Three-Dimensional Aerodynamic Design Optimization of a Turbine Blade by Using an Adjoint Method

Transcription

1 Jiaqi Luo Juntao Xiong Feng Liu Depatment of Mechanical and Aeospace Engineeing, Univesity of Califonia, Ivine, Ivine, CA Ivan McBean Alstom Powe (Switzeland), Baden 54, Switzeland Thee-Dimensional Aeodynamic Design Optimization of a Tubine Blade by Using an Adjoint Method This pape pesents the application of an adjoint method to the aeodynamic design optimization of a tubine blade. With the adjoint method, the complete gadient infomation needed fo optimization can be obtained by solving the govening flow equations and thei coesponding adjoint equations only once, egadless of the numbe of design paametes. The fomulations including imposition of appopiate bounday conditions fo the adjoint equations of the Eule equations fo tubomachiney poblems ae pesented. Two design cases ae demonstated fo a tubine cascade that involves a high tip flae, chaacteistic of steam tubine blading in low-pessue tubines. The esults demonstate that the design optimization method is effective and the edesigned blade yields weake shock and compession waves in the supesonic egion of the flow while satisfying the specified constaint. The elative effects of changing blade pofile stagge, modifying the blade pofile shape, and changing both stagge and pofile shape at the same time ae examined and compaed. Navie Stokes calculations ae pefomed to confim the pefomance at both the design and off-design conditions of the blade designed by the Eule method. DOI:.5/.466 Intoduction In ecent yeas, eseach and application of aeodynamic design optimization ADO technology based on computational fluid dynamics CFD have become vey active. Thee ae a numbe of diffeent stategies with a ange of effectiveness and diffeent levels of computational effot. Compaed with the long hous of manual iteation typically found in the tubine design pocess, optimization can be executed by an automatic compute pocedue based on advanced optimization theoy and thus allow the designe to focus on achieving an optimal flow stuctue and blade pefomance. In geneal, a typical ADO pogam consists of two main components: a flow solve and an optimize. The flow solve is used to analyze the flow field and evaluate aeodynamic pefomance. The optimize is used to identify how to change the geomety to achieve a paticula themodynamic objective. An essential pat of developing a gadient-based optimization method is to find a fast and accuate way to calculate the gadient infomation because it is the most time-consuming pat in the whole design pocess. Although a vaiety of taditional appoaches, such as esponse suface methodology 2,3 and finite diffeence methods 4, ae still widely used in ADO poblems, developing moe efficient and accuate optimization methods is of high pioity to designes because the computational effot fo the taditional methods is popotional to the numbe of design paametes and thus vey costly when the numbe of design paametes is lage. An altenative way is the adjoint appoach advocated by Jameson,5. This method eliminates the explicit dependence of gadient infomation on the vaiation of the flow field though the use of the adjoint equations to the oiginal flow equations. The solution of the adjoint equations is on the same level of effot as one flow solution. By solving the Eule/Navie Stokes NS equations and thei coesponding adjoint equations only once, one can obtain the complete gadient infomation needed fo optimization Visiting Student fom Depatment of Fluid Mechanics, Nothwesten Polytechnical Univesity, Xi an 772, China. Contibuted by the Heat Tansfe Division of ASME fo publication in the JOUR- NAL OF TURBOMACHINERY. Manuscipt eceived May 4, 29; final manuscipt eceived June 26, 29; published online Septembe 27, 2. Edito: David Wisle. independent of the numbe of design paametes. Jameson and co-wokes,5 7 pefomed aeodynamic design optimization fo aifoils, wings, and wing-body combinations. The adjoint method has been subsequently used fo the design and optimization of many complex configuations fo extenal flows. Howeve, at pesent, thee ae only a few epoted applications of the adjoint method fo the design and optimization of tubomachiney blades, fo example, Refs Compaed with the extenal flow ove aicaft wings, tubomachiney flow is much moe complex because of the complex intenal geomety, the elative motion of the blades, and the multistage configuation of the machine. Thee ae many distinct aspects of the design of tubomachiney blades that have not been studied befoe, including the pope choice and implementation of objective functions, constaints, design paametes, and bounday conditions. Fo example, an impotant aspect fo tubomachiney is the specification of themodynamic constaints on a blade ow o stage as dictated usually by a one-dimensional though-flow design of the complete compesso o tubine system, which is usually sepaately optimized in the peliminay design. One may choose the total to static pessue atio and swallowing capacity as constaints and allow the ow o stage loading to change due to changes in the blade shape. Altenatively, one may fix the total to static pessue atio and the ow o stage loading but allow changes in the swallowing capacity. The design pocess should not allow any excess feedom that can lead to false pefomance impovement because of unintended changes in the onedimensional themodynamics athe than a tue impovement in the pefomance of the blade design at the given paticula themodynamics fo the ow o stage. The pesent pape attempts to develop the adjoint appoach fo the design and optimization of tubomachiney blades, in paticula, to addess the above mentioned issues. We pesent unique fomulations of the objective functions, constaints, and design paametes that ae suitable and effective fo tubomachiney blade ows. The inlet and outlet bounday conditions fo tubomachiney flows lead to special bounday conditions fo the adjoint equations. The diffeent bounday conditions and the gadient fomulas coesponding to the diffeent design objectives ae deived and pesented fo the Eule equations. The inte-elations of the objective functions, the constaints, and the design paametes ae Jounal of Tubomachiney Copyight 2 by ASME JANUARY 2, Vol. 33 / 26-

