Development of an Aeroelastic Code Based on an Euler/Navier ...

g

NASA

Technical

Memorandum

Development of an Aeroelastic Code an Euler/Navier-Stokes Aerodynamic

Milind

A. Bakhle,

University Toledo,

Rakesh

of Toledo Ohio

George L. Stefko Lewis Research Center Cleveland,

J. Mark

Ohio

Janus

Mississippi

State

University

Mississippi

State,

Mississippi

November

1996

National Aeronautics and Space Administration

,,

L

107362

Srivastava,

and Theo

G. Keith,

Jr.

Based on Solver

DEVELOPMENT BASED

ON

AN

Milind

EUI_R

OF AN AEROELASTIC / NAVIER-STOKES

A. Bakhle,

CODE

AERODYNAMIC

Rakesh Srivastava, and University of Toledo Toledo, Ohio

Theo

SOLVER

G. Keith,

Jr.

J. Mark Janus Mississippi State University Mississippi State, Mississippi

George L. Stefko NASA Lewis Research Center Cleveland, Ohio

ABSTRACT This

paper

describes

the development

an Euler / Navier-Stokes relevant research in the briefly the

describes

described.

The

of the modeling

Euler

aeroelastic

/ Navier-Stokes extensions.

of the dynamics

code

(TURBO)

The

aeroelastic

of the blade

detailed, along with the grid deformation deformation of the blade. The work-per-cycle

approach approach

stability

used

The plans

is described.

paper

Representative

concludes

for further

code (TURBO-AE)

based

on

unsteady aerodynamic analysis. A brief review of the area of propulsion aeroelasticity is presented. The paper

the original

development

of an aeroelastic

with

results

an evaluation

development

of the

and validation

using

and then

formulation

a modal

the code

development

of the TURBO-AE

thus

is

approach

used to model used to evaluate

to verify

details is

the elastic aeroelastic

are presented. far,

and

some

code.

INTRODUCTION

NASA's

Advanced

Subsonic

Technology

(AST)

program

seeks

to

develop

technologiesto increase the fuelefficiencyof commercial aircraftengines, improve the safetyofengine operation,reduce the emissions,and reduce engine noise.With the development ofnew designs ofducted fans,compressors, and turbines to achieve these goals,a basic aeromechanical requirement isthat there should be no flutteror high resonant blade stresses in the operating regime. In order to verify the aeroelastic soundness

of the design, an

accurate prediction of the unsteady

aerodynamics complex

and structural

geometry,

modeling geometry, modeling

the

of the

dynamics

possibility

unsteady

of the propulsion

of shock

waves

aerodynamics

a

component

and flow

difficult

task.

separation The

new blade materials and new blade attachment of the structural dynamics a difficult problem.

Computational

aeroelastic

several simplifying assuming that the formulation

and

modeling

of fans,

assumptions. Flutter blade row is isolated.

the unsteady

compressors,

calculations This simplifies

aerodynamic

calculations

is required.

concepts,

of the unsteady

aerodynamics.

1990,

and

Kaza

et al., 1989].

The

major limitation

make

the

requires

are typically carried out the structural dynamics considerably.

In the past,

linear compressible small-disturbance potential theory unsteady aerodynamics and aeroelasticity of propfans

the blade

turbines,

For an isolated blade row flutter calculation, the modeling of aerodynamics is the biggest challenge. Many simplifying assumptions the modeling

makes

advanced

and

The

a panel

the unsteady are made in

method

based

on

has been used to model the in subsonic flow [Williams,

of this

analysis

is the neglect

of

transonic and viscous flow effects in the model. Although this analysis requires a fairly small computational effort, the inherent limitation in the model precludes its use in a majority

of practical

applications.

