CONSTRUCTION AND USE OF IMITATING MODEL IN THE PROCESS OF MULTI-CRITERIA OPTIMIZATION OF THE SEPARATE-FLOW NOZZLE

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1 CONSTRUCTION AND USE OF IMITATING MODEL IN THE PROCESS OF MULTI-CRITERIA OPTIMIZATION OF THE SEPARATE-FLOW NOZZLE N.A.Zlenko*, S.V.Mikhaylov*, A.V.Shenkin* * TsAGI Keywords: nozzle, opimum aerodynami design Absra There is onsidered a proedure o an opimum aerodynami design o exhaus nozzles or highbypass-raio urboans (BPR8) using numerial simulaion o visous lows on he base o RANS equaions wih q- urbulene model. Problem o muli-rierion opimizaion o nozzle oniguraion is onsidered in severe saemen wih aking ino aoun real resriions on he aepable range o onrol parameers variaion. In order o simpliy soluion o opimizaion problems and derease volume o parameri alulaions, unional orrelaion beween nozzle haraerisis and he onrol parameers is presened as a muliaor simulaion model. Inroduion Designing boh nozzle and oher elemens o high-bypass urboan (HBT) has obvious ompromise haraer sine here is neessiy o provide sable and eeive work o ruising power plan or all ligh regimes. In suh siuaion, i is raher diiul o hoose deisive rierion (objeive union) beause onsidered objeive unions eiher have opimum in dieren zones o onrol parameer spae or onrol parameers has opposie inluene o hese unions. For example, diminishing ake-o noise gives diminishing he hrus a ruising ligh regime and worsening mass-overall parameers. One o he key iems o proposed mehodology is represenaion o paramerial alulaion resuls as a simulaion model ha deines a unional onneion beween onrol and geomerial parameers and inegral haraerisis o nozzle. Using he simulaion model essenially simpliies soluion o mulirierion opimizaion problem and, simulaneously, diminishes he volume o neessary paramerial alulaions. Eiieny o developed mehodology is demonsraed or shape opimizaion o high-bypass urboan nozzle wih bypass raio BPR~9.. 1 Choie o onrol parameers and opimizaion rieria Perormed analysis o numerial-experimenal invesigaion resuls o opimal geomery o bypass nozzle has shown ha eeive hrus losses are inluened by abou 10 geomerial parameers and ha expeed posiive ee o hoosing hese opimal parameer values is oen omparable wih es errors o hrus haraerisis o models in wind unnel (WT). I is obvious ha mos o geomerial parameers o bypass nozzle elemens an' be hanged independenly. For example, in he ase o approximaely onsan engine lengh, variaion o ouer onour nozzle owl lengh ineviable resuls in hange o gas generaor lengh and, hene, in hange o is angle onraion. Large dierene o pressure drops in dieren onours is haraerisi or HBT wih bypass raio BPR 8. Preliminary esimaions show ha gas low in ouer onour nozzle akes plae a superriial pressure drops (NPR 2 >2) and, in inner onour nozzle, a subriial pressure 1

2 N.A.ZLENKO, S.V.MIKHAYLOV, A.V.SHENKIN drops (NPR 1 <1.8). Ouer onour supersoni je resuls in a ompliaed low sruure wih inerhange o rareaion zones and shok waves. Their posiion inluenes on subsoni je o inner onour is realized. To develop an adequae simulaion model o bypass nozzle hrus haraerisis wih aking ino aoun 10 geomerial parameers, i is neessary o reae a daa bank inluded no less han 3 10 poins. Beause i is impossible o es neessary number o model varians, numerial simulaion mehods o low on basis o Navier- Sokes equaions an be used as a ool o aerodynami designing o bypass oupu devie. Bu, even in his ase, alulaion o ~ nozzle varians is unreal ask. For orre soluion o opimizaion problem, i is neessary o hoose a se o onrol parameers among whole diversiy o onrol parameers. They mus orrespond o low physis, permi o develop an adequae simulaion mahemaial model o nozzle hrus haraerisis and inluene boh on aerodynami and on massoverall haraerisis o bypass nozzle. In he urren work, ollowing geomerial parameers have been hosen as onrol hese, in aordane wih he sheme in