2 YZ X of design paametes that specify the geomety of the blade; and 3 any design constaints on the geomety o the flow. The choice of the above functions and the basic pinciples of the adjoint method will be discussed in Secs Inlet Casing Blade Outlet 2. Choice of the Cost Function. In the pesent wok, two types of cost functions ae used. In the fist, we stive to design the blade such that the pitch-aveaged flow angle at the exit of the blade ow attains a specified distibution. Vey often the designe may have a desied spanwise distibution of the flow angle and would like to achieve that by estaggeing the blade pofiles while maintaining the same aveaged tuning. This is an invese design poblem. We can always convet such an invese design poblem to an optimization poblem by defining a cost function that measues the absolute o least-squaes eo of the designed flow angle distibution fom the desied distibution. We define the following least-squaes eo as the cost function: Hub I = 2 2 d Fig. Configuation of a typical low-pessue tubine stato studied fo the success of the design. One significant contibution of this wok is the development and use of a estaggeing method of the blade pofiles coupled with modification of the blade pofiles. Such a combined appoach not only eflects the pactice of the industy but also is shown hee to be citical fo obtaining bette pefomance. A simple pofile change without changing stagge o a change in stagge without modification of the pofile is shown to be not sufficient to bing an impoved design within the given themodynamic constaint. The details of the mathematical fomulation and the method to estagge and modify the blade pofiles fo an annula blade ow ae pesented in Sec. 2. Section 3 demonstates the ability of the poposed fomulations of the adjoint method to design and optimize a typical tubine blade ow. Two cases ae consideed. The fist is an example of invese design by use of the optimization method, whee a pope spanwise distibution of blade stagge angle is sought to achieve a given spanwise distibution of pitch-aveaged exit flow angle. The second case seeks the edesign of the blade by changing both the local stagge angle as well as the blade pofile to minimize the total pessue loss. Two diffeent constaints ae consideed: aveaged flow tuning and mass flow ate. Both the invese design and the pefomance optimization cases ae shown to be vey effective in achieving the design objectives while satisfying the constaints. Finally, the designed blades in the second case ae checked fo thei pefomance at off-design conditions by both Eule and Navie Stokes computations to assess the obustness of the design. 2 Desciption of Optimization Poblem We apply ou optimization method to a typical tubine blade ow. Figue shows the geomety of the case consideed. This is typical of a stato blade found in a low-pessue steam tubine in tems of the pitch to chod atio ove the blade span, the exit angle, the flat-backed pofile sections, and the high flae angle at the tip of the blade. At the blade hub, the contou is at a constant adius. The flow ove the blade is usually tansonic; the exit Mach numbe is above unity ove most of the blade span. The design optimization is pefomed at the same design conditions. At the inlet, the total pessue, total tempeatue, and inflow angles ae specified. At the outlet, the exit static pessue coesponding to an isentopic Mach numbe of.6 is specified. At the outset of a design optimization task, one has to define thee sets of vaiables o functions: the design objective o cost function to be minimized; 2 the design space, usually in the fom whee the integation is done fom hub to tip at the exit coss section of the blade passage, is the cicumfeentially aveaged flow angle at a given adial location, and is the specified taget flow angle distibution. The second type of objective functions involves cost functions that diectly measue a pefomance paamete of inteest of the tubomachiney blade ow. One might choose to maximize the total exit to inlet pessue atio o the adiabatic efficiency. An equivalent and pehaps moe geneal appoach is to minimize the mass-aveaged entopy geneation acoss a blade ow. Thus, we choose I = s gen = B o s u j N j da Bo u j N j da whee B o indicates the exit bounday of the blade ow, and da is the diffeential aea. The fomulations of the adjoint equations and thei bounday conditions coesponding to the Eule equations fo tubomachiney poblems ae discussed below. 