More recently,a fullpotentialunsteady aerodynamic

analysishas been used with a

modal structuraldynamics approach to model the aeroelasticbehavior of fan blades [Bakhle and Reddy, 1993 and Bakhle et al.,1993]. In this aeroelasticanalysis,the unsteady aerodynamics model is in the time domain. The time domain flutter calculationsuse the method of simultaneous integrationof structuraldynamics and aerodynamics

equations in time. Alternatively,for the frequency domain

calculations,a Fourier analysis isrequired to transform the time domain aerodynamic

flutter

unsteady

forcecoefficients to the frequency domain. The eigenvalue approach is

then used. Although the fullpotential aerodynamic formulation is able to model transonic effectsto a certainextent (limitedto weak shocks),the vorticaleffectsare stillneglected.Thus, for example, the blade tip vortex,or a leading-edge vortex is not

modeled.

Significant computational

effort is required

for such

flutter

calculationsbased on fullpotentialaerodynamics. An aeroelasticanalysis based on the Euler equations [Srivastava and Reddy,

1995] can model

vorticalflows, but

viscous effectsare not taken into account. Recently, other researchers [Hall,1992, He

and

Denton,

1993, Gerolymos

and

Vallet, 1994, Peitsch et al., 1994, and

Carstens, 1994] have also developed inviscidand viscous unsteady analyses forvibratingblades.

aerodynamic

For aeroelastic

problems

in which

flutter with flow separation, boundary-layer interaction), is required. (TURBO-AE).

viscous

effects

play

an important

role

(such

or stall flutter, and flutter in the presence of shock and a more advanced aeroelastic computational capability

This paper describes the development of such an aeroelastic code This aeroelastic code is based on an unsteady aerodynamic Euler /

Navier-Stokes

code

(TURBO),

developed

separately,

which

is

described

in

following section. In this paper, the grid deformation method is described, with the interpolation from the finite-element structural mesh to the CFD and the calculation of the aeroelastic forces and work.

DESCRIPTION

OF EULER

/ NAVIER-STOKES

This sectiondescribesvery brieflythe TURBO

CODE-

the

along mesh,

TURBO

code.Additional detailsregarding the

code are available elsewhere [Janus, 1989 and

Chen,

provides allthe unsteady aerodynamics to the TURBO-AE The TURBO

as

1991]. The

TURBO

code

code.

code was originallydeveloped [Janus, 1989] as an inviscidflow solver

for modeling the flow through multistage turbomachinery. It has the capabilityto handle multiple blade rows with even or uneven blade count, stationaryor rotating blade rows and blade rows at an angle of attack. Multiple blade passages are included in the calculation,when

required. Additional developments

were made

[Chen, 1991] to incorporate viscous terms into the model. The code can now be applied to model realisticturbomachinery configurationswith flow phenomena such as shocks, vortices,separated flow,secondary flows,and shock and boundary layer interactions. The

code is based on a finitevolume

scheme. Flux vector splittingis used to

evaluate the flux Jacobians on the lefthand side ofthe governing equations [Janus, 1989] and Roe's flux differencesplittingis used to form a higher-order TVD

(Total

Variation Diminsihing) scheme to evaluate the fluxes on the right hand side. Newton sub-iterationsare used at each time step to maintain higher accuracy. A Baldwin-Lomax

DEVELOPMENT The

TURBO-AE

dynamics to

be

algebraicturbulence model isused in the code.

OF AEROELASTIC code

of the blade. represented

in

assumes Thus, terms

CODE

a normal

the dynamic of in-vacuum

mode

- TURBO-AE representation

characteristics modes,

of each with

the

of the blade

$tructural

are assumed

associated

natural

frequency and generalized code such as NASTRAN

mass for each mode. Typically, a finite-element analysis is used to calculate the modal data mentioned. No

restriction is placed on the coordinates, modal deflections, mode of interest.

analysis, natural

except that it should provide grid point frequencies, and generalized mass for each

At the current stage of development, the aeroelastic in-phase blade vibrations. That is, the interblade zero.

This

is a limitation

of the

aerodynamics code (TURBO). passage are modeled. A work-per-cycle approach this approach, the motion

aeroelastic

Thus,

code

code (TURBO-AE) models only phase angle is restricted to be and not of the original

in TURBO-AE,

only one

blade

unsteady

and one

blade

is used to determine aeroelastic (flutter) stability. Using of the blade is prescribed to be a harmonic vibration in a

specified

in-vacuum

normal

mode

frequency

is typically

the natural

with

frequency

a

specified

frequency.