Fig. 1: heoreial inlinaion angle o nozzle ouer onour exi seion relaively he nozzle symmery axis,, rounding radius o gas generaor owl near nozzle ouer onour exi seion, R, onraion angle o gas generaor owl a he exi seion o nozzle ouer onour,, onraion angle o onial par o inner onour enral body, Fig. 1. Conrol geomerial parameers or bypass nozzle Conrol parameers have ollowing ranges: been varied in θ max(θ θ min min R min θ min θ θ R R,θ ) θ θ max max max,., θ, (1.1) A lassial problem o nozzle aerodynami design is o searh suh ombinaion o geomerial parameers ha provides maximal oeiien o eeive hrus P and, с e orrespondingly, minimal eeive hrus losses o nozzle P 1 P. I should be с e с e noied ha eiien hrus oeiien o bypass nozzles is deined as relaion beween sum o inner and ouer onour hruss (exluding exernal drag ore o ouer onour) and sum o inner and ouer ideal hruss. As a rule, a designer, during HBT developmen, have o ake ino aoun boh aerodynami and mass haraerisis o projeed objes, i.e. a muli-rierion opimizaion problem o power plan elemens is solved. Only an engine designer an perorm a qualiied alulaion o nozzle mass haraerisis. Thereore, i is useul o apply a simpliied model o nozzle mass haraerisis a he sage o esing he muli-rierion opimizaion mehodology, when he engine designer doesn ake par ino opimizaion proess. In he sope o he urren work, a surae mahemaial model, where nozzle mass (G ) is proporional o is surae area is proposed. Suh model permis o ompue hange o sruural mass in dependene o our onrol parameers. The mass o onsruion inludes he mass o ixed par G ix ha is aken he same or all varians o nozzle, beause i doesn praially depend upon onrol parameers. The mass o onsruion also inludes he mass o a variable par G var ha essenially depends upon onrol parameers: G = G ix + G var (, R,, ) (1.2) Variaion o onsruion mass relaively some basi varian o nozzle G с base ha mass is alulaed using similar mehodology is deined as ollows: 2

3 CONSTRUCTION AND USE OF IMITATING MODEL IN THE PROCESS OF MULTI-CRITERIA OPTIMIZATION OF THE SEPARATE-FLOW NOZZLE G=G /G с base 1 (1.3) For basi nozzle varian, values o onrol parameers a he ener o aessible region o variaion have been aken. 2 Developmen o mahemaial model o bypass nozzle geomery, reaion and auomai modiiaion o alulaion grid All he alulaions in he urren work have been perormed on basis o he ode ZEUS EWT TsAGI [1] using q urbulene model. Calulaions have been perormed only or ruising ligh regime wih Mah number М =0.8. Using he numerial invesigaion resuls or basi oniguraion, a mahemaial model o bypass nozzle geomery ha is based on analyial dependenes has been proposed. Generarixes o axisymmeri nozzles o he inner and ouer onours have been given by sraigh segmens and irular ars. Firs o all, he geomery o ouer onour nozzle has been opimized. The geomery o subsoni par o ouer onour nozzle hasn been varied during he alulaions. Preliminary esimaions show ha HBT wih bypass raio BPR~9 has inner onour nozzle hrus abou 15% o ouer onour nozzle hrus. Thereore, he main aenion has been devoed o opimizaion o ouer onour nozzle. Only onraion angle o ener body has been varied among all geomerial parameers o inner onour nozzle. The diameer o boom ip a enral body was he same or all he varians. 3 Peuliariies o developing he muliaor simulaion model o nozzle To alulae he inegral haraerisis o nozzle, a simulaion model is proposed. This model is one o possible varians or deerminaion o he unional onneion beween onrol geomerial parameers (aors) and inegral haraerisis o nozzle (responses) objeive unions. The simulaion model is analyial dependenies, whih permi o esimae he response value Y wih neessary auray a any poin o aessible region o aor s variaion. The urren work uses mehods o linear regression analysis [2] in developing he simulaion model o nozzle inegral haraerisis. In he ramework o his analysis, model ype is deined as ollows: k Y( X ) b ( X ) b i i ( X ), i 1, (3.1) X, where X is a veor o aors, (X ) is a unional veor; i(x ) are basis unions omponens o veor (X ) ; k is dimension o oeiien veor b. The paper [3] desribes an algorihm or developing he muliaor regression equaion (3.1), when here is no a priori inormaion abou view o unional veor (X ). In aordane wih his algorihm, a irs, iniial alulaions are perormed o obain 1D seion or eah aor x j and o hoose ormulas ha approximae hese seions suiienly y( x j ) bj j ( x j ) (3.2) Aer ha, he view o unional veor (X ) is deermined on basis o ormulas (3.2) using raher simple and well-deined proedures. Beause he simulaion model o nozzle is direed, irs o all, o soluion o opimizaion problems, i seems useul o modiy algorihms rom [3] or hoosing a series o 1D seion ha are neessary o deine 1D approximaion ormulas (3.2). For ha, a he irs sage o daa analysis, i is assumed o perorm a preliminary one-rierion opimizaion using he oordinae desen mehod. I is known ha his mehod deermines he rajeory approahing o exremum using he analysis resuls o 1D seions y x ) and ha spae posiion o eah ( j subsequen seion aors is deined wih aking ino aoun opimal values o aors (op) x j ha have been deermined a he previous sages. During he preliminary opimizaion, several problems are solved a he same momen. Firsly, obained 1D seions are used o deine he orm o equaions (3.2). Seondly, 3

4 N.A.ZLENKO, S.V.MIKHAYLOV, A.V.SHENKIN a subregion wih he bes values o response is hosen in aor spae and graviy ener o iniial daa sample shis o his subregion. I has a good inluene on simulaion model (3.1) adequay. In addiion, qualiied inormaion abou zone o objeive union exremum is useul boh o orre he boundaries o aessible region o aor variaion and in drawing he plan o paramerial alulaions. orresponds o approximaion values and he symbol sar shows posiion o eeive hrus oeiien maximum P. с e 3.1 Preliminary opimizaion o nozzle shape, analysis o 1D seions As an iniial poin or oordinae desen mehod, a ener o aessible region o aor variaion (1.1) has been hosen, where several values o eeive hrus P have been с e alulaed. These values have orresponded o dieren values o angle, i.e. poins o 1D seion P ) have been deermined. Using с эф ( a dialogue sysem APPEX [], an approximaion ormula has been hosen o provide adequae represenaion o 1D seions P ). Obained approximaion ormula has с эф ( been used o deermine he value orresponds o Pс e maximum. Obained (op) (op) ha has deermined a new poin o aor spae, where approximaion ormula or he seion (op) R P с e (R) has been hosen and he value has been deermined. Furher, he proess has been repeaed or remain onrol parameers and. As a resul, a poin in zone wih high values o objeive union P с e has been hosen in aor spae and, a he same ime, orm o approximaion equaions (3.2) has been deined: P( ) 1( x1 ) {1, Sin( x1 ), os( x1 ), Sin(2 x1 ), os(2x1 ), sin(3x1 )}, 2 3 P( R) 2 ( x2 ) {1, Ln( x2 ), Ln ( x2 ), Ln ( x2 )}, P( г ) 3( x3) {1,Sin( x3), os( x3), Sin(2 x3)}, 2 P( ) ( x ) {1, x, x }. к (3.3) Qualiy o 1D seion approximaion an be esimaed in Fig. 2, where he solid line Fig. 2. Approximaion o 1D seions Aer riangulaion o obained poins orresponded o 1D seions P ) and P e с e ( с (R), levels o union P с e ( op) ( op) (, R,, ) an be develop. These levels are shown in Fig. 3. Figure 3 shows ha levels obained using iniial daa (solid lines) and using 1D approximaion (dashed line) are very lose. I onirms orrespondene o hosen approximaion ormulas. Fig. 3. Levels o union P (, R) с e