2.2 The Adjoint Equations and Bounday Conditions. Recognizing its dependence on the flow field solution W and the geomety of the blade ow F, we wite the cost function in eithe Eq. o Eq. 2 in the following geneal functional fom: I = I W,F A diect application of vaiation pinciple to Eq. 3 yields I = I W W + I F F The fist and the second tems on the ight-hand-side of the equation epesent the changes due to the vaiation of flow field and that of the geomety, espectively. The vaiation of the flow field W depends implicitly on the vaiation of the geomety F though the govening flow equations, such as the Eule o the Navie Stokes equations, which may be geneally epesented by the following functional fom: R W,F = The geomety of the blade ow F in pinciple is descibed by continuous functions and theefoe contains an infinite numbe of paametes although vey often F is eplaced with a finite albeit lage numbe of discete design paametes. A diect evaluation of W with espect to F involves epetitive solution of the govening flow equations and thus is vey time consuming if not pohibitive fo lage numbes of design paametes even when a simplified suogate model is used in place of the oiginal flow equations. The key of the adjoint-equation method is to eliminate / Vol. 33, JANUARY 2 Tansactions of the ASME

3 the need fo diectly computing W. Taking the vaiation of Eq. 5, we obtain R = R W W + R F F = Multiplying Eq. 6 by a Lagange multiplie T and then subtacting it fom Eq. 4, we obtain I = T I R + T W W I R F F W F The explicit dependence of I on W is eliminated by setting I R = 8 T W T W which is ecognized as the adjoint equation to Eq. 5 fo the so-called co-state vaiable. Once we obtain by solving the adjoint equation, the needed gadient of I with espect to the geomety change F is detemined by I = G F whee G is the gadient and G = T I R F F The deivative tems in the above equation can be easily and cheaply computed even fo lage numbes of design paametes because we can deive the explicit algebaic dependence of I and R on F. The poblem of finding G is then educed to solving govening flow equation 5 and its adjoint equation 8 each once at a given design cycle independent of the numbe of design paametes. The adjoint equation is simila to the oiginal flow equation in type and dimensions and thus equies the same level of computational effot fo solution. Since in this study all the cost functions ae defined at the exit, we can epesent them as an outlet bounday integal I = B o CdB whee C is a scala function of both flow vaiables and geometic vaiables and depends on the definition of the cost function. A weak fom of the Eule equations is D T i F i dd B T F i db = 2 whee F i =S ij f j, and S ij =JK ij, K ij = x i / j. Adding Eq. 2 to the vaiation of cost function, we have I CdB n i = B o B T T F i db F i dd + D i 3 We beak C into two pats, C f and C g, which epesent the flow vaiation tem and the geometic vaiation tem, espectively. Let C f =c T W on B i and B o. Then Eq. 3 can be witten as I = B i c T n i S ij T A j W db + B o c T n i S ij T A j W db + B w C f n i S ik k+ p db + D T i S ij A j W dd + I g 4 Fom Eq. 4, we can detemine the bounday conditions at the inlet, outlet, and wall. Because of the evesed diection of wave popagation, the method used to detemine the bounday conditions of the adjoint equations is diffeent fom that used fo the Eule equations. Yang et al. 9 discussed the details in thei pape. In this pape, the fomulations of the adjoint equations, detailed bounday conditions, and gadient ae pesented in the Appendix. 2.3 A Simple Restagge Method fo Annula Cascade. Modifications of the geometic design F may be diectly achieved by moving the gid points on the suface of the blade o othe pats of the flow bounday of inteest in a numeical computation. Howeve, doing so may violate cetain geometic constaints and may not eflect the cuent pactice of tubomachiney blade ow design. To simplify the design task, tubine blades have been taditionally epesented by a seies of pofiles stacked togethe to fom a blade. The stacking line is defined along the span of the blade, which joins each of the pofiles togethe though a paticula point. This stacking line may move in space, and the pofiles may otate aound the stacking line. The stacking line, the blade pofile shape at a adial location, and the otation angle of the blade pofile aound the stacking line, i.e., the stagge angle, ae thee majo design paametes often used in cuent pactice. Usually the aeodynamics of blade pofiles ae designed in twodimensional space, in a plane that is quasinomal to the spanwise diection. The shape of the pofile is edesigned to achieve optimal pefomance. A majo pefomance indicato is the distibution of pessue o isentopic Mach numbe ove the blade pofile suface. The thee-dimensional pefomance is usually checked by consideing the blade spanwise distibution of cicumfeentially aveaged aeodynamic paametes. In this way, the pefomance of the design in the so-called S blade-to-blade suface is consideed sepaately fom the pefomance in the meidional S2 steam suface. The design method based on the esolution of S2 and S calculations efeed in Ref. 6 indeed povides easonable solution and is widely used in engineeing. In a fully thee-dimensional design optimization method, howeve, a majo issue is the linkage of the design space to the aeodynamics analysis space, in paticula, whee the endwalls ae noncylindical. Typically, the optimization method makes geometical changes in the CFD mesh, to epesent changes in the boundaies of the fluid domain. This is in contast to the methods used by designes to change the blade shape. To evaluate the