The

for the mode of interest,

vibration

but some

other

frequency can also be used. The aerodynamic forces acting on the vibrating blade and the work done by these forces on the vibrating blade during a cycle of vibration are calculated. blade flow,

If work

is dynamically unstable, leading to an increase

coupled

mode

flutter

In the following two cycle and generalized

Grid

is being

cannot

done

on the blade

by the aerodynamic

forces,

the

since it will result in extraction of energy from the in amplitude of oscillation of the blade. Note that

be modeled

with

this approach.

sub-sections, the grid deformation force calculations are detailed.

scheme

and the work-per-

Deformation

The blade ofa rotorundergoes a rigid-bodyrotationabout the axisof the rotorand a simultaneous elasticdeformation. This ismodeled through grid point motion. This requires the calculationof a new grid at each time step.The grid point motion due to blade rotationconsistssimply ofa rotationof each grid point about the axis of the rotor.There isno relativemotion between grid points.The grid point motion due to the elastic deformation of the blade is superposed on this rotation.The deformation

of the blade does cause relative motion

between

elastic

grid points and

therefore,the term grid deformation isused in thiscontext. The

grid deformation

approach

[Huff, 1989] used

in the TURBO-AE

code is

described here. In this approach, the grid pointson the surfaceof the blade move by

an amount

equal

boundaries

of the computational

in the

interior

to the deformation of the

of the blade.

domain

computational

dependent the blade moving

grid points

on the remaining

The

of the grid

motion

is determined

as a product

coordinate direction) in each computational

and the direction

points

of three motion of is linearly

on the distances of the interior grid point from the boundaries and from surfaces along that computational direction. Thus, points close to the

blade

stationary

do not move.

domain

coefficients (one for each computational points on blade surfaces. The coefficient

The

move

almost

boundaries

here, but can functions.

as much

as points

do not move

be found

in [Huff,

much 1989],

on the blade,

and points

at all. The

coefficients

where

are

they

close

to the

are not presented

referred

to as weighting

The grid deformation is calculatedas described here. First,new locationsof allgrid pointson the blade surfaceare calculated.For a harmonic vibrationof the blade in a selectedin-vacuum normal mode, the displacement of any point on the blade (due to elasticdeformation) X(x,y,z,t) can be written in terms of the generalized coordinate --D

-o

q(t)and the modal deflectionor mode shape function 8(x,y,z,t) as:

:_(x,y,z,t)

= q(t) _(x,y,z,t)

Note that x, y, and Further note that interpolated is described interest.

z represent the modal

the coordinates in a rotating coordinate system. deflection or mode shape function 8 has been

from the finite-element grid to the CFD grid. This interpolation, in the following paragraph, is performed only once for each

The interpolation

of the modal

grid is accomplished

as follows.

in a reference

position.

the

three

nearest

(1)

deflections at these scheme to calculate

At each

deflections

from the finite

The interpolation CFD

finite-element three nearest the interpolated

grid point grid

points

is done

element with

on the blade is

the blade surface,

calculated.

neighbors are used value of the modal

the CFD

undeflected

the distance

Then,

the

to

modal

in a bi-linear interpolation deflection at that CFD grid

point. The interpolated modal deflections 8 are stored and used calculate the motion of each grid point on the blade surface. Next, the new

grid onto

which mode of

at each time

step

to

locations of the interiorpoints are calculated.This requires the

calculationof coefficients based on the reference locationof the interiorgrid point. As

previously mentioned,

the distance of each

interior grid point, from

the

computational boundaries and from the blade surfaces,along the corresponding computational direction,iscalculated.For example, foran interiorgrid point (i,j, k),

the distance /-direction

to the nearest coordinate

blade surface (/-index /-direction coordinate computational

line. Then,

three

points

is simply

(inlet/exit)

the distance

boundary

is calculated

from the interior

Then,

computational a linear

these

distances

coordinate scaling

are used to calculate

directions.

of the motion

The

of the blade

along

the

point to the relevant

corresponding to leading/trailing edge) is calculated line. These two calculations are repeated for the

directions.

in the

i fficonstant

motion

along the other two

the coefficients

of the

interior

grid

surface.

