Impact of Geometric Variability Victor E. Garzon on Axial ... · In this work we present a...

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Victor E. GarzonPostdoctoral Associate

e-mail: [email protected]

David L. DarmofalAssociate Professor

e-mail: [email protected]

Department of Aeronautics and Astronautics,Massachusetts Institute of Technology,

77 Massachusetts Avenue,Cambridge, MA 02139

Impact of Geometric Variabilityon Axial Compressor PerformanceA probabilistic methodology to quantify the impact of geometric variability on compreaerodynamic performance is presented. High-fidelity probabilistic models of geomvariability are derived using a principal-component analysis of blade surface measments. This probabilistic blade geometry model is then combined with a compresviscous blade-passage analysis to estimate the impact on the passage loss andusing a Monte Carlo simulation. Finally, a mean-line multistage compressor model,probabilistic loss and turning models from the blade-passage analysis, is developquantify the impact of the blade variability on overall compressor efficiency and presratio. The methodology is applied to a flank-milled integrally bladed rotor. Results donstrate that overall compressor efficiency can be reduced by approximately 1% dblade-passage effects arising from representative manufacturing variability.@DOI: 10.1115/1.1622715#

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1 MotivationTurbomachinery airfoils must perform reliably and efficient

in severe environments for prolonged periods of time. The optishapes of compressor and turbine airfoils have been the subjea large body of research literature. Advances in numerical mods that allow prescribed velocity distributions for controlled dfusion and supercritical transonic operation have resultedhighly optimized airfoils and ever more efficient compression stems. Despite recent noteworthy advances in manufacturing tniques~e.g., electro-chemical machining, flank milling, etc.!, fin-ished airfoils always exhibit some deviation from their intendshape and size. The effect of such variations on compressorformance is poorly understood but generally thought to be demental.

Probabilistic techniques applied to structural design and ansis have been used in the aerospace industry for more thandecades@1#. In contrast, few similar endeavors have been undtaken inaerothermalanalysis and design of turbomachinery components. Turbomachinery probabilistic aerothermal analysis ispecially challenging because of the highly nonlinear mathematmodels involved. A marked increase in computational requments occurs in direct proportion to the physical complexity ofturbomachinery physics. Until recently, probabilistic treatmentsturbomachinery aerothermal analysis and design have bdeemed prohibitively expensive. The advent of relatively inexpsive parallel hardware has considerably increased the feasibilisuch probabilistic studies.

In this work we present a methodology for estimating the ipact of geometric variability on the aerodynamic performanceindividual blade passages and, by a modeling extension, onaerothermal performance of high-pressure axial compressiontems. The methodology is applied to an integrally bladed rofrom a multistage axial compressor.

The paper is divided into three sections. The first sectioncusses the development of a probabilistic model for the geomvariability present in a set of compressor blade measuremeThe second section illustrates the use of conventional comptional fluid dynamics~CFD! analysis in combination with classical probabilistic simulation techniques to assess the impacgeometric variability on the aerodynamic performance of invidual blade passages. In the third section, a probabilistic m

Contributed by the International Gas Turbine Institute and presented at the Inational Gas Turbine and Aeroengine Congress and Exhibition, Atlanta, GA,16–19, 2003. Manuscript received by the IGTI December 2002; final revision Ma2003. Paper No. 2003-GT-38130. Review Chair: H. R. Simmons.

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stage mean-line compressor model is used to estimate the imof airfoil geometric variability on overall compressor perfomance.

2 Geometric Characterization of Compressor BladeVariability

This section presents results from an application of principcomponent analysis~PCA! to characterizing sets of compressblade surface measurements. As discussed below, high fidmodels of geometric variability for use in probabilistic simultions can be readily constructed from the PCA results. Statisticbased models of variability are clearly superior alternativesheuristic models based on manufacturing tolerances or anecevidence alone, as they represent the ‘‘actual’’ variability foundmeasurements.

2.1 Background. The nominal airfoil geometry is taken tobe defined byp coordinate pointsxj

0PRm, j 51, . . . ,p wherem istypically 2 or 3. We consider a set ofn coordinate measurement$xi , jPRmu i 51, . . . ,n; j 51, . . . ,p% taken, for instance, with acoordinate-measuring machine. The measurements may cspond to single radial locations (m52) or entire spanwise segments (m53). Indexj uniquely identifies specific nominal pointand their measured counterparts. Similarly indexi identifies a dis-tinct set of measured points. The discrepancies in the coordimeasurements can then be expressed as

xi , j8 5 xi , j2xj0, i 51, . . . ,n; j 51, . . . ,p.

Subtracting from these error vectors their ensemble average,

xj51

n (i 51

n

xi , j8 , j 51, . . . ,p,

gives a centered set ofm-dimensional vectors,x5$xi , j5 xi , j2 xj u i 51, . . . ,n; j 51, . . . ,p%. Writing the m-dimensional mea-surements in vector form,X j5@xi , j

T , . . . ,xn, jT #T, thescatter matrix

of setx is given by

S5XTX.

The scatter matrix is related to the covariance matrixC via C5(n21)21S.

It can be shown~see, for instance, Refs.@2# and @3#! that thedirections along which the scatter is maximized correspondnontrivial solutions of the eigenvalue problem,

Sv5lv. (1)

ter-unerch

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ense or copyright; see http://www.asme.org/terms/Terms_Use.cfm

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SinceS is symmetric positive definite, it has in generalmportho-normal eigenvectorsviPRmp, i 51, . . . ,mp with correspondingreal, non-negative eigenvaluesl i , i 51, . . . ,mp. By construc-tion, the variance of the geometric data corresponding to eigvectorvi is l i /(n21). The eigenvector corresponding to the larest eigenvalue gives the direction along which the scatter ofdata is maximized. The eigenvector corresponding to the nlargest eigenvalue maximizes the scatter along a directionnormalto the previous eigenvectors. It is in this sense that the eigentors of S are said to provide anoptimal statistical basis for thedecomposition of the scatter of the data.