effectiveness of the pesent optimization pocedue, it is felt necessay to elate the changes in blade shape to the moe taditional design paametes. As mentioned peviously, pofiles ae changed in thei espective S planes, and the flow in the S2 suface is changed by changes in the stacking line. The hub and tip channel contous ae used to ecut the blade and fom the fluid domain. The application of a pofile estagge in the fluid domain fo a noncylindical casing is not so tivial, in paticula, at the endwalls. Hee, cae has to be taken not to change the channel contou when the estagge is applied. Diffeent sophisticated design tools and softwae systems may be used in pactice. Howeve, a simple pocedue is pesented below to eflect the essence of such a design pocedue. Figue 2 illustates the tansfomation of the Kth spanwise blade section of an annula cascade to an auxiliay suface S. O i is the oigin on the tubine otation axis. l is an auxiliay plane, which includes the stacking line and is paallel to the otation axis. P i is the intesection of the Ith x-slice with the auxiliay plane. A i and B i ae two gid points on the Kth blade section. The blade section is pojected onto an auxiliay suface, which is pependicula to and goes though P i. The points C i and D i with espect to A i and B i satisfy that = Pi P i A i C i, = Pi P i B i D i, whee P i A i and P i B i ae the lengths of acs Pi A i and P i B i. The fist step of this method is to expand the oiginal Kth blade section onto such an auxiliay suface, C i and D i ae on the same x-slice, and each x-slice of which is pependicula to ; and P, P2,..., Pn denotes the aay of local adius fom the leading edge to the tailing edge of the Kth blade section, coesponding to the aay of x-slices x P,x P2,...,x Pn, whee n is the num- Jounal of Tubomachiney JANUARY 2, Vol. 33 / 26-3

4 z l - y A i C i P i D i B i Stacking line Auxiliay suface s K mesh level Cost function -2-3 O i Hub θ Fig. 2 Tansfomation of annula cascade onto an auxiliay suface Design cycles Fig. 4 Cost function vesus design cycle be of gid points on each blade suface. The second step is to igidly otate this auxiliay suface about the otation axis. The vaiation of stagge angle equals the otation angle. Afte igid otation, fom the x-coodinate of C i, which coesponds to C i, we can detemine the new x-slice aay signified by x P,x P,...,x 2 P. Consequently the new adius aay is n signified by P, P,..., 2 P. The details ae shown in Fig. 3. n in Fig. 3 denotes the vaiation of stagge angle. D i is on the same x-slice with C i. The thid step is to ecove the blade on the auxiliay suface onto the new adial. The new points A i and B i still satisfy that = Pi P i A i C i, = Pi P i B i D i and thei coodinates ae detemined by x Ai = x P i, y Ai = P i cos P i C i / P i, z Ai = P i sin P i C i / P i, y x Fig. 3 C i D i Pi C i P i D i x B i = x P i y B i = P i cos P i D i / P i z B i = P i sin P i D i / P i Rigid otation of blade section α By this method, the blade pofile on each section is kept and the axial chod changes. 3 Results and Discussion As mentioned in Sec. 2, two types of design objectives with diffeent design spaces and constaints ae studied. Based on the gadient obtained fom the Eule equations and the adjoint equations, the steepest descent method is chosen as the optimization algoithm to ensue the development of design towad the design objectives. The esults ae pesented and discussed in the following subsections The total to static pessue atio is fixed in all cases. 3. Invese Design to Achieve Given Aeodynamic Load. Flow tuning though a blade ow is popotional to tubine wok with fixed mass flow ate. Simila to the angle of attack fo an aifoil and twist of a wing, the stagge angle of a tubomachiney blade pofile is a key paamete in detemining the flow tuning. In the pesent study, the spanwise stagge angle distibution is chosen as the design paamete. This design case is tested to demonstate the feasibility of using the adjoint method as an invese design tool to achieve an abitay flow tuning distibution by choosing the blade stagge distibution. The invese design poblem is conveted into an optimization poblem by using the cost function specified in Eq.. In ode to ensue the same tubine wok, the oveall mass-aveaged flow tuning of the taget distibution is equal to that of the efeence blade fo the same mass flow ate. Figue 4 shows the cost function change vesus the design cycle. As discussed ealie, each design cycle consists of one flow solution and one solution of the adjoint equation to obtain the gadient of the cost function with espect to the design paametes, which in this case ae the stagge angle distibution along the adial spanwise diection. The cost function monotonically deceases with design cycle and is educed by almost two odes of magnitude within 4 design cycles. Figue 5 shows the exit flow angle distibution befoe and afte the design. The initial distibution dotted line is fom the oiginal efeence design. To test the ability of the code fo invese design puposes, an abitay angle distibution is chosen as the taget exit flow angle distibution, which in this case is a staight-line the solid line in the figue. The placement of the staight-line is such that its one-dimensional aveage angle is the same as that of 26-4 / Vol. 33, JANUARY 2 Tansactions of the ASME

