NASA/TM-1998-208754

Neural Net-Based Redesign of Transonic Turbines for

Improved Unsteady Aerodynamic Performance

Nateri K. Madavan and Man Mohan Rai

Ames Research Center, Moffett Field, California

Frank W. Huber

Riverbend Design Services, Palm Beach Gardens, Florida

National Aeronautics and

Space Administration

Ames Research Center

Moffett Field, California 94035-1000

November 1998

https://ntrs.nasa.gov/search.jsp?R=19990008889 2018-05-11T16:55:52+00:00Z
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Neural Net-Based Redesign of Transonic Turbines for Improved Unsteady

Aerodynamic Performance

NATERIK. MADAVAN, MAN MOHAN RAI, and FRANK W. HUBER*

Ames Research Center

Summary Introduction

A recently developed neural net-based aerodynamic design procedure is used in the redesign of a

transonic turbine stage to improve its unsteady

aerodynamic performance. The redesign procedure

used incorporates the advantages of both traditional response surface methodology (RSM) and

neural networks by employing a strategy called

parameter-based partitioning of the design space.

Starting from the reference design, a sequence of

response surfaces based on both neural networks

and polynomial fits are constructed to traverse the

design space in search of an optimal solution that

exhibits improved unsteady performance. The procedure combines the power of neural networks and

the economy of low-order polynomials (in terms of

number of simulations required and network training requirements). A time-accurate, two-dimensional, Navier-Stokes solver is used to evaluate the

various intermediate designs and provide inputs to

the optimization procedure. The optimization procedure yields a modified design that improves the

aerodynamic performance through small changes

to the reference design geometry. The computed

results demonstrate the capabilities of the neural

net-based design procedure, and also show the tremendous advantages that can be gained by including high-fidelity unsteady simulations that capture

the relevant flow physics in the design optimization

process.

A patent application that covers some of the original ideasin

this report has been filed by NASA.This report has been subThe aerodynamic design of transonic high pressure

(HP) aircraft engine turbines is complicated by the

presence of shocks, wakes, tip leakage, and other

secondary flow effects in the flow field. These

shocks, wakes, and vortical flows are ingested by

downstream stages, resulting in complex interactions with one another and with the flow in these

stages. All of these effects are complicated further

by the inherent unsteadiness of the flow field that

results from the relative motion of the rotor and

stator rows and gives rise to unsteady interactions

both within the HP turbine stages and between the

• HP turbine and the adjacent low pressure (LP) turbine stages. These unsteady interactions may be

large enough to affect the time-averaged features of

the flow. Cooling and heat transfer are also important considerations in the design process, since

most HP turbine blades are typically cooled to

withstand high operating temperatures. The heat

transfer is closely coupled to the unsteady aerodynamics and is often affected greatly by it. However,

heat transfer will not be addressed in this article

since the emphasis here is on aerodynamic design.

Several experimental investigations of transonic

turbines aimed at characterizing shock formation

(ref. 1), unsteady stage interactions (ref. 2), heat

transfer effects (ref. 3), and other physical flow

phenomena involved have been carried out over the

years. Various numerical investigations of these

flow fields ranging from single blade row computations to time-accurate Navier-Stokes computations

in two dimensions (ref. 4, 5, 6) and more recently

in three dimensions (ref. 7, 8) have also added to

our understanding of these flows.

mitted for review towardpresentation at the 35th AIAA/ASME/ .....

SAE/ASEEJoint PropulsionConference and Exhibit,June 20- Modem HP turbines are usually composed of

24, 1999,Los Angeles, CA. either one or two stages. Two-stage turbines are

*RiverbendDesign Services, Palm Beach Gardens, Florida. longer and heavier but are subsonic and usually
moreefficient(ref.8).Single-stageturbinesare

lighterandcompactbutoperateinthetransonic

regimeandsufferefficiencypenaltiesduetoshock

lossesandhighaerodynamicbladeloadings

(ref.8).Typically,weakobliqueshocksoccurat

thestatorandrotorairfoiltrailingedges.Thestator

vaneshockinteractswithadjacentstatorvanesand

downstreamrotorbladesto setupacomplexpatternof directandreflectedshockwaves(seeGiles

(ref.4) andAbharietal.(ref.5)forinstructive

illustrateddescriptions)withinthestage.Therotor

bladetrailingedgeshockontheotherhandinteractswiththedownstreamLPstage.Theefficiency

penaltiesresultingfromtheseshockscanbequite

large.Forexample,Giles(ref.4)notesthatthe

unsteadyshocksareresponsiblefora40%variationin thelift ontherotor,resultinginstructural

vibrations,increasedlosses,andtemporaryleading

edgeboundarylayerseparationontherotorsuction

surface;JennionsandAdamczyk(ref.8)reporta

turbinedesignwhereit wasspeculatedthata5.6%

lossinefficiencywasduelargelyto theHProtor

shockinteractionswiththeLPturbinestatorvanes.

Evenintwo-stagedesignsthataredesignedto

operateinthesubsonicregime,thereisthepotentialfor unsteadyshocksintheflowfieldwithhigh

bladeloadings.Becauseof thedetrimentaleffects

of theseshocks,suchasdegradedaerodynamic

performance, unsteadystresses, fatigue,vibration,

andreducedbladelife,designershaveto payspecial attentionto them.A designoptimization

methodthatwouldhelpthedesignersintheir

effortstomitigatetheeffectsof theseshockswould

serveasaveryusefultool.