The PCA synthesis ofS can be shown to be equivalent to thsingular value decomposition~SVD! of X in reduced form@4#,

X5USVT, (2)

where S5diag(s1, . . . ,smp), s j5Al j , j 51, . . . ,mp, and thecolumns ofV are the corresponding eigenvectors ofS. The stan-dard deviation of the geometric data attributable to thei th mode istherefores i5(n21)21/2s i . The columns ofA5US are calledthe amplitude vectors orprincipal componentsof the data setx.The SVD of X is made unique by requiring that$s j% j 51

mp be anonincreasing sequence.

2.2 Application. As an application of the PCA formalismoutlined above, we consider an integrally bladed rotor~IBR! con-sisting of 56 blades. Surface measurements of 150 bladesfour separate rotors were taken using a scanning coordinmeasuring machine. Each blade was measured at 13 differendial locations. The scanning measurements of each radial stawere condensed to 103 points corresponding to those definingnominal airfoil sections.

For the present application, the 13 separate cross-sectionalsurements were stacked together to form a three-dimensionalresentation of the measured portion of the blade. Using bicuspline interpolation, the nominal geometry, as well as each msured blade, were sliced along a mid-span axial streamline pavarying radius~see next section!. In addition, the coordinate values were scaled by the blade tip radius.

Using the notation introduced above, the resulting setx of two-dimensional centered measurement vectors can be writtenn3mp matrix X wheren5150,m52, andp5103. Figure 1 showsthe modal scatter fractionlk /( i 51

mp l i ~decreasing! and the partialscatter( i 51

k l i /( i 51mp l i ~increasing! of the first six eigenmodes o

S. The first mode contains 82% of the total scatter in the origimeasurements and it clearly dwarfs the scatter fraction ofother modes. The scatter corresponding to the second-mostgetic mode is roughly eight times smaller than the first.

Figures 2 and 3 depict the outlines of the first and third eig

Fig. 1 IBR mid-span section: PCA modes

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modes ofS applied to the baseline geometry and scaled by threspective amplitude, that is,

xi5x01 x1ss ivi . (3)

An additional scaling factors has been used for plotting purposeto help distinguish the effect the eigenmodes from the meanometry. Figure 2 indicates that the main effects of mode 1uniform thickening of the airfoil and azimuthal translation. Mod3, on the other hand, exhibits a thinning of the airfoil on tsuction surface away from the leading edge, with the shape oflatter being maintained. The perturbations to the airfoil noseparticularly noteworthy as the aerodynamic performance of trsonic airfoils is known to be sensitive to leading-edge shapethickness.

Figure 1~a! indicates that the first five modes, when combinecontain 99% of the total scatter present in the sample. The ra

Fig. 2 IBR mid-span section: Scaled mode 1

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Fig. 3 IBR mid-span section: Scaled mode 3

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decrease in relative energy of the higher modes suggests threduced-order model containing only the first few modes maysufficient to represent most of the geometric variability containin the original set of measurements.

The above description of modes 1 and 3 would suggest acomposition into customary geometric features of known aeronamic and structural importance. Table 1 summarizes perchanges in maximum thickness, maximum camber, leading-eradius, chord length and cross-sectional area, as well as traiedge deflection angles, for the first six eigenmodes. In computhe parameters shown in the table, the modes were not scales51) and a positive amplitude was assumed.1 The row labeled‘‘mean’’ contains the changes corresponding to the average aix. From Table 1, no single mode produces a dominant changeparticular feature; rather, each mode contributes to changes ifeatures. It follows that, in characterizing actual geometric vaability, checking only for compliance of individual design toleances based on customary geometric features may not be effesince these features show strong correlation.

2.3 PCA Results versus Number of Samples.In the appli-cation of PCA to compressor rotor blade measurements discuabove, all available samples~150! were used in the analysis. Thisection discusses how PCA results~covariance matrix eigenvalues! vary according to the number of samples being consideGiven n measurement samples, there are (k

n) different ways ofselectingk<n among them.2 Figure 4 depicts convergence trendof the first three eigenvalues of the covariance matrix fork mea-sured samples. The average value and standard deviation ofeigenvalue are computed from min@(k

n),104# random permutationsof the indices$1¯n%. For each random permutation, the eigevalues of the covariance matrix of the corresponding indemeasurements is computed via singular value decompositionFig. 4 the average eigenvalues are shown as solid lines andsinterval about the mean by dash-dot lines.

The uncertainty of the first covariance matrix eigenvalue is vlarge for small sample sizes and decreases monotonically assample size is increased. Since the62s bands are approximatelythe 95% confidence bands, the figures suggest that at leastblades would be required to have 95% confidence that the donant eigenvalues deviate from theirk5n value by no more than10% of the total variance.3

2.4 PCA-Based Reduced-Order Model of Geometric Vari-ability. A reduced-order model of the geometric variabilipresent inx can be motivated as follows. LetZi , i 51, . . . ,mp beindependent, identically distributed random variables fromN(0,1)~normally distributed with zero mean and variance 1!. By linear-ity, the random vector

1Geometric parameters computed withXFOIL @5#.2(k

n) or ‘‘ n choosek’’ is defined for k<n by @6#: (kn)ªn!/ @(n2k)!k! #.

3The total variance of the blade population is defined as( i 51mp s i

2 and for the datapresented here it is roughly 1.231025.

Table 1 IBR mid-span section: geometric features of PCA modes

Mode Max thickness~%D!

Max camber~%D!

LE radius~%D!

TE angle~D deg!

Chord~%D!

Area~%D!

Mean ~from x0) 20.12 20.98 18.33 0.06 20.04 21.03

Fro

mm

ean

1 20.54 20.06 24.22 0.00 20.04 20.772 21.09 0.28 2.15 0.04 20.04 21.273 0.53 0.04 1.83 20.02 0.01 0.854 20.74 0.10 1.34 20.07 0.02 20.875 20.11 0.27 4.23 0.06 0.02 20.016 0.03 20.01 24.11 0.01 0.00 20.08

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Fig. 4 Eigenvalues of covariance matrix versus number ofsamples. Average eigenvalues „—…, Á2s interval about themean „-.-… and Á10% accuracy bands „- - -….

p-



X5x01 x1(i 51

mp

s iZivi

has meanx01 x and the same unbiased estimator of total varianas the set of measurementsx. This suggests a reduced-ordmodel of the form

X5x01 x1(i 51

K

siZivi (4)

whereK,mp is a free truncation parameter. For large enoughn,asK increases the total variance ofX approaches that ofX. In fact,the total scatter of a finite set of instances ofX is bounded by( i 51

K l i<( i 51mp l i .