5 the initial design. Afte the 4 design cycles, a flow field is obtained that yields the exit flow angle distibution as shown by the solid squaes, which closely appoaches the taget distibution except nea the tip, whee the flow appeas to have difficulties to follow the staight-line angle distibution. Notice that an invese design poblem is not always well-posed. In othe wods, thee may not be a stagge angle distibution that can yield the specified taget flow angle distibution. Howeve, the pesent optimization method will always find the closest possible solution to the taget unde the given conditions. Figue 6 shows the change in stagge angles elative to the oiginal design as needed to poduce the designed exit flow angle distibution. This test case demonstates the ability of the adjoint method fo invese design puposes. In pactical applications, the specification of the exit flow angle distibution is left to the judgment of the designe. The choice of the linea distibution is used hee fo demonstation puposes only. 3.2 Optimization fo Maximum Exit Total Pessue. Instead of aiming fo an exit angle distibution, maximizing the exit total pessue may be a moe effective design stategy, given the Flow tuning angle (degee) Fig. 5 initial designed taget Exit flow angle befoe and afte design δβ (degee) Fig. 6 Designed change in stagge angle distibution diect link of total pessue loss with blade ow pefomance. Thee is moe than one choice of the cost function. Papadimitiou and Giannakoglou 3 defined it diectly as diffeence of total pessues between the outlet and inlet. The total enthalpy is constant in an adiabatic tubine stato. The entopy geneation pe unit mass flow ate as defined in Eq. 2 is then a diect measue of total pessue loss. Fo this test case, we include in the design space both the spanwise estagge of the blade pofiles as in the pevious case and the blade pofile shape at each spanwise location. The stacking line, howeve, is still kept unchanged fom the efeence design. The blade pofile is modified by adding a Hicks Henne shape function at each gid point on the blade suface as was done by Wu et al.. The shape functions ae defined as b i x = sin 4 x m i, m i = ln.5 /ln x Mi, i =,...,N whee x is the chodwise coodinate, and x Mi ae gid points on the blade suface that ae selected as contol points of the shape functions. N is the numbe of contol points on each blade suface. Notice that the defomation of each oiginal gid point is along the diection of the mesh line. Optimization of blade ow pefomance without constaining flow tuning acoss the blade would be physically meaningless just as in the case of minimizing dag of an aifoil in extenal flow without constaining its lift coefficient to a fixed value. Fo tubomachiney, an altenative constaint is the mass flow ate. Howeve, specifying both the tuning angle and mass flow ate would be an oveconstaint since tuning angle and mass flow ae almost uniquely elated acoss a blade ow o a stage. We pesent below esults by imposing the two altenative constaints sepaately Maximum Total Pessue With Fixed Tuning. The flow tuning at the exit is chosen as a constaint in ode to achieve a paticula stage chaacteistic fo example, the stage eaction and stage loading whee the design is consideed in a stage. This is accomplished by augmenting the cost function in Eq. 2 by the addition of a penalty function as follows: I = s gen + whee is the mass-aveaged flow tuning = Bo u j N j da Bo u j N j da is the flow tuning angle on each cell face at the exit and is defined as the invese tangent of the atio of tangential velocity to axial velocity. is the taget mass-aveaged exit flow angle, which in this pape is chosen to be that of the oiginal efeence design. The penalty weight paamete may be chosen to adjust the toleance in the enfocement of the constaint. In this study, =2. Figue 7 shows the entopy poduction, stagnation pessue atio, and the adiabatic efficiency of the blade ow vesus design cycle. With 5 design cycles, the adiabatic efficiency is inceased by about.7 of a pecentage point. Figue 8 shows that the tuning angle of flow acoss the blade ow is kept vey close to the fixed value duing the design pocess, indicating that the method stictly enfoced the desied flow angle constaint. The mass flow ate shown in Fig. 8, howeve, shows a slight incease. This is bought about by the inceased efficiency of the blade ow. Figues 9 and show the total pessue atio and adiabatic efficiency distibution in the adial diection befoe and afte the edesign. Impovements have been achieved along the whole spanwise diection. Figue indicates that the flow tuning angle is deceased nea the hub while slightly inceased in the 55 85% blade height aea. Figue 2 shows the spanwise stagge angle change in addition to blade pofile changes needed to achieve this Jounal of Tubomachiney JANUARY 2, Vol. 33 / 26-5