A varietyof formaloptimizationmethodshave

beendevelopedinthepastandappliedtoturbine

design.Theseincludeinversedesignmethods(see,

e.g.,DemeulenaereandVandenBraembussche

(ref.9)),bladeshapeoptimizationprocedures(see,

e.g.,Chattopadhyayetal.(ref.10)),andmultidisciplinaryoptimizationproceduresthatintegratethe

heatransferandaerodynamiceffects(ref.11).The

gasturbineindustryhasalsobeenincorporating

designoptimizationtechniquesin theturbine

designprocessforsometimenow.Thereareseveralreferencesintheliterature(see,e.g.,Tongand

Gregory(ref.12),andSheltonetal.(ref.13))dealing withtheuseof acommerciallyavailableoptimizationenvironment(iSight)in preliminary

designaswellasdesignoptimization.However,

mostof thisworkhasitsbasisintraditionalnumericaloptimizationprocedures.

Morerecently,theauthorshavedevelopedadifferentapproachto turbomachinerybladedesignoptimizationthatisbasedonneuralnetworks

(ref.14,15).Thismethodoffersseveraladvantages

overtraditionaloptimizationprocedures. First,

neuralnetworksareparticularlysuitableformultidimensional interpolationof datathatlackstructure.Theycanprovideagreaterlevelof flexibility

thanothermethodsindealingwithdesignin the

contextof unsteadyflows,partialandcomplete

datasets, combinedexperimentalndnumerical

data,theneedtoincludevariousconstraintsand

rulesof thumb,andotherfeaturesthatcharacterize

theaerodynamicdesignprocess.Second,neural

networksprovideanaturalframeworkwithin

whichasuccessionof numericalsolutionsof

increasingfidelityincorporatingmoreandmoreof

therelevantflowphysicscanberepresentedand

utilizedsubsequentlyfor optimization.Third,and

perhapsmostimportant,neuralnetworksofferan

excellentframeworkfor multidisciplinarydesign

optimization.Simulationtoolsfromvariousdisciplinescanbeintegratedwithinthisframework.

Efficientusecanalsobemadeofparallelcomputingresources. Rapidtrade-offstudiesacrossoneor

manydisciplinescanalsobeperformed.

Whileneuralnetworkshavebeenusedinother

applications,includingaeronautics, for sometime

now,theirapplicationto turbinedesignoptimizationisrelativelynew.Theonlyotherreferencein

thisareaistheworkof Sanz(ref.16),whousesa

neuralnetworktodetermine,fromadatabaseof

inputpressuredistributions,apressuredistribution

thatwouldproducetherequiredflowconditions.

Aninversedesignmethodisthenusedto compute

theairfoilshapethatcorrespondsto thisdesired

pressuredistribution.In otherwork(ref.17),

althoughnotdirectlyrelatedto neuralnetworks,a

turbineaerodynamicdesignmethodisdeveloped

thatisbasedonanevolutionaryoptimizationtechniqueanduses"reinforcement learning"to learn

adaptivelyfromthedesignenvironment.

Thispapereportsoncontinuingworkbythe

authorsindevelopinganeuralnetwork-basedtur-

'1 !i
bomachinerybladedesignmethod.It dealswith

theapplicationof thismethod(ref.t5)totheredesignof atransonicI-IPturbinewiththegoalof

improvingitsunsteadyaerodynamicperformance.

Theturbinechosenhereisatwo-stageconfigurationwithanaggressivedesigncharacterizedby

highturninganglesandhighspecific-workper

stage(ref.18).Ourinterestis inthefirststageof

thisconfiguration.Althoughtheturbineisdesigned

tooperateinthehigh-subsonicregime,anunsteady

analysisshowsverystronginteractioneffectsdue

tothepresenceof anunsteadymovingshockinthe

axialgapregionbetweenthestatorandrotorrows.

It ishypothesizedthatthestrengthof thisshock

canbereducedbyoptimizingtheairfoilgeometries,andtheoverallunsteadyaerodynamicperformanceof theturbinecantherebybeimproved.

Sincetheshockcanonlybediscernedbyan

unsteadyaerodynamicanalysis,atime-accurate

Navier-Stokessolver(ref.19)iscoupledto the

neuralnet-basedoptimizerandprovidesimulation

inputstoit. Theresultspresentedheredemonstrate

thattheneural-netbasedoptimizationmethod

yieldsamodifieddesignthatisveryclosetothe

referencedesignandachievesthesameworkoutput,yethasbetterunsteadyaerodynamicperformancesincetheflowthroughit isshock-free.

Therestofthisreportdealswiththeapplicationof

thedesignoptimizationmethodof RaiandMadavan(ref.15)totheredesignof atransonicHPturbine.Thedesigngoalis toimproveits unsteady

aerodynamicperformance. Detailsregardingthe

redesignprocedureandtheresultsobtainedarediscussedin thefollowingsections.

The Reference Design

The transonic turbine that is considered for redesign in this report is a preliminary design developed by Pratt and Whitney for a new generic gas

generator (G 3) turbine (ref. 18). This turbine is

designed to operate in the high-subsonic regime.