Special attention is required for grid points located between the blade tip and the duct or casing.Typically,the grid pointsin thisregion are very closelyspaced in the radialdirection.Ifitisassumed

that the gridpointson the casing do not move when

the blade deforms, the computational cellscan become

excessively skewed

due to

the motion of the blade tipand the interiorgrid points.To avoid this,the grid points on the casing are allowed to move (along the casing surface)in such a way as to reduce the skewing of the interiorcomputational cells.The location of the grid points in the interiorregion is calculatedby linearinterpolationbetween the blade tip and the casing. This treatment improves the aspect ratioof the computational cellsconsiderably.

Work

and Force

Calculation

To determine aeroelasticstabilityusing the work-per-cycle approach, the blade motion isspecifiedto be harmonic:

q(t)

= qo sin(_t) (2)

where

qo is the amplitude

of motion

and m is the vibration

frequency.

The work-per-cycledone on the blade iscalculatedas:

(3) s or,

W =

pdA s

"Sqoo)cos(o_t)dt

(4)

In the blade f 8

above,

p = p(x,y,z,t)

vibration,

is the

A" is the

integral

vibration.

over

For along

expressions

for

The

with the

blade

the

viscous

calculation

presented

is the

unsteady

surface

area

blade

surface,

calculations,

the

the

pressure.

work

pressure

viscous

integral

over

stresses

is done

inviscid

blade

pointing/nto

_ is the

This

including

vector

on the

must

in the

and

surface

due

to

the

blade

surface,

one

cycle

of blade

be

included

in

the

TURBO-AE

code,

but

the

are

not

viscous

contributions

here.

work-per-cycle

unstable

if the

Finally,

the

Am,n

is an indicator work

done

generalized

-

on the

force

Pn dA

of aeroelastic blade

on the

during

blade

stability. a cycle

The

of blade

Am, n is defined

blade

is dynamically

vibration

is positive.

as:

° _m

(5)

8 where the

Pn is the

unsteady

m th modal

calculation,

deflection.

along

not

presented

the

flutter

pressure

with

here

for

stability

The the

generalized

simple

check

of the work-per-cycle

section,

derived

from

some the

established

in the

included

generalized

in

the

with

mode

can

just

leads

the

the

8m is force

contributions

be

one

and

generalized

viscous

force

an analysis

n th mode

used

to determine

mode,

motion.

are

the

blade

This

will

provides

a

selected

is

calculation.

component

has

tests

results

presented have

presented

the

code.

The

fan

The

inlet

flow

(axial)

grid

with

are

Engine

also

here

rotor

has

Mach

15 points

presented.

under

subsonic

recently

[Smith,

presented

are 32

meant blades

number on the

used blade

fan

NASA

necessary

in future

been

The

(E-cubed)

Engines

technologies effects

been

performance

results

Efficient

GE Aircraft

environmental

program

simple

are

However,

An, n for that

sample

Energy by

demonstrate

The

force

For

vibration

RESULTS

In this

reduce

term. The

blade.

if the

was

terms

pressure

simplicity.

of the

to blade

viscous

flutter

SAMPLE

due

to

turbofan 1993].

in this surface

Details and

calculation

state

the

1980's

efficiencies

to and

A summary

of the

design

and

1983].

of development

of 210.8 is 0.5.

program

in the

regarding

Hager,

the

diameter

in both

E-cubed

higher

engines.

to demonstrate a tip

The

sponsorship

achieve

[Sullivan

with

configuration

rotor.

cm

The

chordwise

(83

CFD and

of

inches). grid

is a

spanwise

directions.

It is assumed

dynamics

data

that

is for a grid with

the

224 points.

run of the code. The code has been To

begin,

a

steady

calculations

solution

are then surface.

are slightly extrapolation

different. In (rather than

Figure

It is to be noted

from the finite-element at these locations

is zero.