3 Impact of Geometric Variability On Blade PassageAerodynamics

3.1 Blade Passage Analysis: MISES.The transonic com-pressor blade analysis was carried out using MISES~multipleblade interacting streamtube Euler solver!, an interactive viscousflow analysis package@7# widely used in turbomachinery analysand design. MISES’ flow solver, ISES can be used to analyzedesign single- or multielement airfoils over a wide range of floconditions. ISES incorporates a zonal approach in which theviscid part of the flow is described by the projection of the steastate three-dimensional~3D! Euler equations onto an axisymmeric stream surface of variable thickness and radius. The resutwo-dimensional equations are discretized in conservative foover a streamline grid. The viscous parts of the flow~boundarylayers and wake! are modeled by a two-equation integral bounary layer formulation@8#. The viscous and inviscid parts of thflowfield are coupled through the displacement thickness andresulting nonlinear system of equations is solved usingNewton-Raphson method@9#. A feature of MISES that is particu-larly relevant to probabilistic analysis is its speed. For the careported herein, typical execution times are three to ten sectrial on a 1.8-GHz Pentium 4 processor.

The aerodynamic performance of an isolated compressor rpassage may be summarized by the changes in total enthalpyentropy in the flow across the passage, i.e., the amount of wdone on the fluid and the losses accrued in the process. Thependence of total enthalpy change,Dht , on tangential momentumchanges across the blade row is described by the Euler turequation,

Dht5v@r 1u1 tanb12r 2u2 tan~b12q!#,

where v, r, and u denote wheel speed, radius, and axial flovelocity, and the subscripts 1 and 2 denote inlet and exit, restively. For small radial variation and axial velocity ratio (u2 /u1)near unity, the enthalpy change depends primarily on the amoof flow turning,qªb22b1 .

An appropriate choice for a measure of loss in an adiabmachine is entropy generation@10#. The increase in entropy results in a decrease of the stagnation pressure rise when compwith the ideal~isentropic! value. In what follows, the loss coefficient is defined as the drop in total pressure at the passagescaled by the inlet dynamic pressure,

Ãª

pT2

0 2 pT2

pT12p1

. (5)

HerepT2

0 is the ideal~isentropic! total pressure at the passage eand pT2

is the mass-averaged total pressure at the passageDetails of MISES’ cascade loss calculation can be found in Apendix C of Ref.@9#.

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3.2 Probabilistic Analysis. Computed loss coefficient anturning values are taken to be functions ofn independent variablesrepresenting the geometry of the flow passage andm variablesrepresenting other flow parameters,

Ã5Ã~x,y!, q5q~x,y!,

wherexPRn denotes the vector of geometric parameters anyPRm contains the remaining parameters. BothÃ andq are deter-ministic functions ofx and y, i.e., for givenx and y, there is aunique corresponding value ofÃ.

Consider next a continuous random vectorX with joint prob-ability density functionf X . For fixed flow parametersy, the ex-pected value ofÃ(X,y) is defined by

mÃªEx@Ã~X,y!#5E

RnÃ~x,y! f x~x!dx (6)

and the variance ofÃ(X,y) is given by

sÃ2ªVar~Ã~X,y!!5E

xb~Ã~X,y!2mÃ!2c. (7)

Similar expressions follow for meanmq and variancesq2 of

turning.In general, the functional dependence ofÃ on the geometric

parametersx is too involved to allow for a closed form evaluatioof the integrals in definitions~6! and ~7!. However, numericalapproximations can be obtained via probabilistic analysis teniques. One such technique, the Monte Carlo method@11–13#, isapplied here to estimating the effect of geometric and inlet flcondition variability. Garzon and Darmofal@14# applied and com-pared other probabilistic analysis techniques~e.g., response surface methodology, probabilistic quadrature! to assessing the impact of geometric variability on aerodynamic performance. In tinvestigation, while the mean aerodynamic performance couldreasonably estimated using lower-fidelity probabilistic analytechniques, the accurate prediction of the aerodynamic pemance variability required Monte Carlo simulations~MCS!. Thus,in the present work, we rely solely on Monte Carlo simulationsprobabilistic analysis.

One of the attractive features of MCS is that parallelizationconcurrent calculations can be readily implemented in shamemory parallel computers as well as across networks of hetgeneous workstations. In the present context, each function evation consisted of grid generation, flow-field analysis and poprocessing steps that were automated and parallelized ucommand scripts. All probabilistic simulations reported in thpaper were carried out on a 10-node Beowulf cluster at the MAerospace Computational Design Laboratory. Each nodeequipped with dual 1.8-GHz Xeon processors. For the preapplication, 2000 trials required about one hour of computing tiusing all ten nodes.

3.3 Application. The nominal rotor reported here was paof the sixth stage from an experimental core axial compresssystem. The following operating conditions were assumed inthrough-flow analysis: mass flow rate of 20 kg/sec, wheel spee1200 rad/sec (U tip5301 m/sec) and axial inlet flow~no swirl!. Inaddition, the stage inlet static temperature and pressure were tto be 390 K and 200 kPa resulting in an inlet Mach number0.43.

The axisymmetric viscous flow packageMTFLOW was used toperform the initial through-flow calculation.MTFLOW implementsa meridional streamline grid discretization of the axisymmeEuler equations in conservative form. Total enthalpy at discrflow field locations and constant mass along each streamtubeprescribed directly. The localized effects of swirl, entropy genetion, and blockage due to rotating or static blade rows can alsmodeled@15,16#.

The inlet relative Mach number and flow angle were taken to0.90 and 62.6 degrees, respectively, and the Reynolds num

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based on inlet tip radius was 33106. Figure 5 shows the pressurdistribution on the suction and pressure surfaces of the airAfter a short precompression entry region, a shock appears onsuction surface followed by mild compression until about twthirds of the chord length; from there the flow is further decelated until the trailing edge is reached. On the concave sideadverse pressure gradient exists until about midchord, followeda plateau. The baseline loss coefficient and turning were compby MISES to be 0.027 and 14.40 degrees, respectively.