Table 1 lists all the relevant flow and geometry

parameters for the turbine, a two-stage configuration that is characterized by very high turning

angles (160 deg. in the rotor passage) and high specific work per stage. Further, low-convergence airfoil shapes are used for the rotor blades. All these

features made the design process for this turbine

very critical. In particular, the 160 deg. turning

angle was well above most existing designs.

Because this design was so far beyond the range of

their existing data base, the designers were unsure

of the effects of unsteady interactions on turbine

performance. A post-design unsteady time-accurate analysis of the flow was performed (ref. 6) as a

final evaluation of the design. This analysis

revealed significant unsteady effects and an

unsteady shock on the suction surface of the stator

that spanned the gap region and impinged on the

rotor blades as they passed by the stator airfoils.

The position of this unsteady moving shock on the

stator suction surface and its strength oscillated

periodically in time at blade-passing frequency.

The shock is entirely due to the stator-rotor interaction and any analysis that does not account for this

interaction will fail to indicate the presence of the

shock (ref. 6). On the basis of these findings, a

design modification that increased the axial gap

between the stator and rotor rows (from 30% of

mean chord to 75% of mean chord) was recommended. Unsteady analysis of this modified design

showed that the flow through the stage was shockfree. The uncooled stage efficiency of the modified

design was also higher, and the overall performance level was closer to that expected by the

designers. The reference design in this report is the

original design without the axial gap modification.
Table1.Geometryandflowparameters forthe

referenceandmodifiedesigns. All anglesare

measuredfromtheaxialdirection.

ReterenceModltled Parameter Design Design

Numberofstatorvanes 38

Numberofrotorblades 52

0.455

38

52

Pressureratioacrossstage

UnitReynoldsNumberatstator

inlet(perinch)

RPM

Ratioofspecificheats

490,000

24,000

1.3699

Stator 0.0° inflowangle

Statoroutflowangle 83.i°

Rotor-relativeinflowangle 79.4°

Rotor-relativeoutflowangle -82.0°

StatorinflowMachnumber 0.0585

0.455

490,000

24,000

1.3699

0.0°

83.2°

79.4°

82.0°

0.0587

The Redesign Procedure

General Objective

In this report we use our recently developed neural

net-based turbomachinery airfoil design procedure

to improve the reference design by successfully

mitigating the effects of the unsteady shock. We

accomplish our redesign objective by optimizing

the shape of the airfoils while maintaining the original axial gap (30% of mean chord). Our purpose is

to demonstrate the capabilities of our method in

unsteady design and also to show the tremendous

advantages that can be gained by including highfidelity unsteady simulations that capture the relevant flow physics in the design optimization process.

Airfoil Geometry Parameterization

Geometry parameterization and prudent selection

of design variables are among the most critical

aspects of any shape optimization procedure. Since

this study focuses on airfoil redesign, the ability to

represent various airfoil geometries with a common set of geometrical parameters is essential.

Variations of the airfoil geometry can be obtained

then by smoothly varying these parameters. Geometrical constraints imposed for various reasons,

structural, aerodynamic (e.g., to eliminate flow

separation), etc., should be included in this parametric representation as much as possible. Additionally, the smallest number of parameters should

be used to represent the family of airfoils.

The method used for parameterization of the airfoil

geometries is described in Rai and Madavan

(ref. 15) and is reviewed here for completeness.

Figure 1 illustrates the method for a generic airfoil.

Some salient features of the method are noted:

1. The leading edge is constructed using two different ellipses, one for the upper surface and one for

the lower surface. The eccentricity of the upper

ellipse and the semi-minor axes of both ellipses are

specified as geometric parameters (%, tu , and

t l ), respectively. All other related parameters can

be determined analytically. The major axes of both

ellipses are aligned with the tangent to the camber

line at the leading edge. This tangent is initially

aligned with the inlet flow but is allowed to rotate

as the design proceeds. The angles c_u and a t

determine the extent of the region in which the

leading edge is determined by these ellipses. The

two ellipses meet in a slope-continuous manner.

2. The trailing edge can also be constructed in a

similar manner with the major axes of the ellipses

aligned with the tangent to the camber line at the

trailing edge. However, in this study the trailing

edge was defined using a single circle. The angles

I]u and 13t determine the extent of the region in

which the trailing edge is determined by this circle.

3. The region of the upper surface between the

upper leading edge ellipse and the trailing edge circle is defined using a tension spline. This tension

spline meets the leading edge ellipse and the trailing edge circle in a slope-continuous manner.

Additional control points for the tension spline that

are equispaced in the axial direction are introduced

as necessary. These points provide additional control over the shape of the upper surface. The lower

surface of the airfoil between the lower leading

edge ellipse and the trailing edge circle is obtained

in a similar manner.

71 !i:
ta

O

d

0

8 _

d

_ _

0

9

h_

9

0

0

-0.25

I

I I I

0.00 0.25 0.50

x/c

1

I

0.75 1.00 1.25

Figure 1. Schematic of a generic airfoil showing location of nodal points on the airfoil surface and the

defining angles used in the parameterization of the airfoil geometry.

A total of 13 geometric parameters were used to

define the airfoil geometries in the current study.

These parameters are listed below:

(1) Leading edge and trailing edge airfoil metal

angles (2 parameters).

(2) Eccentricity of upper leading edge ellipse (1

parameter).

(3) Angles defining the extent of the leading edge

ellipse (2 parameters).

(4) Angles defining the extent of the trailing edge

circle (2 parameters).

