinae of poin, y, he coninuiy equaion is wrien beween he segmen [O ] upsream he shock wave and he sonic line [B]: y yb Vy ab a (5) cos which leads o: y y B V 1 a 1 a a cos Since a shock wave is isenhalpic, we can wrie: V a M 1 1 (1 M ) 1 (1 ) (6) (7) a a (8) I is imporan o noice ha a choice is possible a ha Eq. (8) in he way he sagnaion pressure raio is calculaed. In he presen case, a normal shock is considered a poin : a 1 1 M a 1 1 M (9) I would also be hinkable o consider an oblique shock o evaluae he hermodynamic sae along he sonic line. This is a he same ime a source of uncerainy in his mehod and a degree of freedom for he user (playing on his raio enables o fi FD or measuremens bu i is raher difficul o give a formula which works for all ypes of blades). The seps of he deachmen disance calculaion are hus: From he value of he inle Mach number M, is calculaed wih: 1 arcsin (10) M and he values of he deviaion and shock angle are deduced from he shock relaions. oin B which belongs o he profile is idenified from is angen B which has o be equal o. Eqs. (3), (4) and (6) are hen used o calculae he deachmen disance d: B B A d e x x (11) This procedure, iniially hough for symmeric isolaed profiles, leads o a geomeric represenaion of he subsonic zone (in grey in Fig. 6), beween he hyperbola and he sonic line. The par of he shock wave confining his subsonic region is responsible for mos of he oal pressure losses. Therefore, i is crucial o le i fully exend upsream he blades. The consequences of a mixing plane cuing he subsonic pocke are examined in he following.

7 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor omparison beween Analyical and Numerical Resuls The analyical mehod previously described for sared regimes enables o draw he shape of he deached bow shock and he subsonic zone. Fig. 7 superimposes he analyical shape (in black) on he FD resuls wihou any mixing plane (he yellow line is he iso-conour M = 1). Qualiaively, he subsonic area is well approximaed. Assuming ha he major par of he losses are due o he par of he shock confining he subsonic par, i is seen ha he analyical calculaion gives a prey good evaluaion of he minimal disance o be pu beween he mixing plane and he leading edge (d min in Fig. 7). However, wo weaknesses should be poined ou. Firsly, he model is very sensiive o he upsream Mach number, in paricular in he low supersonic Mach numbers region (beween M = 1.0 and M = 1.1). Fig. 8 shows he ypical evoluion of he deachmen disance as a funcion of he upsream Mach number. Furhermore, when he upsream Mach number decreases, he shape of he subsonic pocke ends o become more ellipic. Fig. 9 shows ha wih M = 1.1, he deachmen disance and he orienaion of he shock wave are sill well esimaed bu Moeckel s mehod canno reproduce he numerical resul near he leading edge. Secondly, he choice in he calculaion of he loss across he shock leads o he hermodynamic sae of he sonic line, because in he end, i direcly drives he value of he deachmen disance. Since Eq. (9) is non linear, he higher he upsream Mach number, he more imporan he way he loss is calculaed becomes. The main benefis of Moeckel s model are ha only he geomery of he profile and he upsream Mach number are needed. As a consequence, his is a ool which can easily be employed in a pre-design phase. I should be underlined ha he sonic blockage is he mos favorable case because when he mass flow decreases, he compressor will pass from sared d min Fig. 7 Applicaion of Moeckel s mehod wih M 1.3. Fig. 8 Evoluion of he deachmen disance for a given profile. Fig. 9 Applicaion of Moeckel s mehod wih M 1.1.

8 98 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor regime o unsared regime. Thus, he shock paern moves upsream, so ha he impac of he mixing plane can only become sronger. 6. Shock Loss redicion Alhough curren blade design relies on numerical opimizaion, including ransonic bladings [9], many analyical shock loss models exis, as described in Refs. [10, 11], for example. Bloch, openhaver and O Brien also give an ineresing approach based on Moeckel s mehod [1], adaped o unsared regime. Since he oher objecive of his paper is o analyically forecas he change in pressure loss for a given mixing plane locaion compared wih he reference case (wihou any mixing plane), he shock loss associaed o a sared regime has o be evaluaed. Fig. 10 shows he relaive oal pressure calculaed in he reference case (wihou mixing plane): he low relaive sagnaion pressure zone a he leading edge plane corresponds o he projecion of he sronges par of he shock on he leading edge plane in he mean flow direcion. In order o be consisen wih he numerical fields, only he par of he shock wave which is downsream he mixing plane locaion is kep and his porion of he upper branch of he hyperbola is projeced in he plane of he leading edge (Fig. 11). The losses are hen calculaed in he plane of he leading edge as if he fluid going hrough he bow shock was seeing a normal shock and he res of he incoming flow remained unperurbed. The losses are evaluaed wih he coefficien K defined as: K 1 (1) 1 where, is he lengh of he projeced hyperbola, is he pich and / 1 refers o he oal pressure raio across a normal shock (see Eq. (9)). The value of K is of course overesimaed, since a par of he incoming fluid acually goes across an oblique shock and he bow shock exending upsream becomes rapidly evanescen. Bu i enables o evaluae he shock loss in he mos unfavorable siuaion. Noe Secion 1 Secion max min Fig. 10 Relaive sagnaion pressure in a blade-o-blade surface. Mixing plane Incoming flow Incoming flow LE Isenropic par : no pressure loss Normal shock loss Bow shock cu by he mixing plane Fig. 11 Loss calculaion for a given Mach number wih a given mixing plane locaion.

