Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, January 1996A9618413, AIAA Paper 96-0452

Turbine blade design using parallel processors

Stephen C. BrawleyU.S. Navy, Naval Postgraduate School, Monterey, CA

Garth V. HobsonU.S. Navy, Naval Postgraduate School, Monterey, CA

AIAA 34th Aerospace Sciences Meeting and Exhibit, Reno, NV Jan 15-18, 1996

The geometry of a turbine blade designed for an energy efficient engine was further optimized to reduce viscous lossesseven percent. A Navier-Stokes flow solver was used to evaluate the viscous losses of multiple turbine bladegeometries for design operating conditions. A gradient-based optimization scheme was implemented on the IBM SP2parallel computer. Multiple processors simultaneously computed the performance of numerous turbine bladegeometries to greatly increase the speed and efficiency of the optimization process. (Author)

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TURBINE BLADE DESIGN USING PARALLEL PROCESSORS

Stephen C. Brawley, Lecturer*, and Garth V. Hobson*, Associate ProfessorDepartment of Aeronautics and Astronautics

Naval Postgraduate School, Monterey ,CA 93943

AbstractThe geometry of a turbine blade

designed for an energy efficient enginewas further optimized to reduce viscouslosses seven percent. A Navier-Stokesflow solver was used to evaluate theviscous losses of multiple turbine bladegeometries for design operatingconditions. A gradient-basedoptimization scheme was implementedon the IBM SP2 parallel computer.Multiple processors simultaneouslycomputed the performance of numerousturbine blade geometries to greatlyincrease the speed and efficiency of theoptimization process.

NomenclatureCioss measure of total pressure loss

across turbine bladef objective functionn number of design variablesq positive scalar valueG gradient vector 5f75XX vector of design variables

IntroductionComputational fluid dynamics or

CFD has become a valuable engineeringtool in both aerodynamic analysis and

design. Design methods can be classifiedas either inverse or optimizationtechniques. Optimization methods varythe geometry of an initial body tominimize an objective function basedupon design performance criteria. If thecalculation of the objective functionwhich requires a CFD solution of anaerodynamic geometry is independent ofthe optimization method, the designerhas the freedom to choose an appropriateflow solver. For example, if the designcriterion is to minimize viscous losses, aNavier-Stokes flow solver rather than apotential flow solver can be used.

A new grid and a new flow fieldsolution are required after everygeometric perturbation in theoptimization of an aerodynamic shape.The main disadvantage of aerodynamicoptimization schemes is the amount ofcomputer processing time requiredbecause of costly flow field calculationsover various geometries.

The problem of designing an airfoilto match a desired pressure distributionwas first addressed by Lighthill in 1945.1He developed an inverse designtechnique to match a target pressuredistribution for incompressible potential

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flow around an airfoil by conformallymapping the profile onto a unit circle.Sanger designed a compressor bladeusing the optimization code CONMIN(Constrained Function Minimization) anda potential flow solver with boundary-layer estimations.2 CONMIN, which isbased upon the gradient method, wasused to vary independent variablesdescribing the geometry of the airfoil.

In order to significantly increase thespeed of the design process, theprocessing time required for the totalnumber of flow field calculations must bereduced. An efficient flow solver isrequired to quickly evaluate theperformance of various aerodynamicgeometries. Furthermore, parallelprocessors have been proven to greatlyreduce the processing time ofaerodynamic design via optimization byevaluating multiple geometries inparallel.3'4

The objective of this research was tofurther minimize the viscous losses offlow over a previously optimized turbineblade. Flows in turbomachinery arehighly complex and can be dominated byshock waves and viscous effects. Mostturbomachinery blade designs have reliedupon subsonic or transonic potentialanalysis which may not have provided thebest solution for the design criteria. Thistest demonstrates the practicality ofutilizing an efficient Navier-Stokes flowsolver with a gradient-based optimizationscheme for turbine blade design.

Parallel Optimization SchemeAn optimization routine has been

developed which divides the requiredperformance evaluations among multipleprocessors. The parallel optimizationscheme, when coupled with a flowsolver, evaluates the aerodynamic

performance of numerous geometriessimultaneously and greatly decreases thetime required for design.

For turbine blade design, the designermust first select the desired performancecriteria. Next, design variables are usedto describe the geometry of the blade.After an initial CFD solution andperformance evaluation are calculated,multivariable calculus determines adirection to vary the independentvariables to optimize the performancecriteria. The performance of multipleturbine blade shapes are then evaluatedusing CFD, and the geometry whichcomes closest to the desired performancebecomes the baseline solution to vary forthe next iteration. If there are nolimitations to the allowable computerprocessing time, this process is repeateduntil a turbine blade geometry is foundwhich matches or optimizes the desiredperformance.

For a gradient-based or steepest-descent optimization scheme, theobjective function f is described by a setof n design variables, X, placed in vectorform

= f(x1,x2,x3,...Xn) (1).