(5) Airfoil thickness values at the leading edge (2

parameters).

(6) Airfoil y-coordinate values (see fig. 1) at midchord on the upper surface and lower surfaces

(2 parameters).

(7) Airfoil y-coordinate values (see fig. 1) at

intermediate points on the upper surface (2

parameters).

These parameters were adequate to obtain an accurate representation of the reference airfoils, and

acceptable modified shapes required by the optimization procedure could be obtained by varying

these parameters.

Unsteady Aerodynamic Analysis

Unsteady aerodynamic analyses of the turbine

stage configurations required during the redesign

process were obtained using the ROTOR-2 computer code (ref. 19). This code solves the unsteady,

two-dimensional, thin-layer Navier-Stokes equations for rotor-stator configurations in a time-accurate manner. Three-dimensional effects of streamtube contraction are also modeled. The computational method used is a third-order-accurate, iterative-implicit, upwind-biased scheme that solves the

time-dependent, Reynolds-averaged Navier-Stokes

equations. Details regarding the solution methodology can be found elsewhere (ref. 19).
Theflowdomainisdiscretizedusingasystemof

patchedandoverlaidgrids;thegridsattachedtothe

rotorairfoilscanmoverelativetothegridsattached

to thestatorairfoilto simulatetherotormotion.

Figure2showsthestatorandrotorairfoilcross

sectionsatmidspanforthereferenceturbine

design.Thereferencedesignhas38airfoilsinthe

statorrowand52intherotorrow.Tosimulatethis

flowatleast19statorairfoilsand26rotorairfoils

wouldhavetobemodeledasasystem.Thecomputationalexpenseof suchasimulationcanbe

reducedconsiderablybymodifyingthenumberof

statorairfoilsto39,sincethiswouldpermitasimulationwiththreestatorandfourrotorairfoilsasa

systemwithperiodicityconditionsto accountfor

therestof theairfoils.Themodificationof thestator airfoilcountisaccomplishedbyrescalingthe

statorgeometrybyafactorof 38/39andkeeping

thepitch-to-chordratiothesameastheactual

design.Thisrescalingisrelativelyminorandisnot

expectedto significantlyaltermostof therelevant

featuresof theflow.

Figure2alsoshowsthegridsystemusedtodiscretizetheflowdomain.Eachairfoilhastwogrids

associatedwithit: aninner"O" gridthatcontains

theairfoilandanouter"H'"gridthatconformsto

theexternalboundaries. Fortheanalysesperformedhere,eachinnerO-gridhas151pointsin

thecircumferential directionand41pointsinthe

wall-normaldirection.EachouterH-gridhas100

pointsintheaxialdirectionand71pointsinthe

transversedirection.Forthesakeof clarity,only

someof thegridpointsareshowninfigure2.

Thedependentvariablesareinitializedto

freestreamvaluesandtheequationsof motionare

thenintegratedtoconvergence, subjectothe

boundaryconditions.Theflowparametersthatare

specifiedarethepressureratioacrosstheturbine

airfoil(ratioof exitstaticpressuretoinlettotal

pressure),theinletMachnumber,andtheinletflow

angle.

Rotor

_ Direction of

Rotor Motion

Stator

Inner 0-(

Figure 2. Turbine geometry (at midspan

reference design) and computional grid used.

of

Optimization Problem Formulation

The goal of the redesign is to improve the unsteady

aerodynamic performance of the turbine by optimizing the shape of the stator vane (the rotor blade

geometry is kept the same). This is accomplished

by formulating an objective function that minimizes the unsteady amplitudes Pi on the stator

vane subject to the constraint that the tangential

force on the airfoil does not change from the reference design by more than 1%. The pressure ampli-

;1 !!
tude Pi is used as a measure of the unsteadiness in

the flow field and is defined as the difference

between the maximum and minimum pressures

occurring over a complete cycle at each point on

the airfoil surface. (For the stator vane, a cycle is a

rotor pitch; for the rotor blade, a cycle is a stator

pitch.) Thus, the pressure amplitude ,bi is defined

as:

17i = (Pi, max- Pi, min)cycle (l)

In the current redesign the goal is to improve

unsteady aerodynamic performance by eliminating

the shock. The presence of the unsteady shock in

the reference design results in large unsteady pressure amplitudes. Thus the pressure amplitudes are

directly related to the shock strength. Hence it is

assumed that a reduction in the unsteady amplitudes on the stator vane will result in a weakened

shock. The results obtained demonstrate the validity of this assumption.

Neural Net-Based Redesign Procedure

The redesign procedure used here is based on Rai

and Madavan (ref. 15). The procedure uses a

sequence of response surfaces based on both neural

networks and polynomial fits to traverse the design

space in search of the optimal solution. A technique called parameter-based partitioning of the

design space is used, where the functional dependence of the variables of interest (e.g., pressure)

with respect to some of the design parameters is

represented using neural networks, and the functional dependence with respect to the remaining

parameters is represented using polynomials. The

power of neural networks and the economy of loworder polynomials (in terms of number of simulations required and network training requirements)

are thus effectively combined. The method

(ref. 15) can be viewed as a variant of Response

Surface Methodology (ref. 20, 21), or RSM, where

the response surfaces are constructed using both

neural networks and polynomials. Traditional RSM

uses only low-order polynomials in constructing

the response surfaces.