9 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 99 ha in case of no mixing plane, he model is equivalen o considering a normal shock over he whole pich. This choice keeps he presen ool simple, bu i would be possible o make a more sophisicaed model: Firs, by improving he shock wave approximaion. A good example is given by Oavy [13], by coupling Levine s mehod [14] o predic he unique incidence seen by he blade profile wih Moeckel s deachmen disance calculaion. By discreizing he shock wave, so ha he flow angle and he pressure loss change from a normal shock on he blade axis o an oblique shock and inegraing he resul on a pich o he neighboring blade. For a given profile, i is possible o plo he loss as a funcion of he upsream Mach number in he differen configuraions (Fig. 1). The case wihou mixing plane corresponds o he shock loss across a normal shock. Depending on is locaion, he mixing plane has no more impac beyond a cerain Mach number. For example, wih a mixing plane locaed in Secion (b) (green curve), i can be seen ha beyond M = 1.46, he mixing plane has no more influence. I means ha if he upsream Mach number is higher han his value, he seady resuls can be considered reliable. For a given mixing plane locaion, he discrepancy compared o he reference case firsly increases wih he Mach number. Indeed, he lengh coninuously increases bu he pressure loss is increasing faser due o he shock wave. Then he upper branch of he hyperbola is more and more sraighened up ogeher wih a decrease in he deachmen disance unil is projecion covers he whole pich. A ha sep, here is no more difference wih or wihou a mixing plane and a seady RANS simulaion can be considered as reliable (Fig. 13). This analyical ool has been applied o he hree mixing plane posiions and o he reference case wih no mixing plane. The inpu daa are he circumferenially-averaged Mach numbers in Secion (b) coming from he FD wih no mixing plane and he geomery of he fron par of he blades. (a) (b) Fig. 1 (a) Evoluion of K for a given profile; and (b) zoom from Fig. 1a. M = 1.0 M = 1.50 M = 1.80 Fig. 13 Effec of he mixing plane on he bow shock for differen upsream Mach numbers.

10 100 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor Fig. 14 Analyical and numerical loss as funcion of he upsream Mach number for differen mixing plane locaions. Fig. 14 compares he analyical and numerical values obained for he losses. Differen secion heighs beween h 1 and h have been esed, so ha boh he profile and he Mach number were changing. The values of K are ploed as a funcion of he upsream Mach number and are calculaed as: where, K 1 π 0 π 0 1 V rd n n V rd In Fig. 14, he analyical evoluion corresponding o case (a) is he same as he reference one for Mach numbers greaer han This means ha he major par of he hyperbola is conained downsream he mixing plane for he corresponding secion heighs. According o his crierion, any mixing plane should be locaed upsream Secion (a) (Fig. 3) for he presen impeller. Neverheless, despie he good qualiaive agreemen beween analyical and numerical resuls in Fig. 7, we can observe quaniaive discrepancies in he shock loss. Firs of all, he simulaion wihou mixing plane describes a differen loss evoluion around M = 1.0 han hose wih a mixing plane. This is a well-known problem wih mixing planes in general: he loss is radially redisribued. The influence of he locaion of he mixing plane is clearly visible in he numerical curves. The slopes are no so far from he analyical ones. Bu here is a sor of offse beween he numerical and he analyical resuls. I is probable ha for he low Mach numbers, he shock loss is low compared o he viscous loss. To compare properly he wo curves families, i would be necessary o ake from he Navier-Sokes simulaions only he shock loss, as done in Ref. [1] by subracing he fricion loss from measuremens. The discrepancies are also due o he srong hypoheses made in he analyical mehod, which consiss in a wo-dimensional approach and due o he choice made for he loss calculaion. Real blade profiles are also ofen cambered and no symmeric in order o produce lif, which is no aken ino accoun in he presen model. And finally, he numerical loss a he highes Mach numbers (close o he secion heigh h ) is suddenly increasing, near he shroud boundary layer. I is likely ha fricion loss dominaes shock loss in his area. Thanks o his simple model, he order of magniude of he under predicion of he losses due o he inroducion of a mixing plane is easily evaluaed. I has been esed ha his approach gives accepable resuls from inle Mach number larger han 1.1. Once again, he major drawbacks of his mehod are ha i gives wrong predicions for lower Mach numbers and ha he shock formulas which are used in i are very sensiive. This is maybe one of he reasons why Bloch e al. [1] had o ake ino accoun an effecive leading edge radius in heir model dedicaed o predic he shock loss hrough he lower branch of Moeckel s hyperbola, in supersonic compressor cascades operaing in unsared regime. Indeed, hey increased he leading edge hickness unil he analyical resuls