5 CONSTRUCTION AND USE OF IMITATING MODEL IN THE PROCESS OF MULTI-CRITERIA OPTIMIZATION OF THE SEPARATE-FLOW NOZZLE 3.2 Developmen o muli-aor simulaion model ype Aording o [3], approximaion ormulas (3.3) are he basis or developmen o muli-aor regression equaion. Le s deine he symbol as an operaion ha is used o wo unional veors and permis o obain a veor, whih omponens are various pairwise produs o iniial veor omponens. Then, ollowing unional veor (X ) is deined or he simulaion model (3.1) ( X ) 1( ) 2( R) 3( г ) 3( к ) (3.) I should remember ha suh approah o developmen o simulaion model ype gives raher ompliaed resuling regression equaion (3.1) (k=288 or ormulas like (3.3)). I an inlude erms ha inessenially inluene on response behavior. Bu, i only signiian erms remain in he equaion a he sage o regression oeiien alulaion, hen he lengh o resuling regression equaion an be essenially diminished and, appropriaely, he volume o iniial daa ha are neessary o develop he simulaion model an be dereased. One o he responsible momens o developing he simulaion model is o hoose a plan o paramerial alulaions, i.e. deerminaion o he number and posiion o alulaion poins in aor spae. Mehods o experimen mahemaial planning [5] permi o hoose an experimen plan or regression model o given ype, so as neessary saisial properies o his model are provided and, a he same ime, he volume o iniial daa sample is minimal. When he inal ype o simulaion model an be deined, mehods o experimen mahemaial planning [5] an be used. To develop he plan o paramerial alulaions, he urren work proposes o use so-alled LP τ sequenes [6], whih permi he mos uniorm disribuion (in erms o saisis) o alulaion poins in aor spae. Paramerial alulaion plans developed using LP τ sequenes has ollowing advanages: - values o eah aors are varied a nonreurren levels. I permis o obain more ull represenaion o aor inluene on he response and o alulae aepable saisial properies o resuling regression equaion; - he algorihm o developing LP τ sequenes provides expansion o iniial daa sample wihou neessiy o realulaed he poins inluded ino plan a he previous sage; - during he muli-rierion opimizaion, i is possible o hoose approximaely eiien poins ha are esimaion Pareo s poins. The plan o paramerial alulaions has been hosen wih aking ino aoun he limiaions (1.1) ha deine he aessible region o aor variaion. The disribuion o poins inside he aessible region o aor spae has been deined using LP τ sequenes and has been added by angular poins ha deine limiing values o aors. General number o plan poins is equal o 78. To deermine he inal orm o simulaion model (3.1), he urren work uses a sep-by-sep regression proedure ha permis essenial diminishing he number o erms in he inal regression equaion. In he ramework o his proedure, he mos onsiderable erms o basi equaion (3.) are inluded ino developed regression equaion sequenially unil growh o muliple orrelaion oeiien r and simulaneous derease o mean-square deviaion ss beome negligible. The nex sage o sep regression proedure ompleion is o esimae he adequay o obained varian o simulaion model. Beause he dispersion an be deined or resuls o numerial alulaions, i isn possible o perorm sri veriiaion o regression relaion, as i is aeped in ramework o regression analysis [3]. In he urren work, when a soluion abou adequay o obained simulaion model is made, boh approximaion auray o plan poins and prediing properies o model or onrol daa sample are aken ino aoun. The resuls o omparing he levels drawn using iniial daa and regression relaions are also aken ino aoun. As a onrol sample, daa ha have been obained a he sage o 1D seion analysis P x ) are onsidered. In с e ( j Fig., dashed lines show he approximaion resuls o onrol daa sample using muli-aor simulaion model. I is visible ha approximaion urves are in good orrespondene wih alulaion resuls and 5