The method of Rai and Madavan (refl 15) uses

polynomial approximations on multidimensional

simplexes. An s-dimensional simplex is a spatial

configuration of s dimensions determined by s+ 1

equispaced vertices, on a hypersphere of unit

radius, in a space of dimension equal to s. (By this

definition, a two-dimensional simplex is an equilateral triangle that is circumscribed by a unit circle.)

This approach assumes that the local variation of

the design objective function can be accurately represented using low-order polynomials, which is

very often the case. The polynomial fit on this simplex together with the trained neural network represents a composite response surface. The

optimization procedure then uses a sequence of

such composite response surfaces to traverse

through the design space in search of the optimal

solution.

Following Rai and Madavan (ref. 15), parameterbased partitioning of the design space is accomplished in the following manner. Since the variation of the unsteady pressure amplitudes along the

airfoil surfaces is typically far more complicated

than the variation with small changes in geometric

parameter values, a neural network is used only to

represent unsteady pressure amplitude variation in

physical space. The three-layer neural network

(with two hidden layers) shown in figure 3 is used

for this purpose. The first node in the input layer is

a bias node (input of 1.0). The second set of nodes

is used to specify the physical location. Since we

are dealing with two-dimensional geometries only,

physical location is specified by a single parameter: the axial location on the airfoil surface.

Figure 3 shows a third set of input nodes that are

not activated in this study, but may be used in cases

where the functional behavior of the pressure

amplitudes with some of the geometric parameters

is "complex" and one wishes to use the neural network to represent this behavior.

The variation of the unsteady pressure amplitudes

with the geometry parameters is approximated

using simple polynomials. Since a linear variation

is assumed, the points at which the pressure amplitude data are determined are located at the vertices

of a simplex of dimension equal to the number of

geometry parameters,
Imo_x kmax

mmax

"Complex" _

Geomet_"__ _

Parameters _ax

Physical_...p! /__

Location _ J/Output

/ /errO 

Hidden Layers

Figure 3. Schematic of the three-layer feedforward neural network used in this study.

The optimization strategy used here to redesign the

turbine airfoil geometry starting from the reference

design can be summarized as follows:

1. Populate the design space in the vicinity of the

reference geometry. The reference design geometry serves as the centroid of the first simplex in

the optimization process. A simplex in design

space is constructed around this centroid and

unsteady aerodynamic analyses (computational

fluid dynamics (CFD) simulations) at each of the

vertices (a linear variation is assumed) are

obtained.

2. Train the neural networks and compute the

polynomial coefficients to define the composite

response surface. The input nodes of the neural

nets will typically contain parameters that correspond to the physical location on the airfoil and

those geometric parameters that give rise to

"complex" surface pressure variations. The neural nets are trained and the polynomial coefficients that define the pressure variation within

the simplex are computed. The trained neural

networks in combination with the polynomial fit

then constitutes the composite response surface

(ref. 15).

3. Search the region of the design space represented by the composite response surface. A conjugate gradient method was used in this study to

perform this constrained search. Geometrical and

other constraints can be incorporated within this

search procedure easily. In addition, constraints

that limit the search procedure to the volume of

the simplex are also incorporated in the search.

4. Relocate the simplex. If the local optimum

obtained in the previous step lies on the boundaries of the simplex then this point is chosen as

the new centroid and steps 1-4 are repeated until

the search culminates inside the simplex. However, the process can be stopped at any time

when the design is deemed adequate.

5. Validate the design. As a final step in the process the unsteady aerodynamic analysis is carried

out for the geometry corresponding to the optimal design to determine the adequacy and quality of the design.

Implementation Details

The optimization procedure was initiated from the

reference design. The process focused on the suction surface of the stator vane. Although 13 geometric parameters were used to represent the stator

vane, only 5 of these parameters that were related

to defining the suction surface were considered. A

linear variation in the parameters was assumed,

resulting in a five-dimensional simplex (with six

vertices) at each design optimization step. The process of constructing new simplexes and searching

for the local optimum was repeated 3 times.

Each of the six 3-layer nets (representing the six

vertices of the simplex) had two input nodes, one

for the bias and one for the axial location, and one

output neuron. The first and second hidden layers

had 15 and 7 neurons, respectively, for a total of

136 connection weights. Thus, the total number of

connection weights for all six nets was 816. During

the training process the training error was reduced

by about four orders of magnitude from the initial

value. Further details regarding the training process

can be found in Rai and Madavan (ref. 14, 15).

/t 11
Results

The neural net-based redesign method was used to

optimize the unsteady performance of the reference

turbine. This optimization yielded a modified

design. Detailed comparisons with the reference

design are presented in this section.

Comparison of Stator Vane Geometry for

Reference and Modified Designs

Figure 4 compares the stator vane geometry for the

reference and modified designs. It is worth noting

that the geometry of the modified design obtained

at the end of the optimization process is very close

to that of the reference design. The suction surface

has been thinned out in the aft region, and the location of the point where the maximum thickness

occurs (the airfoil "crown") has moved slightly

downstream. Since the geometry modifications are

slight, the effect on flow angles and other mean

flow parameters is small. However, the impact on

the unsteady flow features through the turbine

stage is quite substantial, as the following results

will show.

Comparison of Flow Parameters for

Reference and Modified Designs

Table 1 compal:ed the flow parameters for the reference and modified designs. The differences

between the overall flow parameters in the two

cases are very small. This is to be expected, since

the geometry has been modified very slightly from

the reference design.