The

The results

run in the viscous

is obtained

performed.

on the blade

tip gap

for

Note

surface, surfaces. However, can CFD

that

whereas

data

the interpolated

data

grid and

interpolated disagreement

that

this

is one

and CFD

grids

from the two grids

there

is potential

for errors

deflections corresponding to the of the deflection at each grid grid is provided

on a mean

is for the CFD grid on the two distinct

deflections in the blade

of the areas

the finite-element

aeroelastic

where the two planforms do not match, is used to transfer the modal deflections

on the finite-element

Overall, the interpolated some differences are noted

be noted

planforms

grid to the CFD grid, and hence

the original

The

the finite-element

Figure 2 shows the original and interpolated modal first mode. The quantity plotted is the magnitude point.

are for an inviscid

configuration.

that the blade

the regions interpolation)

presented

structural

mode for other configurations.

this

1 shows

finite-element

blade

match the original deflections. tip region. Referring to Figure 1, it

in which

grid do not match.

the blade Figure

planforms

3 shows

modal deflections for the second mode. In this can be seen between the original and interpolated

from

the original

case, data.

the and

no significant

Aeroelastic calculationshave been performed for the firsttwo modes

separately.

These two modes have natural frequencies of about 72 Hz and 164 Hz respectively. Aeroelastic calculationswere performed for these modes at their natural vibration frequencies,using 100 computational time steps per cycleof blade vibration.Note that, the time step used in the two calculationsis not the same. In Figure 4, the instantaneous work isplottedagainst the time forblade vibrationin the firstmode. The amplitude of blade vibration used in this calculation results in a maximum deflection of the blade of about 0.9% of the blade tip diameter. eight cycles of blade vibration. The variation of instantaneous

The code was run for work with time shows

that

is seen

the

sinusoidal,

flow

has

indicating

become

periodic

the absence

in time. of significant

the absence of higher harmonic content). flowfield. Also, the results indicate that does not result

in any non-linear

The

variation

non-linear

This may the selected

effects

to be

nearly

(as evidenced

by

be attributed to the subsonic amplitude of blade vibration

effects.

In Figure 5, the variationofthe cumulative work, from the beginning of the current vibration cycle,is plotted.This quantity starts at zero at the beginning of each

vibration

cycle

of the

work-per-cycle.

This

After

eight

work

is being

Figure also the

it can

done/_y

of the

the

be

blade

that

blade,

the

and

the

at the

cumulative end

of each

work-per-cycle

this

shows

work cycle

remains

that

the

blade

done

is the

of vibration.

negative. is stable

Thus,

under

the

after

flow

the

each

cycle

on a slightly

to periodicity

work-per-cycle

of vibration.

different

in time.

does

not

scale.

For

vary

the

This

information

It is presented

configuration

much

afar

is

to show

analyzed,

the

fourth

it

cycle

of

vibration.

7 shows

of

blade,

the

generalized that

result

reached

the

variation

given

by

force

indicates

All blade

in work

being

the

are

are

maximum

periodic

significantly

to the

comparatively behavior

vibration

work-per-cycle requires

a periodic

10

negative

value.

vibration. smaller mode slightly,

in half

amplitude. Note

Hence,

the

amplitude. are

stable. even

after

that

The It

blade

eight

just

motion that

the

one

mode,

this

the

motion

will

of the

is seen

valid

for

tip

conclusion

work-per-cycle

be

cycles

noted of blade

that

that the

vibration.

in the

normalized

work-per-cycle

indicates

work

is seen

to be

This

is in

this

is expected

4) for

a

non-linear to be linear

the be

a

amplitude noted

that

the

approach

second

mode

of the only

one.

that

values

in

8 shows

(Figure

of

since

vibration

is not

mode

9, where

amplitude,

the

Figure

content).