6 entopy total pessue efficiency initial designed p,e /p & η s gen (E-2) Designcycles Fig. 7 Total pessue, efficiency, and entopy vesus design cycle with tuning angle constaint p,e /p Fig. 9 Spanwise total pessue distibution design. The edesign inceases the stagge angle acoss the whole span, especially nea the middle. This effectively closes the blade passage. Figue 3 details the blade pofile befoe and afte the edesign at the spanwise gid level k=7. Case is the oiginal blade. Case 2 is the edesigned blade with both pofile modification and estagge. Case 3 is the blade with only the estagge fom Case 2 and Case 4 is that with only the pofile modification fom Case 2. The flow fields and pefomance of Cases 3 and 4 ae calculated and compaed in ode to study and compae the effects on the flow by the individual changes in stagge and the blade pofile shape, and the combined effect of the two. Table lists the computed dimensionless entopy poduction, mass flow ate, exit flow angle, and total pessue atio fo the fou cases. Figue 4 compaes the isentopic Mach numbe ove the blade fo the fou cases. The oiginal design suffes fom a high supesonic suction peak and theefoe causes moe stagnation pessue loss though the blade ow. The high Mach numbes ae also seen in the compaisons of the Mach numbe contous in the blade passage and those on the blade suction suface shown in Figs. 5 and 6, espectively. By estaggeing the blade pofiles Case 3, one educes the suction peak. Howeve, the inceased stagge closes the blade passage and consequently educes the mass flow ate because most pat of the blade passage is aleady choked in this case. The educed mass flow ate coupled with inceased stagge esults in significant flow tuning and, in fact, a slight net decease in total pessue because of the inceased entopy geneation pe unit mass as listed in Table. In ode to maintain the same tuning angle specified in the constaint, the optimize modifies the blade pofile to maintain the same total loading on the blade. By compaing the blade pofiles of Case and Case 4 o those of Case 2 and Case 3 in Fig. 3, we see significant thinning of the blade on the suction side. Contay to the effects of the stagge angle, the change in the blade pofile opens the thoat of the blade passage and thus leads to inceased mass flow ate and inceased inlet Mach numbe. The combination of the estagge and the change in the pofiles deceases the suction peak while at the same time inceases the loading on the font potion of the blade with the net esult of educed shock loss while maintaining the same flow tuning. Figue 7 shows the pitch-aveaged total pessue contous in the meidional plane. The edesigned blade shows educed total pessue loss in the supesonic egion of the channel. Optimization by eithe estagge o blade pofile change alone Nomalized mass flow ate mass flow ate flow tuning Design cycles Fig. 8 Tuning angle and mass flow ate vesus design cycle with tuning angle constaint 76 Flow tuning angle (degee) initial designed η (%) Fig. Spanwise adiabatic efficiency distibution 26-6 / Vol. 33, JANUARY 2 Tansactions of the ASME

7 case case 2 case 3 case y initial designed Flow tuning angle (degee) Fig. Spanwise tuning angle distibution x Fig. 3 Blade pofile at K=7 was pefomed but esults showed little gain in pefomance when a constaint on eithe mass flow ate o flow tuning was imposed. This is undestandable fom the pevious analysis on the effect of the two types of design paametes: stagge angle and blade pofile shape. The situation is much simila to that of minimizing dag of an aifoil in extenal flow by changing the aifoil shape with fixed lift. As the aifoil shape is changed, its angle of attack must usually be adjusted in ode to maintain the same lift. At pesent the optimization is only pefomed by using the Eule equations at the design point. In ode to assess the pefomance of the designed blade Case 2 in eal viscous situations and also away fom the design point, Navie Stokes calculations ae pefomed fo the initial and the edesigned blades at both design and off-design conditions. Figue 8 compaes the computed total pessue atio and adiabatic efficiency vesus the nomalized mass flow ate though the blade ow. The mass flow ate in the figue is nomalized by the choked value of the efeence design by the Eule equations. The Navie Stokes esults show educed total pessue, adiabatic efficiency, and mass flow ate compaed with the coesponding Eule solutions as expected due to the inclusion of viscous losses and blockage. Both the Eule and Navie Stokes esults show inceased pefomance and mass flow ate of the edesigned blade compaed with the efeence blade. Because of the oveall bette behavio of the flow field of the edesigned blade, the elative eduction in mass flow ate due to viscous blockage evealed by the Navie Stokes computations is also less fo the edesigned blade. Figue 9 shows the impovements in adiabatic efficiency of the edesigned blade ow based on the Eule and the Navie Stokes evaluations. Both calculations eveal pefomance gains ove the whole ange of computed opeating pessue atios even though the design optimization was pefomed at the design point coesponding to an exit isentopic Table The effects of stagge and blade shape Case s gen 2 ṁ 3 deg p,e / p case case 2 case 3 case M is Fig δβ (degee) Spanwise distibution of stagge angle change x Fig. 4 Blade isentopic Mach numbe distibution at K=7 Jounal of Tubomachiney JANUARY 2, Vol. 33 / 26-7