6 N.A.ZLENKO, S.V.MIKHAYLOV, A.V.SHENKIN mean-square deviaion is abou ~0.005%. I is raher aepable or praial appliaions. Fig.. Muli-aor approximaion o 1D seions ss For he purpose o urher inrease o simulaion model qualiy, he plan o paramerial alulaions has been added by poins rom onrol daa sample and he sep regression proedure has been repeaed wih aking ino aoun he ombined sample o iniial daa. Suh proedure boh enlarged he number o reedom degrees a he sage o esimaion o regression oeiiens and improves prediive properies o simulaion model a he mos ineresing subregion o aor spae near maximum o eeive hrus oeiien. The solid line in Fig. orresponds o eeive hrus oeiien values alulaed using inal varian o nozzle simulaion model. Algorihms o bypass nozzle geomery using simulaion model As i has been noied above, replaemen o disree series o numerial alulaion resuls by raher simple analyial expression boh simpliies soluion o dieren problems, inluding opimizaion problems, and exludes neessiy o addiional numerial alulaions even in he ases, when problem ormulaion is essenially hanged. Examples o ombined using obained simulaion model and ormulas (1.2)-(1.3) or soluion o opimizaion problems in dieren ormulaion are presened below..1 Convenional one-rierion opimizaion The problem o onvenional exremum searh is ormulaed as ollows. In he admissible region o varying he geomerial parameers (aors) (1.1), i is neessary o deermine he poins, where eeive hrus oeiien and relaive hange o nozzle weigh ahieve heir maximal and minimal values. For soluion o his problem, dire mehod, so-alled Box s omplex-mehod, is used. This mehod permis o solve he problem o exremum searh wih aking ino aoun limiaions o boh he irs (1.1) and he seond ype. Figure 3 demonsraes levels o he union P (, R in relaive oordinaes с e ) θ θ op (, R). The symbol rhomb shows he posiion o eeive hrus maximum and he symbol irle shows he poin orresponded o minimum o nozzle mass inremen..2 Muli-rierion opimizaion o bypass nozzle Usually, in developmen o real onsruions, i is neessiy o ake ino simulaneously aoun several onradiory requiremens. Figure 5 shows ha minimal nozzle weigh and eeive hrus maximum are realized in dieren poins o onrol geomerial parameer spae. I is obvious ha, in suh siuaion, i is impossible o design a nozzle wih minimum possible mass and wih maximal hrus eiieny. Thereore, i is neessary o searh ompromises, i.e. o solve a muli-rierion opimizaion problem. Quaniy o works and is qualiy, when suh problems are solved, are in dire relaion wih mehod o obaining he haraerisis o designed arile and wih he number o onsidered varians. Analyial represenaion o nozzle haraerisis (in he orm o simulaion model) permis o solve in sri mahemaial ormulaion almos all opimizaion problems. Examples o muli-rierion opimizaion problem soluions using bypass nozzle simulaion model are presened below..2.1 Deerminaion o a unied objeive union One o possible varians o solve a mulirierion problem is o hoose a unied deisive rierion ha is a ombinaion o iniial 6

7 CONSTRUCTION AND USE OF IMITATING MODEL IN THE PROCESS OF MULTI-CRITERIA OPTIMIZATION OF THE SEPARATE-FLOW NOZZLE objeive unions. As a resul, he problem is slighly simpliied and redued o one-rierion opimizaion. As suh rierion, a weighed sum o P P (, R,, ) eeive с e 1 с e hrus losses and G G(, R,, )/ G 1 nozzle relaive mass hange is proposed: base F(, R,, ) P wg, (.1) с e where w is a weigh oeiien. Limiing siuaion or he union (.1) is: w=0 one-rierion opimizaion o nozzle aerodynamis, w>1 one-rierion opimizaion o onsruion mass. Generally, he value o oeiien w depends upon HBT bypass raio, upon uel rae, required engine hrus and ligh duraion. I means ha he oeiien w or eah onree airra modiiaion mus be deined separaely, during he proess o obje invesigaions. For eah given value o w, he opimal orm o he nozzle is deined by geomerial parameer values,,,, so as minimum o union R (.1) is ahieved. Figure 5 shows values o in opimal poin in dependene o P с e weigh oeiien w. Fig. 5. Dependene o P с upon weigh oeiien e w in he opimal poin Char presened in Figure 5 permis boh o hoose opimal nozzle geomery and o deine is haraerisis or any w wihou using he simulaion model and wihou any addiional alulaions. Bu, when orm o equaions (.1) is hanged, he proedure o onvenional onerierion opimizaion mus be repeaed..2.2 Compromise urve alulaion One o ommonly used varians o solving he muli-rierion