It should

a linear

work-per-cycle

calculation

response

mode.

results

instantaneous

cause

in Figure

either

second

that

variation

first

for blade

seen

smaller

the

The

diameter).

not necessarily

be

the

harmonic

for the

The

the

diameter.

cycles,

higher

investigated. This

in

tip

deflection.

blade

the

should

The

seen

lagging

Although

obtained

(0.1%

is

can

be

vibration

blade

or five

by the

work-per-cycle It

a force

vibrating

time.

four

variation

amplitude.

the

may

with

of blade

0.2%

the

evidenced

response,

shows

(All).

is a verification

blade

of about

is being

approach

the

about

maximum

is reduced

smaller

after

larger

small

This

blade

It

analysis since

amplitude

against

linear

for a sufficiently

an

plotted

and

shown.

an

blade.

for

nearly

is unknown

on the

calculation.

blade

(as

also

stable,

on the

for

in time

force

For

are

presented

non-linear

contrast

Figure

done

work

is

motion.

vibrations

of the

instantaneous

becomes

(2),

the

performed

deflection

generalized

Equation

work-per-cycle

results

Calculations

of the

lags

the

from

Finally,

the

seen

5, although

that

Figure

not

the

of the

seen

be

cycle,

as a symbol

work-per-cycle

in Figure

convergence

can

end of the

analysis.

6 shows seen

At the

is represented

cycles,

conditions

blade.

the

are blade

work-per-cycle

converges by the partly

a

vibrations continues

to

for

the

a small

amplitude

of

result

of

the

in the

second

to

change

CONCLUDING

An

REMARKS

aeroelasticanalysis code named

TURBO-AE

has been developed. The starting

point forthe development was an Euler / Navier-Stokes unsteady aerodynamic code named

TURBO.

Routines

have

been

developed

to interpolate the structural

deflectionsfrom the finite-elementgrid to the CFD grid.Grid deformation routines have been developed to calculate a new grid for the deformed blade at each time step. Routines have been developed for the calculationof work forces. These configuration.

routines have

been verified by running

and generalized

the code for a realistic

Results have been presented to show the working ofthe code fortwo modes ofblade vibration.Dynamically, the blade is seen to be stable in both the firstand second modes. The results for the firstmode show that the force acting on the blade is linearat the selectedamplitude of vibration.The resultsfor the second mode show that a non-linear response is obtained for a fairlysmall amplitude of vibration. However, a reduction ofamplitude resultsin a linearresponse, as expected. The calculationsfor the second mode

illustratean advantage of the work-per-cycle

approach, namely, it remains valid even ifthe force is non-linear.However, the work-per-cycle approach suffersfrom an assumption that the aerodynamics and the structuraldynamics can be decoupled.

A major limitationof the code isthat itiscurrently restrictedto the analysis of inphase blade motions. In a propulsion component, fan, compressor, or turbine,itis necessary to consider the various interbladephase angles at which fluttercan occur. Hence, in the future,the code will be extended to allow the analysis of arbitrary interblade phase

angles. This can be accomplished either by including multiple

blade passages in the calculations,or by using a singleblade passage with time (or phase) shiftedboundary conditions.Also, it is necessary that the TURBO-AE

code

be exercised to evaluate itsabilityto analyze and predict flutterfor conditions in which viscous effectsare significant. This isalsoplanned forthe future.

ACKNOWLEDGEMENTS The

TURBO

Whitfield,

code was J.

Mark

Mississippi

State

and guided

by John

developed Janus,

University.

by a team

Jen-Ping

of researchers

Chen,

The development

F. Groeneweg

and Dennis

10

and

which

Timothy

of the TURBO

included

W.

Swafford,

code

L. Huff of the Propeller

was

David

L.

all

at

supported

and Acoustics

Technology

Branch,

by

Lawrence

J. Bober

and

Kestutis C. Civinskas of the

Turbomachinery Technology Branch, and by Anthony J. Strazisar and Eric R. McFarland of the Turbomachinery Flow Physics Branch, allat the NASA Lewis Research Center. The development of the aeroelasticTURBO-AE

code is also being

supported by the above individuals,and by Peter G. Batterton and John E. Rohde of the Advanced Center. The

Subsonic Technology (AST) Project Office at NASA authors would

like to gratefullyacknowledge

Lewis Research

the support provided.