The noise model employed in the probabilistic analysis isPCA-based model described in a previous section. The moddefined by Eq.~4!. The convergence criterion used for the MonCarlo simulations was

usÃN 2sÃ

N2nu,«,

where the superscripts indicate the number of samples takenthe present study,«51025 and n510 were used. Results fromnumerical experimentation suggested thatN52000 trials weretypically sufficient to achieve the required tolerance for the creported here.

Figure 6 shows histograms of loss coefficient and turning. Tabscissa represents the values of the output variable, whileordinate indicates the relative number of trials that fall within eaof the equal-length intervals subdividing the abscissa. In the liof large number of trials,N→`, the outline of the histogram baplot approaches the continuous distribution of the output variaThe two vertical dashed lines indicate the nominal~baseline! andmean values. The estimated mean loss coefficient is abouthigher than the baseline~noiseless! value, while the mean turningis about 1% lower than nominal. The standard deviation of lcoefficient is 0.0008, which is less than 3% of the mean loss.the turning, the standard deviation is 0.087 degrees which is oabout 0.6% of the mean turning.

The small impact of geometric variability on aerodynamic pformance is not surprising given the small geometric variabipresent in the measurement samples. Production airfoils, mfactured with processes that are less tightly controlled thancurrent flank-milled IBR case, should be expected to exhhigher levels of geometric variability. The Appendix illustrates tdifferences in shape variability between two IBR manufactuwith point and flank milling, respectively. As discussed in tAppendix, the flank-milled IBR data being studied has appromately ten times less variability than can occur in other commmanufacturing processes.

To explore the impact of increased manufacturing noise amtude on the aerodynamic performance statistics, a series of M

Fig. 5 IBR Pressure Coefficient, M1Ä0.90, axial velocity-density ratio „AVDR…: 1.27



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Carlo simulations were performed with various levels of geomric noise. In those simulations, the geometric noise modelmodified to take the form

X5x01 x1a(i 51

K

s iZivi , (8)

wherea is a geometric variability amplitude. Figure 7 summarizthe Monte Carlo estimates of mean and standard deviation ooutputs of interest fora51, . . . ,8. In thefigure, the horizontaldashed line indicates the loss and turning corresponding toaverage geometry,x01 x. Similarly, the baseline loss and turninare indicated by solid lines. At the original noise level,a51, theimpact of the average geometry dominates the difference incoefficient and turning from the baseline values, i.e., the geomric variability about the average geometry has little effect on‘‘mean shift.’’ For a noise amplitude ofa52, the effect of thegeometric variability becomes noticeable; in the case of thecoefficient, the noise contributes about half of the total shift. Fa54, the contribution of the average geometry to turning meshift is about half of the total. Fora.4, the shift from nominal inboth loss and turning is dominated by the variability of the blameasurements, rather than by the average geometry. Ata55 theloss mean shift is about 23% of the nominal value, an increasa factor of six froma51. The standard deviation of loss coefficient increased by a factor of 6 from 0.0008 ata51 to 0.005 ata55. The turning mean shift ata55 is roughly twice as large a

Fig. 6 IBR: Loss and turning histograms



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the baseline value, while the standard deviation of turningcreased by a factor of 5. One implication of the increase in retive importance of the scatter is that controlling the manufacturprocess by ‘‘re-centering’’ the target geometry may not be sucient to effectively improve the mean performance.

The mean loss coefficient and turning depicted in Fig. 7 dovary linearly with geometric noise amplitude in the vicinity ofa51. Rather the amount of curvature indicates a higher-orderpendence. The increase in loss and turning variability—a msured by their estimated standard deviation—increases nearlyearly with geometric noise amplitude, at the approximate rates0.001/a for loss and 0.1/a degrees for turning. This behavior cabe explained by considering a quadratic approximation to the lcoefficient; namely,

Ã~x!5Ã01c1x1c2x2,

wherex is a noise variable. In particular, consider a centered nmal variableXPN(0,asX

2) wherea is the noise amplitude multi-plying sX . Then, the expected value ofÃ is

E@Ã~X!#5Ã01c1E@X#1c2EbX2c5Ã01c2a2sX2.

Thus the mean-shift in loss coefficient is seen to depend quadcally on the amplitude of the noise. The variance of the assumquadratic loss is

Var~Ã0~X!!5c12a2sX

2F112S c2asX

c1D G .

The nondimensional quantityc2asX /c1 , is the ratio of the changein loss due to the quadratic term relative to the linear term fo12s ~after amplification bya! noise. Thus if the impact of thequadratic terms at this noise level is small compared to the linterms, we would expect to see a largely linear dependence onstandard deviation of the loss with respect to the noise amplita. This linear dependence ona of the standard deviation of theturning angle is clearly seen in Fig. 9. The loss standard deviais also fairly linear, but some curvature can be seen indicatingthe quadratic terms are more important in the loss behavior.

Further understanding of how the amplitude of the noise ipacts the cascade aerodynamic performance can be seen in Fand 9 which show the cumulative distribution functions~CDF!4 ofloss and turning for values ofa ranging from 1 to 8. The

4The distribution functionF:TR°@0,1# of a random variableX is defined byF(b)5P$X<b%, i.e., the probability thatX takes on a value smaller than or equal tb.

Fig. 7 IBR: Mean and standard deviation versus noise ampli-tude

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nominal and average-airfoil values are indicated by dasheddash-dot vertical lines, respectively, and the arrows indicatedirection of increasinga. Figure 8 shows how the high-end ‘‘tailsof the loss distributions become thicker asa increases. Fora51the probability that the loss coefficient will take on values smathan nominal is only about 15%, while ata58 that probabilityhas dropped nearly to zero.

By comparison, the behavior of the turning distribution wiincreasing noise amplitude does not show a significant impacmean turning~Fig. 9!. The CDF of turning seem to all cross in thvicinity of the nominal value, indicating that the mean shiftsmall compared to the variability. This behavior of loss and tuing has been consistently observed in a variety of compreapplications studied previously@17#. In particular, the mean loss ialways increased as a result of geometric variability whilemean turning is relatively unaffected.