8 Initial.99 Designed.994 Initial Designed Fig. 5 Mach numbe contous at K=7 Mach numbe of.6. The diffeence between the adiabatic efficiency gains pedicted by the Navie Stokes and the Eule evaluations is small less than. pecentage point and almost a constant except nea the vey low back pessue point whee the flow is fully choked. This is not supising fo the type of steam tubine blades as consideed hee fo the following thee easons. Fist, the oveall pessue gadient in a tubine blade ow is favoable. Thus, viscous sepaation does not happen easily in a easonably well designed blade ow. The additional loss due to viscous effect is mostly due to wall skin fictional loss, which stays elatively constant. Second, the flow in such a tubine is mostly supesonic and theefoe losses ae mostly fom shocks o stong compession waves, which the Eule equations can accuately esolve. Thid, the low-pessue steam tubine blades have high aspect atios. Theefoe, endwall and seconday flow losses ae small. As the back pessue is deceased to appoach choking, the diffeence in efficiency gain between the viscous and inviscid pedictions Fig. 7 Pitch-aveaged total pessue contous becomes negligible because the losses ae dominated by shocks. Despite the emakable success of the above design optimization based on the Eule equations, the need is ecognized fo a Navie Stokes based optimization method fo poblems whee viscous effects may be significant. Futue wok will extend the adjoint method to the Navie Stokes equations. Both the Eule and Navie Stokes calculations show consistent pefomance gains ove the complete off-design ange by the design that is optimized only at the design condition and as such we can only claim optimality at the design condition. Fo this test case of steam tubine blades, this conclusion is not supising because of the oundness of the leading edge, which ensues good toleance to off-design inflow angles. To add obustness in the Initial.9 Designed.8.9 Eule_oiginal p Eule_oiginal η Eule_designed p Eule_designed η N-S oiginal p N-S oiginal η N-S designed p N-S designed η p,e / p η Fig. 6 Mach numbe contous on blade suction suface Nomalized mass flow ate Fig. 8 Compaisons of total pessue atio and efficiency of both efeence and designed blades 26-8 / Vol. 33, JANUARY 2 Tansactions of the ASME

9 δη_eule δη_n-s δη (%) Nomalized mass flow ate Flow tuning angle (degee) An efficient design optimization method based on the adjointequation appoach is pesented fo tubomachiney blade ow design. Gadient infomation of the cost functions is obtained by solving the Eule equations and the coesponding adjoint equations only once, independent of the numbe of design paametes. The method is demonstated to pefom both invese design and diect optimization of pefomance paametes with constaints fo a high-speed steam tubine nozzle blade ow. Cae is taken to apply appopiate themodynamic constaints, to ensue that the obseved design impovements ae valid in a typical design pocess. In the invese design case, the designe may specify a desied spanwise distibution of exit flow angle. The design optimi p b /p Fig. 9 Vaiations of adiabatic efficiency gains pedicted by the Eule and NS flow calculations futue one needs to conside multipoint design, which can be included in the pesent method by using a weighted cost function. Nevetheless, a check of the designed blade at off-design conditions like the one pefomed hee seves as a good pactice Maximum Total Pessue With Fixed Swallowing Capacity. We conside optimization unde the constaint of fixed mass flow ate instead of fixed tuning angle. This can be achieved by choosing the following cost function that contains a penalty function on the mass flow deviation fom the specified design mass flow ate: I = s gen + ṁ ṁ whee ṁ is the specified mass flow ate and ṁ is the computed mass flow ate duing the design. Figue 2 shows the entopy poduction, total pessue atio, and adiabatic efficiency vesus design cycle fo the optimization. The geneal behavio of the optimization pocess is simila to that when the tuning angle is used as the constaint. Figue 2 shows that in this case the mass flow ate is indeed kept to be almost constant and the tuning angle slightly inceased. Computations show that it is not possible to impove the blade pefomance if.98 both the flow angle and the mass flow ate must be kept unchanged this is somewhat intuitive, as the blade loading and swallowing capacity constain the pefomance to a cetain level. An impoved blade design will eithe incease the flow tuning while maintaining the same mass flow ate o incease the mass flow ate if the tuning angle is kept the same. Figue 22 shows the stagge angle change in the edesigned blade. The geneal behavio is the same as that shown in Fig Conclusions mass flow ate flow tuning Design cycles Fig. 2 Tuning angle and mass flow ate vesus design cycle with mass flow constaint entopy total pessue efficiency p,e / p & η s gen (E-2) Design cycles Fig. 2 Total pessue, efficiency, and entopy vesus design cycle with mass flow constaint δβ (degee) Fig. 22 Spanwise distibution of stagge angle change Jounal of Tubomachiney JANUARY 2, Vol. 33 / 26-9

10 zation code successfully estagges the blade pofiles to achieve the exit flow angle within a few tens of design cycles. In the diect optimization test cases, the mass-aveaged entopy poduction acoss the blade ow is chosen as the cost function. The optimization minimizes this cost function by modifying the blade pofiles and estaggeing the pofiles. Constaints ae imposed by adding a penalty tem in the cost function. Two diffeent constaints ae consideed in the investigations. The fist is to maintain a fixed mass-aveaged exit flow angle. The second is to maintain a fixed total mass flow ate. The optimization inceases the adiabatic efficiency of the blade by about.6 of a pecentage point with eithe fixed with less than.