Static Pressure Variation on Airfoils

Figure 5 shows the time-averaged static pressure

variation on the stator vane. The reference pressure, Pt, ref, in this case is the total pressure at the

stator inlet. The static pressure is time-averaged

over a stator cycle which corresponds to the rotor

blades moving by a distance equal to that between

adjacent rotor blades (i.e., rotor pitch). The major

difference between the time-averaged pressures on

the reference and modified designs is along the

suction surface where the loadings are quite different. Also, the sharp pressure minimum toward the

trailing edge of the stator vane in the reference

design has been smoothed out in the modified

design.

u2.

I I I I I

tO

0

r_

o.

o,

o

"7. _ Modified Design

- -- - Reference Design

I I I I I

"7,

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.S

x/c

Figure 4. Comparison of the stator vane geometries

for the reference and modified designs.

Q

I I I !

_ d

0

Modified Design

- -- - Reference Design

_. I t I |

0

0.0 0.2 0.4 0.6 O.B 1.0

x/c

Figure 5. Comparison of time-averaged pressure

distributions on the stator vanes for the reference

and modified designs.
Thevariationof time-averagedpressureson the

rotor blades is compared for the reference and

modified designs in figure 6. The reference pressure, Pt, ref, in this case is the relative total pressure at the inlet to the rotor row, and the timeaveraging is performed over one rotor cycle which

corresponds to the rotor blades moving by a distance equal to that between the stator blades (i.e.,

stator pitch). Since the rotor blade geometry was

not modified, the difference in time-averaged pressures between the reference and modified designs

is quite small and is limited to the vicinity of the

leading edge of the blade. This small change is a

result of the flow field being altered by the modified stator vane.

I | I I

Q

,,.21_

d

0

-- Modifi

0

0.0 0.2 0.4 x/c 0.6 0.8 1.0

Figure 6. Comparison of the time-averaged

pressure distributions on the rotor blades for the

reference and modified designs.

Unsteady Pressure Amplitudes on Airfoils

A quantitative measure of the unsteadiness in the

flow can be obtained from the unsteady pressure

amplitudes on the surfaces of the stator and rotor

airfoils. The pressure amplitudes ,b are defined as

the difference between the maximum and minimum pressures occurring over a complete cycle at

each point on the airfoil surface (see eqn. 1.)

The pressure amplitudes on the stator vanes for the

reference and modified designs are shown in

figure 7. The abscissa on figure 7 is the axial distance x (normalized by the stator axial chord, c)

along the stator vane measured from the leading

edge (x/c = -1.0) along the suction surface to the

trailing edge (x/c = 0.0) and then back to the leading edge along the pressure surface (x/c = 1.0). It

is evident from the figure that the high unsteady

interaction effects in the reference design have

been reduced substantially in the modified design.

In particular, the maximum pressure amplitude that

is located at the trailing edge of the vane has been

reduced by about 30%. As noted earlier (ref. 6), the

large pressure amplitudes are caused by the presence of an unsteady moving shock in the gap

region. The reduced pressure amplitudes in the

modified design indicate that the strength of this

sh_k has been reduceddrastically and itsdetrimental effects have been mitigated.

0

tO

0

0

0

,:r5o

6

0

I I I

-- Modified Design

- -- - Reference Design

,

-1.0 -0.5 0.0 0.5 1.0

x/c

Figure 7. Comparison of the pressure amplitude

distributions on the stator vanes for the reference

and modified designs.

The pressure amplitudes on the rotor blades for the

reference and modified designs are shown in

figure 8. Unlike in figure 7, the abscissa on figure 8

is the axial distance x (normalized by the rotor

axial chord, c) along the rotor blade measured from

the trailing edge (x/c = -1.0) along the suction

surface to the leading edge (x/c = 0.0) and then

10

rq 1 ]:
backto thetrailingedgealongthepressuresurface

(x/c = 1.0).Althoughthetime-averagedpressure

ontherotorbladeishardlyaffectedbythestator

vanemodification,theunsteadypressuresonthe

rotorbladesareconsiderablyreducedin themodifieddesign.Attheleadingedge,thereductionis

againabout30%.

t5

0

9 °

..g

q

0

I I

Modified Design

- -- - Reference Design

I

-1.0 -0.5 0.0 0.5 1.0

X/C

Figure 8. Comparison of the pressure amplitude

distributions on the rotor blades for the reference

and modified designs.

It is important to note that the reduction of

unsteady effects in the modified design is due primarily to the weakening of the shock. The

unsteadiness due to potential flow interactions and

wake/blade interactions between the stator vanes

and rotor blades continues to be present since the

axial gap between the vanes and blades was not

changed in the optimization process.