The impact of geometric variability on boundary layer thicness is illustrated in Fig. 10 for noise levela55. The figure showsnominal and mean momentum thickness (u/c) on the suction andpressure sides~indicated in the plots by SS and PS, respective!.The dashed and dot-dashed lines indicate the nominal valui.e., without geometric noise—while the solid lines indicate tmean values from Monte Carlo simulation. The error bars indicto a one-standard-deviation interval about the mean. The discancy between nominal and mean momentum thickness valumore pronounced on the pressure side, as is its variability.

Fig. 8 IBR: Impact of geometric variability on loss coefficientdistribution, aÄ1,2, . . . ,8

Fig. 9 IBR: Impact of geometric variability on turning distribu-tion, aÄ1,2, . . . ,8

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growth in mean momentum thickness relative to the nominaseen to occur most significantly at the leading edge~notably atabout 5% chord! on the pressure surface.

As discussed by Cumpsty@18#,5 the momentum thickness itsedoes not necessarily point to the mechanism by which lossescreated. A more appropriate quantity to consider is the boundlayer dissipation coefficient, defined by

Cd85E0

d t

rUe2

]

]y S u

UeDdy, (9)

where t stands for shear stress,Ue is the boundary layer edgevelocity, r stands for density,d is the boundary layer thicknessand u is the component of the flow velocity along the dominaflow directionx ~herex and its normal complementy are boundarylayer coordinates!. As shown by Denton@10# the cumulative valueof rUe

3Cd8 over the interval 0<x8<x,

E0

x

rUe3Cd8dx8, (10)

is a measure of the rate of entropy generation per unit span6 in theboundary layer.

Figure 11 shows cumulative values ofrUe3Cd8 as per Eq.~10! at

geometric noise levela55. The rate of entropy generation iabout three times higher on the suction side than on the presside. The nominal-to-mean shift is more pronounced on the psure side, as is the variability. As shown in the figure, the meshift and variability in entropy generation rate increase rapidlythe first 10% chord and change little aft of the 25% chord lotion. This indicates that loss variability is accrued primarily at tleading edge and agrees with the observed growth of meanmentum thickness in this region.

While the PCA-based probabilistic model optimally describthe geometric variability, the impact of the geometric modesthe aerodynamic performance depends not only on the magniof the underlying geometric variability, but also on their aerodnamic sensitivity to that geometric perturbation. To quantify trelative importance of the PCA modes on aerodynamic permance, we have performed Monte Carlo simulations with increing values ofK ~i.e., increasing numbers of PCA modes!. Figure12 presents statistics of loss and turning according to the numof PCA modes used in the geometric noise model~denotedK inEq. ~8!! for noise amplitudea55. For each value ofK, a MonteCarlo simulation withN55000 trials was performed. Baselin

5Section 1.5; see also Denton@10#.6It is assumed here that the process takes place at constant temperature.

Fig. 10 IBR: Effect of geometric variability on momentumthickness. Mean indicated by solid lines, one-standard devia-tion interval by error bars. †SS‡: suction side, †PS‡: pressureside.



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and average-geometry values are denoted by constant dalines, while the values corresponding toK5mp ~all modes! areshown by a solid line. The average-geometry contribution to mloss constitutes a relatively small fraction of the total shift fronominal, as pointed out earlier. The geometric scatter of thesix modes is responsible for about 90% of the total mean shifloss coefficient. Similarly, the first six modes taken together pduce close to 90% of the turning mean shift obtained whenmodes are considered. The first six modes are also the most iential on loss coefficient variability, as indicated by its standadeviation plot. The first two modes clearly dominate turning anvariability. Table 2 shows the percent differences between thetistics of the reduced-order and full-model simulations. Usionly the first PCA mode, mean loss is underpredicted by 15%the error in standard deviation of loss and turning is 56 and 71respectively. It takes 15 modes to reduce the error in standdeviation of loss coefficient to 7%. BeyondK515, comparisonsstop being meaningful due to lack of resolution in the MonCarlo simulation.

Fig. 11 IBR: Effect of geometric variability on boundary layerentropy generation as per Eq. „10…. Mean indicated by solidlines, one-standard deviation interval by error bars. †SS‡: suc-tion side, †PS‡: pressure side.

Fig. 12 IBR mid section: Statistics versus number of PCAmodes, aÄ5



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4 Effect of Geometric Variability on Overall Compres-sor Performance

In this section, the impact of airfoil geometry variability ooverall compressor performance is estimated. Compressorciency and pressure ratio are obtained by exercising a multismean-line compressor model in combination with probabilisloss and turning models for the IBR blade discussed above.

A compressor stage mean-line model was derived from thelowing two observations. First, given the rotor total pressure lcoefficientf r , the flow turningq r , the outlet area, and the upstream flow conditions, the rotor outlet state is described bynonlinear system

FªDTT2v

cp~r 2V2 sina22r 1V1 sina1!50, (11)

Gª

mATT2

PT2A2 cosa2

2Ag

R

M2

S 11g21

2M2

2D ~g11!@z~g21!#50,

(12)

HªV2@sina21cosa2 tan~b12q r !#2vr 250, (13)

where

TT2~DTT!5TT1

1DTT ,

V2~DTT ,M2!5M2S gRTT2

11g21

2M2

2D 1/2

,

PT2~DTT!

5PT1RF 12f r

1

2gM1R

2

S 11g21

2M1R

2 D g/~g21!G S TT2

TT1R

D g/~g21!

.

In the above equations,P, T, M, A, andR stand for temperaturepressure, Mach number, area, and gas constant, respectiveadenotes absolute andb relative flow angles; the subscripts 1,and 3 denote rotor inlet, rotor exit/stator inlet, and stator exit; athe subscriptsT and R denote ‘‘total’’ and ‘‘relative’’ quantities,respectively;v stands for wheel speed,m for mass flow rate, andr for radius. The stator is described by similar equations withv50, loss coefficientfs , and turningqs .