% eo tuning angle o fixed mass flow ate. Examination of the flow field shows that this gain in efficiency is obtained by educing the supesonic suction peak in the nozzle passage though a combination of pofile modification and estaggeing of the blade pofiles. Restaggeing o blade pofile change alone is not sufficient to poduce significantly impoved design without violating the themodynamic constaints. When the blade pofile is changed, estaggeing is necessay to maintain the oveall loading of the blade such that the tuning angle constaint is satisfied. Acknowledgment The authos would like to thank Alstom Powe fo suppoting this eseach wok though a eseach contact to UCI. J. Luo also eceived suppot fom the China Scholaship Council fo his studies at UCI. Nomenclatue a speed of sound A i Jacobian matices, A i = f i / W B boundaies of the domain c p constant pessue specific heat D the computational domain domain f i inviscid flux G gadient fo optimization I cost function K ij tansfomation functions between the physical domain and the computational domain, K ij = x i / j ṁ mass flow ate M is isentopic Mach numbe n i unit nomal vecto in the domain, pointing outwad fom the flow field N j unit nomal vecto in the physical domain, pointing outwad fom the flow field p,e exit total pessue R g gas constant s gen entopy geneation pe unit mass flow ate s entopy, s= R g ln p,e / p u t tangential velocity u x axial velocity W consevative flow vaiables, W=, u, v, w, E T x i coodinates in the physical domain mass-aveaged exit flow tuning angle weight of the penalty function i coodinates in the computational domain co-state vaiables, =, 2, 3, 4, 5 T Appendix: Bounday Conditions, Adjoint Equations, and Vaiation of Cost Functions Bounday Conditions of Flow Solve. Inlet Bounday Conditions. The total pessue p, total tempeatue T, and the two inflow angles ae specified..2 Outlet Bounday Conditions. The back pessue p is specified..3 Inviscid Wall Bounday Conditions. The nomal velocity component is zeo as follows: U N =.4 Peiodic Bounday Conditions. w i,b2 = w i,b, i =,2,5 w 3,B2 = w 3,B cos w 4,B sin w 4,B2 = w 3,B sin + w 4,B cos whee B and B2 ae two peiodic boundaies, and is the pitch angle of annula cascade. 2 Adjoint Equations. The final expession of the adjoint equations in unsteady fom is t A i T x i = A 3 Inlet Bounday Conditions of Adjoint Equations: n =, n 2 =, n 3 = C f =, c T = The details of the inlet bounday condition ae shown by Wu and Liu. 4 Outlet Bounday Conditions of Adjoint Equations: n =, n 2 =, n 3 = 4. Invese Design Case. The cost function of the invese design case can also be witten in discete fom KL I = k k 2 A2 k= 2 whee KL is the numbe of flow tuning s components. The outlet bounday conditions ae A j+,k + A,k K j+ k = c j+ + K j+c, j =,2,3,4 A3 whee A =n i S ij A j and K =, K 2 = 2u u j u j, K 3 = 2u 2 u j u j, K 4 = 2u 3 u j u j, K 5 = 2 u j u j A4 c c 2 5 = c 3 c 4 c g 2 = u t u 2 2 x + u, t S k g 2 S k g 3 S k g 4 k k S + k S u k k k S 2 + k S 2 u k k k S 3 + k S 3 u k g 3 = u x sin u 2 2 x + u, t g 4 = u x cos u 2 2 x + u t 4.2 Optimal Design Case. The outlet bounday conditions ae 26- / Vol. 33, JANUARY 2 Tansactions of the ASME

11 A j+,k whee + A,k K j+ k = sign c,j+ + K j+c, + c s,j+ cs, c s,2 c s,3 c s,4 c, c,2 c,3 c,4 c,5 = + K j+c s,, j =,2,3,4 c p u j S j s s gen S m c s,5 = f s s gen S s s gen S S + u j S j g 2 S 2 + u j S j g 3 S 3 + u j S j g 4 5 Inviscid Wall Bounday Conditions N k k+ =, k =,2,3 A5 6 Resultant Vaiation of Cost Functions I = I g C g db n i S ij = B + B T f jdb T S ijf j dd W D i whee A6 Refeences f j = uj u u j u 2 u j u 3 u j u j E Jameson, A., 23, Aeodynamic Shape Optimization Using the Adjoint Method, Lectues at the von Kaman Institute. 2 Samad, A., and Kim, K. Y., 28, Multiple Suogate Modeling fo Axial Compesso Blade Shape Optimization, J. Popul. Powe, 24 2, pp Jang, C. M., Li, P., and Kim, K. Y., 25, Optimization of Blade Sweep in a Tansonic Axial Compesso Roto, JSME Int. J., 48 4, pp Hage, J. O., Eyi, S., and Lee, K. D., 993, Design Efficiency Evaluation fo Tansonic Aifoil Optimization: A Case fo Navie-Stokes Design, AIAA Pape No Jameson, A., 988, Aeodynamic Design Via Contol Theoy, J. Sci. Comput., 3 3, pp Jameson, A., 995, Optimum Aeodynamic Design Using CFD and Contol Theoy, AIAA Pape No Kim, S., Alonso, J. J., and Jameson, A., 24, Multi-Element High-Lift Configuation Design Optimization Using Viscous Continuous Adjoint Method, J. Aic., 4 5, pp Deye, J. J., and Matinelli, L., 2, Hydodynamic Shape Optimization of Populso Configuations Using a Continuous Adjoint Appoach, AIAA Pape No Yang, S., Wu, H., and Liu, F., 23, Aeodynamic Design of Cascades by Using an Adjoint Equation Method, AIAA Pape No Wu, H., Yang, S., and Liu, F., 23, Compaison of Thee Geometic Repesentations of Aifoils fo Aeodynamic Optimization, AIAA Pape No Wu, H., and Liu, F., 25, Aeodynamic Design of Tubine Blades Using an Adjoint Equation Method, AIAA Pape No Wu, H., 25, Aeodynamic Optimization of Intenal and Extenal Flows Using the Adjoint-Equation Method, Ph.D. thesis, Depatment of Mechanical and Aeospace Engineeing, Univesity of Califonia, Ivine, Ivine, CA. 3 Papadimitiou, D. I., and Giannakoglou, K. C., 26, A Continuous Adjoint Method fo the Minimization of Losses in Cascade Viscous Flows, AIAA Pape No Wang, D., and He, L., 2, Adjoint Aeodynamic Design Optimization fo Blades in Multi-Stage Tubomachines: Pat I Methodology and Veification, ASME J. Tubomach. 32, p.2. 5 Wang, D., He, L., Wells, R., and Chen, T., 2, Adjoint Aeodynamic Design Optimization fo Blades in Multi-Stage Tubomachines: Pat II Validation and Application, ASME J. Tubomach. 32, p Havakechian, S., and Geim, R., 999, Aeodynamic Design of 5 Pecent Reaction Steam Tubines, Poc. Inst. Mech. Eng., Pat C: J. Mech. Eng. Sci., 23, pp. 25. Jounal of Tubomachiney JANUARY 2, Vol. 33 / 26-