In order to minimize the objectivefunction, the ratio of the design variablesare varied in the direction of the negativeof the gradient vector, G. The new setof design variables is then calculatedfrom the relation

k+l _ vk— X - q (2),

where k is the optimization cyclecounter. The positive scalar q must begiven minimum and maximum limits bythe designer. The gradient vector is

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calculated using central-differenceapproximations with constantperturbations for each design variable:

f(x1+Ax)-f(x1-Ax)2Ax '"" (3).

f(xn + Ax)-f(xn-Ax)v '2Ax

After the gradient is calculated, theobjective function becomes a function ofonly q as shown in equations (1) and (2).A directional search to define new valuesfor the design variables is conducted tominimize the objective function. In thedirectional search, q is varied and theobjective function is evaluated until aminimum is found. The gradientcorresponding to the new set of designvariables is then calculated for the nextdirectional search to minimize theobjective function.

Numerous objective functionevaluations for turbine blade optimizationare necessary for the gradientcalculations and line searches. Eachobjective function evaluation of a turbineblade geometry requires one CFDsolution for a single design point.Parallel processors are used tosimultaneously calculate the flow fieldsover multiple turbine blade geometriesfor the estimation of the gradient vectorsand in directional searches to minimizethe objective functions.

The parallel computer used for thisresearch is the IBM SP2 located atNASA Ames Research Center. The SP2contains 160 RISC System/6000processor nodes. Each 66.7 megahertzprocessor has 256 megabytes of memory.The improved performance of the SP2over previous parallel supercomputersallows for more advanced optimizationproblems such as viscous flow and three-dimensional analysis.

Central-difference estimations of thegradient vector require twice the numberof objective functions as the number ofdesign variables for an application asshown by equation (3). By using 2nprocessors for an objective functiondefined by n design variables, all of theperformance calculations can beconducted in parallel.

In order to quickly locate a minimumobjective function, all processors areutilized for function evaluations in thedirection of search. Each processor isdesignated a unique value of q at equalintervals between its maximum andminimum values and calculates theobjective function based upon designvariables defined in equation (2). Next,all objective functions are compared in aglobal operation and the design variablescorresponding to the minimum objectivefunction become the design variables forthe next optimization cycle. The moreprocessors which are used in anapplication provide for a more thoroughdirectional search and increase theprobability of converging to a globalminimum instead of a local minimum.Conducting the gradient estimations andline searches in parallel greatly increasesthe speed and efficiency of the designprocedure.

Baseline Airfoil and Design VariablesThe baseline turbine blade used for

this optimization was designed for aGeneral Electric energy efficient engineand has been analyzed computationallyand experimentally at NASA Lewis.5Eight points on the upper surface andeight points on the lower surface of theblade were used as the design variables.The x-axis coordinate was kept the sameand the y-axis coordinate was varied atthese locations to minimize the viscous

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losses across the blade. The points onthe surface of the blade within 10%chord of the rounded leading and trailingedges were not varied in order tomaintain adequate volume and to ensurethe optimization scheme would notproduce sharp or flat leading or trailingedges. The design locations were evenlydistributed across the remaining surfacesof the suction and pressure sides, and aspline fit defined the shape of the bladeafter each variation. For each linesearch, the maximum variation of adesign variable was limited to 1% of themaximum thickness of the baseline blade.Additionally, each design variable wasperturbed 0.04% of the maximumthickness for the estimation of thegradient vector.

A modification to the GRAPE gridgeneration program was used to generatea 250 x 60 C-type grid around thegeometries evaluated in the designprocess.6 Figure 1 shows the gridaround the baseline turbine blade row.

Flow SolverThe selected performance criterion

required a Navier-Stokes flow solver toevaluate the viscous losses of the internalcascade flow. Chima developed a multi-stage Runge-Kutta scheme for quasithree-dimensional flows inturbomachinery.7 This efficient Navier-Stokes code was developed forturbomachinery design and analysis. Thethin-layer assumption was invoked toeliminate the streamwise viscousderivatives, which reduced theprocessing time and allowed for thecomputation of separated flows. Thealgebraic eddy-viscosity modeldeveloped by Baldwin and Lomax wasused for the evaluation of turbulencewithin the boundary layers.8

A five-stage Runge-Kutta schemewas applied to this problem. The flowfield around each cascade bladegeometry was initialized based uponfreestream conditions prior to eachperformance evaluation. Theconvergence of the density residuals aftereach flow field iteration is shown inFigure 2, which revealed a decrease inconvergence rate after 1200 iterations forthe baseline solution. The overallreduction in the residuals was almostthree orders of magnitude. For all CFDsolutions conducted in the optimizationdesign, 1200 flow field iterations wereperformed.

Performance CriteriaThe explicit Navier-Stokes flow

solver was used to evaluate a losscoefficient for the viscous losses throughthe cascade. The loss coefficient, GIOSS,was based upon the loss of the mass-averaged total pressure from the cascadeinlet to the cascade exit. The objectivefunction to be reduced in theoptimization scheme was set equal to

Qoss an value of 0.561corresponding to the baseline geometry.As shown in Figure 3, the value for theloss coefficient used as the objectivefunction decreased to a minimum afterapproximately 980 flow field iterationsand then oscillated around a mean value.