Equations~11!–~13! form a nonlinear system in the variableDTT , M2 , and a2 , which was solved numerically usingNewton-Raphson solver. Equation~11! is simply a restatement othe Euler turbine equation for calorically perfect gases. Equa~12! is the flow parameter formula for quasi-one-dimensional flof calorically perfect gases~Fligner’s formula!. Equation ~13!states that the absolute and relative tangential velocities arelated via the wheel speed.

The probabilistic mean-line calculations estimate only the ipact of blade-to blade flow variability—caused by geometnoise—on compressor performance. Geometric variability leadto three dimensional flow effects~e.g., tip-clearance leakage, parspan losses, off-design radial imbalances, end-wall losses,!

Table 2 Percent difference in statistics from reduced-ordermodel „with K modes … and full model „KÄmp … simulations, aÄ5

K mÃ sÃ mq sq

1 214.5 256.4 1.08 271.15 25.2 220.0 0.43 29.310 22.2 25.7 0.19 25.215 21.4 23.0 0.12 22.420 20.8 22.5 0.07 21.4

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Fig. 13 IBR: Loss coefficient and turning angle versus incidence, aÄ5

ys

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are not considered. The present approach therefore is likelunderestimate the performance variability of actual compresin the presence of geometric variability.

4.1 Loss Coefficient and Turning Angle Models. Concep-tually, the loss coefficient and turning angle obtained from blapassage analyses may be taken to be deterministic functionvarious geometric and flow parameters such as inlet flow aninlet Mach number, etc. In particular, let

f5f~a,x! and q5q~a,x!,

wherea is inlet flow incidence andx is a vector of parameterdescribing amplitudes of geometric noise modes such as tdescribed above. In the present application, ‘‘incidence’’ is takto mean the difference between the nominal inlet flow angle~herethe minimum-loss angle! and the actual flow angle of the incoming stream. Other geometric and flow parameters are assumbe fixed and their functional dependence is not explicitly modeFurthermore, we separatef into nominal and noise terms,

w~a,x!5w0~a!1Df~a,x!, (14)

then define

Dmf~a!ªEX

@Df~a,X!#, sf2 ~a!ªVar

X~Df~a,X!!.

In generalDmf(a) and sf2 (a) may not be available in close

form. Instead, letDmf(a) and sf2 (a) be models of loss ‘‘mean

shift’’ ~i.e., the difference between loss variance, respectivThen

EX

@f~a,X!#5EX

@f0~a!1Df~a,X!#'f0~a!1Dmf~a!,

VarX

~f~a,X!!5EX

@~Df~a,X!2Dmf~a,X!!2#'sf2 ~a!.

For fixeda, it is further assumed thatDf(a,X) is normally dis-tributed; that is,

Df~a,X!PN~Dmf~a!,sf2 ~a!!.

Models of Dmf(a), sf2 (a), Dmq(a), and sq

2 (a) were ob-tained from computed statistics of loss and turning at fixed valof incidence for isolated blade passages using the MISES bpassage analysis in a Monte Carlo simulation. Finally, an identargument was applied to obtaining models for turning angle.

, OCTOBER 2003

0 to 129.5.32.121. Redistribution subject to ASME lic

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Figure 13 shows loss coefficient MCS results. The geomenoise assumed in the simulations was the PCA-based modebefore, with noise amplitudea55. The output statistics werecomputed for each fixed value ofa from N52000 trials. In Fig.13, the solid line connects the computed nominal loss coeffic~i.e., in the absence of geometric noise!; the dash-dot line con-nects the values computed for the average-geometry airfoil;dashed line indicates the MCS mean values; and the errorshow two-standard-deviation intervals centered at the mean.figure shows a typical ‘‘loss bucket’’ shape with minimum nomnal loss approximately at zero incidence. The loss coefficientcreases more steeply for positive values of incidence as doevariability. No points are plotted fora.1 degree where numericaconvergence of the Monte Carlo simulations was deficient~lessthan 80% of the MISES’ runs in the Monte Carlo simulation coverged!.

The computed data points illustrated above are used to depiecewise-cubic interpolating splines with zero-second-derivaend conditions. To avoid dangerous extrapolation outside thecidence range for which computed data points were availaadditional points were added ata52, 3, and 4 by linearly ex-trapolating the nominal loss and turning and replicating the meshift and variance values corresponding to the highest compa.

4.2 Probabilistic Six-Stage Compressor Model. The start-ing point for the probabilistic analysis was a six-stage compresmodel with nominal pressure ratiop0510.8 and polytropic effi-ciency e050.96. The nominal rotor and stator loss coefficienwere f r5fs50.03, and the nominal rotor turning wasq r514.4 deg. The high nominal efficiency is due to the absenceend-wall and tip-clearance losses in the model.

Stator nominal loss and turning, as well as their mean shiftstandard deviation, were taken directly from the IBR incidenmodels discussed above. Stators are generally required to promore flow turning than rotors. Therefore it should be expectedthe stators will exhibit higher exit flow variability~e.g., more flowdeflection! than the rotor. As a conservative estimate the sanominal, mean-shift and variance models for loss coefficient wused for the stator and for the rotor. The stator turning variabimodel was obtained by scaling the rotor model to the stator nonal turning—effectively using the same standard-deviation verincidence model as for the rotor.

Monte Carlo simulation results (N52000) show a 0.2% drop



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from nominal polytropic efficiency to the mean value, and a 0.decrease in total pressure ratio for the base noise level. Atbaseline noise level the primary contribution to performanceviations comes from the average geometry rather than fromgeometric variability.

The geometric variability present in the IBR coordinate mesurements is quite small, due to the use of a highly contromanufacturing process. The ‘‘small’’ geometric noise in the msurements translates to small loss and turning variability, whicturn result in small compressor performance uncertainty: the sdard deviation of polytropic efficiency and pressure ratio0.04% and 0.02, respectively. The impact of increased noise lis reported next.

4.3 Impact of Geometric Noise Amplitude. Presumably,as the amount of geometric variability increases, so shouldimpact on compressor performance. This subsection attempquantify that trend within the limitations of the current mean-limodel. Table 3 summarizes the polytropic efficiency and ovepressure ratio statistics for three noise variability levels:a51, 2,and 5.