11 Applicaion of an Analyical Mehod o Locae a Mixing lane in a Supersonic ompressor 101 mached he experimenal ones. This reminds us of he difficuly of implemening a shock loss model ha fis all ypes of profiles, under various operaing condiions. 7. onclusion Seady sae numerical simulaions performed wih a mixing plane approach show ha he resuls, in erms of mass flow and losses, significanly depend on he mixing plane posiion. The operaing poin chosen here corresponds o he sonic blockage of he compressor bu for near-surge poins, i would be even more imporan. The fac ha his sudy akes place near he blockage enables o propose an analyical mehod in order o forecas his change in performance. The validiy of his analyical mehod is checked by comparing is resuls wih he numerical ones in he enry zone of a ransonic compressor. Analyical and numerical resuls show good agreemen. This ool may be useful for ransonic compressor design: firs, o have an idea of he minimal disance ha should be pu beween he mixing plane and he leading edge of he blades, and hen o know how represenaively he seady simulaions can be expeced. Acknowledgemens We would like o hank Turbomeca which suppored his sudy, ogeher wih ONERA which collaboraed on he numerical simulaion. This work was graned access o he H resources of INES under he allocaion 013-a6356. References [1] Lichfuss, J. J., and Sarken, H Supersonic ascade Flow. rogress in Aerospace Sciences 15: [] Kanrowiz, A The Supersonic Axial-Flow ompressor. NAA echnical repor. [3] hauvin, J Supersonic Turbo-Je ropulsion Sysems and omponens. AGARD repor No. 10. [4] Trébinjac, I., Oavy, X., Rochuon, N., and Bulo, N On he Validiy of Seady alculaions wih Shock-Blade Row Ineracion in ompressors. In roceedings of he 9h Inernaional Symposium on Experimenal and ompuaional Aerohermodynamics of Inernal Flows, ISAIF9-06. [5] ambier, L., and Gazaix, M. 00. elsa: An Efficien Objec-Oriened Soluion o FD omplexiy. resened a 00 he 40h AIAA Aerospace Science Meeing and Exhibi, Reno, USA. [6] Smih, B. R redicion of Hypersonic Shock Wave Turbulen Boundary Layer Ineracions wih k-l Two-Equaions Turbulence Model. resened a 1995 he 33rd AIAA Aerospace Sciences Meeing and Exhibiion, Reno, USA. [7] Rochuon, N Analysis of he Three-Dimensional Unseady Flow in a High ressure Raio enrifugal ompressor. h.d. hesis, École cenrale de Lyon. [8] Moeckel, W. E Approximae Mehod for redicing Form and Locaion of Deached Shock Waves. NAA echnical repor. [9] Burguburu, S., Toussain,., Bonhomme,., and Leroy, G Numerical Opimizaion of Turbomachinery Bladings. Journal of Turbomachinery 16 (1): [10] König, W. M., Hennecke, D. K., and Foner, L Improved Blade rofile Loss and Deviaion Angle Models for Advanced Transonic ompressor Bladings: ar II A Model for Supersonic Flow. Journal of Turbomachinery 118 (1): [11] Schobeiri, M. T Advanced ompressor Loss orrelaions, ar I: Theroreical Aspecs. Inernaional Journal of Roaing Marchinery 3 (3): [1] Bloch, G. S., openhaver, W. W., and O Brien, W. F A Shock Loss Model for Supersonic ompressor ascades. Journal of Turbomachinery 11 (1): [13] Oavy, X Laser Anemomery Measuremens in an Axial Transonic ompressor. Analysis of he Unseady eriodic Srucures. h.d. hesis, École cenrale de Lyon. [14] Levine, The Two-Dimensional Inflow ondiions for a Supersonic ompressor wih urved Blades. WAD echnical repor