Figure 4 illustrates the Machcontours around the baseline turbinestator blade for an inlet total-to-exitstatic pressure ratio of 0.685. Theresulting flow field was subsonic with amaximum Mach number approaching0.9.

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ResultsThe purpose of the optimization

application was to design a turbine bladeto minimize viscous losses at its designoperating conditions. Thirty-twoprocessors of the IBM SP2 parallelcomputer were utilized requiring 257different flow field solutions and twohours of processing time. If thisoptimization was performed on a singleRISC processor, approximately 64 hoursof processing time would be required tocalculate the CFD solutions of thevarious geometries.

The Mach contours around theoptimized blade are shown in Figure 5.The region of maximum Mach numberextends farther along the suction side ofthe optimized blade than for the baselineblade. Furthermore, the boundary layerof the optimized blade necks down atapproximately 80% chord on the suctionside. Conversely, the boundary layer ofthe baseline blade continues to growdownstream of the maximum Machnumber which results in a larger wakethan with the optimized blade. Figures 6and 7 illustrate the differences betweenthe baseline and optimized blades. Onlyslight differences are observed in theflow field on the pressure sides of thebaseline and optimized blades eventhough the perturbations are of similarmagnitudes on both the pressure andsuction sides.

A convergence history of the fouroptimization cycles are presented inFigure 8. The decrease of the losscoefficient was less each optimizationcycle, and the steepest-descent searchfailed to decrease the loss coefficient onthe fourth cycle. The optimizedgeometry resulted in a 7% reduction inthe viscous loss coefficient.

ConclusionsThis work demonstrates the

practicality of utilizing quasi three-dimensional Navier-Stokes flow solverswith parallel optimization schemes fordesign improvements of turbine blades.The use of multiple parallel processorssuch as on the IBM SP2 greatlyincreased the speed and efficiency ofaerodynamic optimization by conductingnumerous CFD solutions in parallel forestimations of derivatives of the gradientvector and for directional searches tominimize an objective function. Thememory and speed capabilities ofrecently developed parallel processorsalso allow for more complexaerodynamic optimization using three-dimensional Navier-Stokes flow solvers.Furthermore, the design of turbine bladesfor multiple operating points can beconducted in approximately the sameamount of time as single-point designproblems if enough processors areavailable in a parallel application.

References1. Lighthffl, M. J (1945), 'A NewMethod of Two DimensionalAerodynamic Design1, ARC, Rand M2112.

2. Sanger, N. L. (1982), The Use ofOptimization Techniques to DesignControlled Diffusion CompressorBlading1, NASA TM-82763.

3. Brawley, S. C. andHobson, G. V.(1995), 'Airfoil Design Utilizing ParallelProcessors, Part I: Theory1, AIAA 95-0125.

4. Brawley, S. C. and Hobson, G.V.(1995), 'Airfoil Design Utilizing Parallel

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Processors, Part II: Applications', AIAA95-0126.

5. Goldman, L. J. and Seasholtz, R. G.(1982), 'Laser AnemometerMeasurements in an Annular Cascade ofCore Turbine Vanes and ComparisonWith Theory1, NASA Technical Paper2018.

6. Sorenson, R. L. (1980), 'A ComputerProgram to Generate Two-DimensionalGrids About Airfoils and Other Shapes

by Use of Poisson's Equation1, NASATM-81198.

7. Chima, R. V. (1986), Tnviscid andViscous Flows in Cascades with anExplicit Multi-Grid Algorithm', AIAAJournal, Vol. 23, No. 10.

8. Baldwin, B. S. and Lomax, H.(1978), Thin-Layered Approximationand Algebraic Model for SeparatedFlows', AIAA 78-257.

Figure 1: Baseline Turbine Blade and 250 x 60 Grid

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BASELINE ANNULAR TURBINE

10 ,0 500 1000Iteration

1500 2000

Figure 2: Convergence History of Baseline Flow Field Solution

BASELINE ANNULAR TURBINE

<D 0 jo 2U

COoraO

1-

0 500 1000Iteration

1500 2000

Figure 3: Convergence of Loss Coefficient in Baseline Flow Field Solution

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CONTOUR LEVELS0,00000O.OSOOO

0.2120.00 DEG1.73xlO**52.00xlO**3250x60

MACHALPHAReTIMEGRID

Figure 4: Mach Contours Around Baseline Turbine Blade

CONTOUR LEVELS

Figure 5: Mach Contours Around Optimized Turbine Blade

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Figure 6: Comparison of Baseline and Optimized Turbine Blade Geometries

Figure 7: Leading Edges of Baseline and Optimized Turbine Blade Geometries

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u.o/ •

Occ 4

0.55-

« 0.54-V)o

Q 0.53-

0.51-

n s -

f\\^^^

— — —— | ———————— ̂

0 1 2 3 4

Optimization Cycle

Figure 8: Convergence History of Parallel Optimization Scheme

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