For a52, the mean shift in polytropic efficiency and pressuratio are 0.3% and 0.7%, respectively. The standard deviatiopolytropic efficiency increases roughly by a factor of 2 when copared with itsa51 counterpart. At thea55 level, the standarddeviation of efficiency has risen to about 0.2%, a sevenfoldcrease froma51. At this level of noise, the mean shift in polytropic efficiency also becomes noticeable at;1% from nominal.When comparing the impact of geometric noise amplitudepressure ratio variability, the increase is nearly linear with nolevel, i.e., compared with thea51 case, the standard deviationpressure ratio ata52 increases by a factor of 2 and ata55 by afactor of 6.

4.4 Multiple-Blade Rows. In the calculations reportedabove, it was assumed that for a given bladed row, all passbehaved identically, i.e., that a single passage can be considerrepresent each bladed row. In this section, multiple passagesblade row are considered. Loss and turning values for eachsage are sampled from a normal distribution according to iincidence. Thus, for each passage instance, loss and turning vare in general different but have the same statistics prescribethe loss and turning models. The corresponding system of sequations@Eqs.~11!–~13!# is solved for each passage. The outconditions are area-averaged to initialize the inlet conditionsthe next rotor or stator and the calculation is marched throughcompressor.

Table 4 shows polytropic efficiency and pressure ratio statis

Table 3 Impact of geometric noise amplitude on compressorperformance. e0 and p0 are the efficiency and pressure ratiofor the nominal compressor „no noise ….

a e0 me se3100 p0 mp sp

10.963

0.961 0.03810.79

10.73 0.0222 0.959 0.083 10.71 0.0435 0.951 0.275 10.59 0.111

Table 4 Six-stage compressor, IBR airfoil-based loss and turn-ing models, 80 blade passages per row. e0 and p0 are the effi-ciency and pressure ratio for the nominal compressor „nonoise ….

a e0 me se3100 p0 mp sp

10.963

0.961 0.03310.79

10.73 0.0052 0.960 0.065 10.72 0.0105 0.953 0.197 10.61 0.029



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for the six-stage compressor model reported above but withindependent blades per row~rotor or stator!. Statistics for threelevels of geometric variability are reported. In comparing Tableto Table 3, it is seen that ata51, the mean shifts for the baselinmultiple-blade calculation are the same as for the single-blcase. The efficiency and pressure ratio standard deviations inmultiple-blade case are 13 and 77% lower, respectively, thathe single-blade case. At thea55 noise level the efficiency meashift is one percentage point, in contrast the 1.2% drop seenthe single-blade calculation. The standard deviations of efficieand pressure ratio have decreased by 30 and 75%, respectcompared to the single-blade results.

The reduction in efficiency and pressure ratio mean shift canexplained in part by considering a deterministic compressor wloss and turning given by the mean values obtained from MoCarlo simulation. As the number of blade passages is increathe mean values of efficiency and pressure ratio converge to tobtained for mean loss and turning models. Table 5 showsresulting polytropic efficiency and pressure ratio values usmean loss and turning for the three noise levels considered. Cparing the values in Table 5 to the mean polytropic efficiency apressure ratio in Table 4 and taking into account their reduvariability, it can be concluded that the contribution of the mevalues of loss and turning~for given incidence! dominates themean shifts of polytropic efficiency and pressure ratio.

5 ConclusionsIn this paper, we developed and applied a probabilistic meth

ology to quantify the impact of geometric variability on compresor aerodynamic performance. The methodology utilizesprincipal-component analysis~PCA! to derive a high-fidelityprobabilistic model of airfoil geometric variability. This probablistic blade geometry model is then combined with a compreible, viscous blade-passage analysis to estimate the aerodynperformance statistics using Monte Carlo simulation. Finallyprobabilistic mean-line multistage compressor model, with probilistic loss and turning models from the blade-passage analysdeveloped to quantify the impact of the blade variability on copressor efficiency and pressure ratio.

The methodology was applied to a flank-milled integrabladed rotor~IBR! with blade surface measurements taken bycoordinate measuring machine. The PCA model of the geomevariability demonstrates that 99% of the geometric scatter canmodeled with approximately the five strongest PCA modes. Hoever, subsequent aerodynamic analysis of the blade passageonstrates that approximately 15 modes are needed to modelof the overall aerodynamic impact on loss and turning.

At the blade passage level, the overall impact of the variabiin the flank-milled IBR was found to be very low causing only4% shift ~i.e., increase! of the mean loss compared to the lossthe design-intent blade, and an even smaller impact on the mturning. The source of the change in mean performance wastributed to variability in the loss and turning. However, compasons of the level of geometric variability in the flank-milled IBdata to other manufactured blades showed the flank-milled dathis study to have at least five times less variability than comonly observed in other situations. Thus a study of the aeronamic impact of the variability was also performed at increasnoise levels.

Table 5 Six-stage compressor, mean loss, and mean turning„no variability …

a51 a52 a55e p e p e p

0.961 10.73 0.960 10.71 0.954 10.62

OCTOBER 2003, Vol. 125 Õ 701


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Fig. 14 Measured deviations for sample point and flank-milled IBR

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At five times the actual IBR geometric noise level~which isconsidered representative of many manufacturing processes!, themean loss was approximately 20% larger than the nominal losparticular, for this application we note that:

• The majority of the mean shift in loss arises from the blavariability and only a small portion is due to errors in the megeometry. Thus simply re-targeting the manufacturing processnot have a substantial impact on the mean aerodynamic pemance of the blades.

• The major source of the increased mean and variance opassage loss can be traced to the leading-edge of the bladecifically in the first 5% of the chord on the pressure surface. In tregion, substantial increases in mean dissipation and subsequboundary layer momentum thickness were observed as a resuthe geometric variability.

• Mean turning is not greatly impacted by geometric variabity. At all the noise levels studied, the mean turning is similarthe nominal blade turning, though the variation of the turniincreases with geometric noise level.

The impact of the blade variability was then studied for a mtistage compressor using a mean-line model for a notionalstage compressor. As observed in the blade-passage analysactual noise in the flank-milled data has a small impact oncompressor efficiency~a 0.3% decrease in mean efficiency fronominal! and pressure ratio~a 0.7% decrease in mean pressuratio from nominal!. However, at the fivefold geometric noislevel, the compressor mean efficiency drops by 1% indicatinggeometric variability could have a significant impact on the copressor performance. Furthermore, the majority of this mean sin compressor efficiency can be accounted for by the mean shthe passage aerodynamic behavior.