Instantaneous Contours in the Flow

Figures 9 and 10 compare the instantaneous pressure contours in the flow for the reference and

modified designs, respectively. These contours

show the overall features of the time-averaged

pressure distributions on the vane and blade surfaces shown earlier. For example, there is very little

pressure variation on the forward half of the stator

vane in both designs, and most of the flow expansion occurs on the latter half of the vanes. The

major difference between the reference and modified designs is the unsteady shock in the gap

region. This shock can be seen clearly in the reference design, while the flow in the modified design

appears shock-free. In the reference design, the

shock lies on the vane surface and impinges upon

the rotor blades as they pass by the vanes. This

unsteady shock and its motion is one of the causes

of the large time variations in the vane and blade

surface pressures seen in figure 7 and figure 8,

respectively. This shock is entirely due to the interaction between the stator and rotor airfoils. The

slight change in the stator vane geometry on the

suction side of the modified design effectively

weakens the shock strength. It is important to note

that figure 9 and figure 10 represent different

instances in the rotor blade passing cycle. The time

instances correspond to the rotor position when the

instantaneous Mach number in the flow field is

maximum and was chosen to represent the worstcase scenario.

Instantaneous Mach number contours are shown in

figure 11 and figure 12 for the reference and modified designs, respectively. The maximum instantaneous Mach number is noted to be 1.33 in the

reference design and 1.13 in the modified design.

While pressure contours, in general, highlight only

the inviscid aspects of the flow, Mach number contours also highlight the viscous aspects, such as

boundary layers and wakes. The shock-wake interaction in the reference design can be clearly seen in

figure 11. Despite the unusually high turning

angles, the contours show no indication of boundary-layer separation.

]1
Figure 9. Instantaneous pressure contours in the

flow for the reference design.

Figure 10. Instantaneous pressure contours in the

flow for the modified design.

12

'1 '!I
\

Figure I 1. Instantaneous Mach number contours in

the flow for the reference design.

\

Figure 12. Instantaneous Mach number contours in

the flow for the modified design.

13
Computing Time Requirements References

The time required to compute the unsteady CFD

simulations represents almost all of the computing

time required. The time required to train the neural

nets and search the design space is negligible in

comparison. The redesign was accomplished in

three optimization steps, with seven (six vertices

plus the centroid) CFD simulations being required

at each step. Each CFD simulation required about 5

hours of single-processor CPU time on a Cray C90. The total computing time required for the redesign was thus about 100 hours.

Concluding Remarks

A recently developed method for aerodynamic

design that incorporates the advantages of both traditional RSM and neural networks has been

applied to the redesign of a transonic turbine to

improve its unsteady aerodynamic performance.

The redesign procedure employs a strategy called

parameter-based partitioning of the design space

and uses a sequence of response surfaces based on

1. Sieverding, C. H.: Experimental Data on Two

Transonic Turbine Blade Sections and Comparison with Various Theoretical Methods. In

Transonic Flows in Turbomachinery, VKI Lecture Series 59, Von Karman Institute, RhodeSt.-Genese, Belgium, 1973.

2. Ashworth, D. A.; LaGraff, J. E.; Schultz, D. L.;

and Grindod, K. J.: Unsteady Aerodynamic

and Heat Transfer Processes in a Transonic

Turbine Stage. Journal of Engineering for Gas

Turbines and Power, vol. i07, 1985, pp. 1022-

1030.

3. Guennette, G. R.; Epstein, A. H.; Giles, M. B.;

Haimer, R.; and Norton, R. J. G.: Fully Scaled

Transonic Turbine Rotor Heat Transfer Measurements. ASME Journal of Turbomachinery,

vol. 111, 1989, pp. 1-7.

4. Giles, M. B.: Stator/Rotor Interaction in a

Transonic Turbine. AIAA Journal of Propulsion and Power, vol. 6, 1990, pp. 621-633.

5. Abhari, R. S.; Guenette, G. R.; Epstein, A. H.;

and Giles, M. B.: Comparison of Timeboth neural networks and polynomials to traverse Resolved Turbine Rotor Blade Heat Transfer

the design space in search of the optimal solution. Measurements and Numerical Calculations.

This approach results in response surfaces that

have both the power of neural networks and the

economy of low-order polynomials (in terms of

number of simulations needed and network training requirements). By using high-fidelity, timeaccurate Navier-Stokes simulations to steer the

optimization process, the relevant physics of the

flow field is included at every stage of the redesign

process. The use of such unsteady simulations is

mandatory in the current study, since the moving

shock in the reference design could not be simulated accurately by any other means.

The application of this design method to a reference transonic turbine yielded a new design with a

slightly different geometry. Results shown in this

report indicate that the unsteady shock in the reference design has been eliminated in the modified

design. This leads to much lower unsteady pressure

amplitudes on the airfoil surfaces and hence

improved aerodynamic performance. These results

demonstrate the capabilities of the neural net-based

design method.

ASME Journal of Turbomachinery, vol. 114,

1992, pp. 818-827.

6. Rangwalla, A. A.; Madavan, N. K.; and

Johnson, P. D.: Application of an Unsteady

Navier-Stokes Solver to Transonic Turbine

Design. AIAA Journal of Propulsion and

Power, vol. 8, no. 5, 1992, pp. 1079-1086.

7. Rangwalla, A. A.: Unsteady Navier-Stokes

Computations for Advanced Transonic Turbine

Design. AIAA Paper No. 94-2835, 30th

AIAA/ASME/SAE/ASEE Joint Propulsion

Conference, Indianapolis, IN, June 1994.

8. Jennions, I. K.; and Adamczyk, J. J.: Evaluation of the Interaction Losses in a Transonic

Turbine HP Rotor/LP Vane Configuration.

ASME Journal of Turbomachinery, vol. 119,

1997, pp. 68-76.

9. Demeulenaere, A.; and Van den Braembussche, R.: Three-Dimensional Inverse

Method for Turbomachinery Blade Design.