opimizaion problem is o plo a ompromise surae, whih is deined as a se o eeive or Pareo s poins, in he spae o deisive rieria (objeive unions). In hoosing he opimal geomery o bypass nozzle, a se o Pareo s poins is monoone dereasing urve ha is an envelope or he region o admissible poins in he plane o deisive rieria на плоскости deisive rieria (, G). P с e There are dieren mehods o plo he ompromise urve or Pareo s ron. The urren work proposes ollowing algorihm o deermine he poins ha belong o he ompromise urve. As an objeive union, eeive hrus losses P, R,, ) are с e ( aken. Limiaions (1.1) ha deermine he admissible region o geomerial parameers are added by a limiaion o he seond ype: G(, R,, ) G. (.2) lim Aer ha, a onvenional exremum (minimum) o he union P, R,, ) is searhed с e ( or eah given Glim using Box omplexmehod. I is assumed ha, during he searh o exremum, inermediae values o P, R,, ) and G,,, ) are с e ( ( R alulaed using simulaion model developed in advane and ormulas (1.2)-(1.3) or esimaion o nozzle mass. Admissible values o limiaion o he seond ype Glim (.2) belong o he segmen ( Gmin, Gmax ). Is boundaries have been deined in advane, a he sage o onerierion opimizaion. The resuls o ompromise urve ploing using he algorihm above are presened in Fig. 6, where he symbol «*» designaes Pareo s poins obained or dieren values o Glim and he symbol «rhomb» orresponds o alulaion poins used in designing he simulaion model. 7

8 N.A.ZLENKO, S.V.MIKHAYLOV, A.V.SHENKIN Fig. 6. Compromise urve Figure 6 shows ha obained Pareo s poins belong o monoone dereasing urve and all plan poins are above hem. Hene, his urve an be deined as an envelope or admissible region o objeive unions P, R,, ) and G,,, ). As i с e ( ( R has been menioned above, one o he advanages o he paramerial alulaion plan, whih has been ormulaed using LP τ sequenes, is uniorm disribuion o poins in he region o admissible values. I permis o hoose approximaely eeive poins among a se o iniial daa. Suh poins, whih are marked by he blak olor in Fig. 6 and onneed by he dashed line, an be reaed as a zero approximaion or he ompromise urve. I should be noied ha suh preliminary esimaion o Pareo s poin posiion an be perormed a early sages o daa analysis, beore developmen o he simulaion model and he muli-rierion opimizaion proedure. Sample o Pareo s poins an be approximaed by a pieewise smooh urve using he dialogue sysem APPEX []. The solid line in Fig. 6 orresponds o he resuls o Pareo s poin approximaion using his ormula. I should remember ha eah poin o ompromise urve orresponds o onree values o boh geomerial parameers and objeive unions. Reerenes [1] Vlasenko V.V., Mikhaylov S.V. Code ZEUS or alulaion o non-saionary lows in ramework o в RANS and LES approahes. Maerial o XX shoolseminar "Aerodynamis o airras ". 2009, pp [2] Dreiper N., Smih G. Applied regression analysis. Mosow : Saisis, [3] Zlenko N.A. Using he regression analysis mehods or developing he simulaion model o loal ields. Tehnis o air lee. 2009, Vol. LXXXIII, 1. [] Bobsov V.A., Zlenko N.A. Auomaed dialogue sysem or approximaion o experimenal daa. TsAGI's Proeedings. 1993, [5] Voznesensky V.A. Saisial mehods or planning he experimen in ehnial and eonomi invesigaions. Mosow : Finanes and saisis, [6] Niederreier H. Quasi-Mone Carlo mehods and pseudo-random numbers. Bullein Amer. Mah. So. 1978, Vol. 8, 6, pp Cona Auhor Address sergey.mikhaylov@sagi.ru Copyrigh Saemen The auhors onirm ha hey, and/or heir ompany or organizaion, hold opyrigh on all o he original maerial inluded in his paper. The auhors also onirm ha hey have obained permission, rom he opyrigh holder o any hird pary maerial inluded in his paper, o publish i as par o heir paper. The auhors onirm ha hey give permission, or have obained permission rom he opyrigh holder o his paper, or he publiaion and disribuion o his paper as par o he ICAS 201 proeedings or as individual o-prins rom he proeedings. 8