Future work will include studies of other blades including thowith larger manufacturing variability. Also, three-dimensional efects such as tip clearance are currently being investigated

BER 2003

.5.32.121. Redistribution subject to ASME lic

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nally, robust aerothermal blade design has been applied to mgate the impact of the geometric variability and will be reportedsubsequent publications@17#.

AcknowledgmentsThe authors wish to thank Professor M. Drela, Professor

Greitzer, and Professor I. Waitz for their helpful comments asuggestions. Thanks also to Mr. Jeff Lancaster for helping usquire geometric variability data. The support of NASA Glenn Rsearch Center through Grant No. NAG3-2320 is thankfully aknowledged.

Appendix: Comparison With Production Manufactur-ing Variability

The flank-milled IBR considered in the main text exhibitegeometric variability which is uncommonly low for productiohardware. Point milling is a well understood and widely usmethod for manufacturing compressor blades. In this techniquball cutter removes material from a block of metal followincomputer-controlled paths. The main disadvantages of point ming are the time required to cut an entire blade surface in sevpasses and the resulting scalloped surface finish@19#. An alterna-tive to point milling that is starting to become practical is flanmilling, whereby a conical tool is used to cut the entire surfacea blade from the blank material in a single pass@19#. Flank mill-ing poses a more challenging tool control problem than point ming, but can potentially be more time and cost effective. Anotadvantage of flank milling is that it produces a better surfafinish than point milling, requiring less time for surface polishinThe geometric measurements reported above correspondedIBR manufactured via tightly controlled flank milling.

Figure 14 shows plots of measured deviations of two prodtion compressor rotor blades, one manufactured with point milland the other with flank milling. Deviations in chord length an



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leading-edge thickness at various spanwise locations are shThe deviations have been scaled with respect to the nominal swise average chord. Largest positive and negative deviationeach spanwise station are indicated by dashed lines, which inprovides a rough measure of variability in each measured dimsion. Table 6 shows maximum deviations~per unit chord! inter-vals in chord length, leading and trailing-edge thickness formeasurements shown in Fig. 14. The point-milled IBR exhibroughly 18 times more variability in chord length than the flanmilled rotor. The variability in LE and TE thickness measurments for the point-milled IBR is roughly six times that of thflank-milled rotor. This comparison provides a justification for thigher geometric variability levels considered in the main text

References@1# Lykins, C., Thompson, D., and Pomfret, C., 1994, ‘‘The Air Force’s Applic

tion of Probabilistics to Gas Turbine Engines,’’ AIAA paper 94-1440-CP.@2# Preisendorfer, R. W., 1988,Principal Component Analysis in Meteorology an

Oceanography, Elsevier, Amsterdam.@3# Jolliffe, I. T., 1986, Principal Component Analysis, Springer Verlag, New

York.

Table 6 Spanwise maximum deviation intervals „per unitchord … for point- and flank-milled IBR measurements

Dimension Point (3103) Flank (3103) Ratio

Chord length 8.6 0.49 18LE thickness 2.7 0.49 6.7TE thickness 3.8 0.57 5.6



wn.pan-s atturnen-

heitsk-e-ee

-

@4# Trefethen, L. N., and Bau, D., 1997,Numerical Linear Algebra, Society forIndustrial and Applied Mathematics, Philadelphia, PA.

@5# Drela, M., and Youngren, H., 2001,XFOIL 6.9 User Guide, Dept. of Aeronau-tics and Astronautics, Massachusetts Institute of Technology, 77 MassachuAve, Cambridge MA 02139.

@6# Ross, S., AFirst Course in Probability, 1997, Fifth Ed., Prentice Hall, UpperSaddle River, NJ.

@7# Drela, M., and Youngren, H., 1998,A User’s Guide to MISES 2.53, Dept. ofAeronautics and Astronautics, Massachusetts Institute of Technology, 70sar ST, Cambridge MA 02139.

@8# Drela, M., 1985, ‘‘Two-Dimensional Transonic Aerodynamic Design aAnalysis Using the Euler Equations,’’ Ph.D. thesis, Massachusetts InstitutTechnology.

@9# Youngren, H., 1991, ‘‘Analysis and Design of Transonic Cascades With Spter Vanes,’’ Masters thesis, Massachusetts Institute of Technology.

@10# Denton, J. D., 1993, ‘‘The 1998 IGTI Scholar Lecture: Loss MechanismsTurbomachines,’’ ASME J. Turbomach.,115, pp. 621–656.

@11# Hammersley, J. M., and Handscomb, D. C., 1965,Monte Carlo Methods,Methuen & Co., London, England.

@12# Thompson, James R., 2000,Simulation: A Modeler’s Approach, John Wiley &Sons, Inc., New York.

@13# Fishman, George S., 1996,Monte Carlo: Concepts, Algorithms and Applications, Springer Verlag, New York.

@14# Garzon, V. E., and Darmofal, D. L., 2001, ‘‘Using Computational Fluid Dnamics in Probabilistic Engineering Design,’’ AIAA paper 2001-2526.

@15# Drela, M., 1997,A User’s Guide to MTFLOW 1.2, Dept. of Aeronautics andAstronautics, Massachusetts Institute of Technology, 70 Vassar ST, CambMA 02139.

@16# Merchant, A., 1999, ‘‘Design and Analysis of Axial Aspirated CompressStages,’’ Ph.D. thesis, Massachusetts Institute of Technology, 77 Massasetts Ave, Cambridge MA 02139.

@17# Garzon, V. E., 2003, ‘‘Probabilistics Aerothermal Design of Compressor Afoils,’’ Ph.D. thesis, Massachusetts Institute of Technology.

@18# Cumpsty, N. A., 1989,Compressor Aerodynamics, Longman, London.@19# Wu, C. Y., 1995, ‘‘Arbitrary Surface Flank Milling of Fan, Compressor an

Impeller Blades,’’ ASME J. Eng. Gas Turbines Power,117, pp. 534–539.

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