ASME Journal of Turbomachinery, vol. 120,

1998, pp. 247-255.

14
10. Chattopadhyay, A; Pagaldipti, N.; and Chang,

K. T.: A Design Optimization Procedure for

Efficient Turbine Airfoil Designl Journal of

Computers and Mathematics with Applications, vol. 26, no. 4, 1993, pp. 21-31.

11. Talya, S. S.; Rajadas, J. N.; and Chattopadhyay, A.: Multidisciplinary Optimization of

Gas Turbine Blade Design. AIAA Paper

No. 98-4864, 7th AIAA/USAF/NASA/ISSMO

Symposium on Multidisciplinary Analysis and

Optimization, St. Louis, MO, September 1998.

12. Tong, S. S.; and Gregory, B. A.: Turbine Preliminary Design Using Artificial Intelligence

and Numerical Optimization Techniques.

ASME Paper No. 90-GT-148, 1990.

13. Shelton, M. L.; Gregory, B. A.; Lamson, S. H.;

Moses, H. L.; Doughty, R. L.; and Kiss, T.:

Optimization of a Transonic Turbine Airfoil

Using Artificial Intelligence, CFD, and Cascade Testing. ASME Paper No. 93-GT-161,

1993.

14. Rai, M. M.; and Madavan, N. K.: Application

of Artificial Neural Networks to the Design of "

Turbomachinery Airfoils. AIAA Paper No. 98-

1003, AIAA 36th Aerospace Sciences Meeting, Reno, NV, January 1998.

15. Rai, M. M., and Madavan, N. K.: Aerodynamic

Design Using Neural Networks. AIAA Paper

No. 98-4928, 7th AIAA/USAF/NASA/ISSMO

Symposium on Multidisciplinary Analysis and

Optimization, St. Louis, MO, September 1998.

16. Sanz, J. M.: Development of a Neural Network

Design System for Advanced Turbo-Engines.

Fourth U.S. National Congress on Computational Mechanics, San Francisco, CA, August

1997.

17. Goei, S.; and Hajela, E: Turbine Aerodynamic

Design Using Reinforcement Learning Based

Optimization. AIAA Paper No. 98-4774, 7th

AIAA/USAF/NASA/ISSMO Symposium on

Multidisciplinary Analysis and Optimization,

St. Louis, MO, September 1998.

18. Johnson, E D.; and Huber, E W.: GGGT Preliminary Design. United Technologies Pratt

and Whitney Memo Pass 57, West Palm

Beach, FL, 1990.

19. Rai, M. M.; and Madavan, N. K.: Multi-Airfoil

Navier-Stokes Simulations of Turbine RotorStator Interaction. ASME Journal of Turbomachinery, vol. 112, 1990, pp. 167-190.

20. Montgomery, D. C.: Design and Analysis of

Experiments. John Wiley and Sons, New York,

1997.

21. Myers, R. H.; and Montgomery, D. C.:

Response Surface Methodology--Process and

Product Optimization Using Designed Experiments. John Wiley and Sons, New York, 1995.

15
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1. AGENCY U_E ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

November 1998 Technical Memorandum

4. TITLE AND SUBTITLE i5. FUNDING NUMBERS

Neural Net-Based Redesign of Transonic Turbines for Improved

Unsteady Aerodynamic Performance

6. AUTHOR(S)

Nateri K. Madavan, Man Mohan Rai, and Frank W. Huber*

7. PERFORMINGORGANIZATIONNAME(S)ANDADDRESS(ES)

Ames Research Center

Moffett Field, CA 94035-1000

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

519-40-12

8. PERFORMING ORGANIZATION

REPORT NUMBER

A-9900398

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA/TM-1998-208754

11. SUPPLEMENTARY NOTES

Point of Contact: Author, Ames Research Center, MS T27A-1, Moffett Field, CA 94035-1000

(650) 604-4427

*Riverbend Design Services, Palm Beach Gardens, Florida.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified -- Unlimited

Subject Category 07 Distribution: Standard

Availability: NASA CASI (301) 621-0390

12b. DISTRIBUTION CODE

13. ABSTRACT (Maximum 200 words)

A recently developed neural net-based aerodynamic design procedure is used in the redesign of a

transonic turbine stage to improve its unsteady aerodynamic performance. The redesign procedure used

incorporates the advantages of both traditional response surface methodology (RSM) and neural networks

by employing a strategy called parameter-based partitioning of the design space. Starting from the reference

design, a sequence of response surfaces based on both neural networks and polynomial fits are constructed

to traverse the design space in search of an optimal solution that exhibits improved unsteady performance.

The procedure combines the power of neural networks and the economy of low-order polynomials (in terms

of number of simulations required and network training requirements). A time-accurate, two-dimensional,

Navier-Stokes solver is used to evaluate the various intermediate designs and provide inputs to the optimization procedure. The optimization procedure yields a modified design that improves the aerodynamic performance through small changes to the reference design geometry. The computed results demonstrate the

capabilities of the neural net-based design procedure, and also show the tremendous advantages that can be

gained by including high-fidelity unsteady simulations that capture the relevant flow physics in the design

optimization process.

14. SUBJECT TERMS

Turbine, Design optimization, Unsteady aerodynamics

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OF REPORT OF THIS PAGE

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OF ABSTRACT

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