This work would not have been possible without this support. The authors would also liketo thank John J. Adamczyk of the Lewis Research Academy, NASA Lewis Research Center forhis helpfulsuggestions and guidance.

REFERENCES Bakhle, M. A., and Reddy, T. S. R., 1993, "Unsteady Aerodynamics Propfans Using a Three-Dimensional 1633.

Bakhle, M.A.

Full-PotentialSolver",IAA

et al., 1993, "Unsteady

PotentialEquation", ALAA

and Flutter of

Aerodynamics

and

Paper AIAA-93-

Flutter Based

on the

Paper AIAA-93-2086.

Carstens,V., 1994, "Computation

of Unsteady

Transonic 3D-Flow

in Oscillating

Turbomachinery Bladings by an Euler Algorithm with Deforming Grids", Proceedings of the 7th International Symposium on Unsteady Aerodynamics and Aeroelasticityof Turbomachines, Fukuoka, Japan, September 25-29. Chen, J.P.,

1991,

"Unsteady

Solutions for Turbomachinery

Three-Dimensional

Thin-Layer

Navier-Stokes

in Transonic Flow", Ph.D. Dissertation,Mississippi

State University,Mississippi. Gerolymos,

G.A.,

Vibrating Hall,

Cascade

K.C.,

He, L., and

Denton,

I.,

1994,

'T'alidation

ASME

Paper

"Calculation

Using

Solutions

Vallet,

Aerodynamics",

1992,

Turbomachinery 0665.

Viscous

and

of

the Linearized

J.D.,

1993,

for Unsteady

Three-Dimensional Harmonic

Around

GT-92.

11

3D

Euler

Methods

for

94-GT-294.

Euler

"Three-Dimensional Flows

of

Vibrating

Unsteady Equations",

Time-Marching Blades",

AIAA

Flows

in

Paper

92-

Inviscid ASME

Paper

and 93-

Huff, D.L., 1989, "Numerical Analysis Sections", AIAA Paper AIAA-89-0437. Janus, J.M., Dissertation,

1989, "Advanced Mississippi State

of

Flow

3-D CFD Algorithm University, Mississippi.

Through

for

Kaza, K. R. V. et al., 1989, "Analytical Flutter Investigation Model", Journal of Aircraft, Vol. 26, No. 8, pp. 772-780. Peitsch,

D., Gallus,

Turbomachinery Unsteady September

H. E., and Weber, Bladings",

Aerodynamics 25-29.

S., 1994,

Proceedings and

Smith, L., 1993, "NASA/GE ASME Paper 93-GT-315.

of the

Aeroelasticity

Fan

"Prediction

and

7th

Oscillating

Turbomachinery",

Ph.D.

of a Composite

Propfan

of Unsteady

International

of Turbomachines,

Compressor

Cascade

Research

2D Flow in

Symposium Fukuoka,

on

Japan,

Accomplishments",

Srivastava, R. and Reddy, T. S. R. , 1995, "AeroelasticAnalysis of Ducted Rotors", presented at the Exposition. Sullivan,

1995

T. J. and Hager,

of the General

Electric

International Mechanical

R. D.,

/ NASA

1983,

E-cubed

Williams, M. H., 1990, "An Unsteady Propellers",NASA CR-4302.

Engineering

"The Aerodynamic Fan", AIAA Paper

Design

and

and Performance

AIAA-83-1160.

Lifting Surface Method

12

Congress

for Single Rotation

Figure

Figure

1" Typical

2: Original

finite-element

and CFD grids,

and superimposed

and interpolated modal deflections for first mode; data is for two blade surfaces.

13

planforms.

interpolated

Figure

3: Original

and interpolated

interpolated

3

modal

deflections

data is for two blade

I

I

I

I

2

4

6

I 8

for second

mode;

surfaces.

2

i'

i° -I

-2

-3 0

I0

Time / vibration period

Figure

4: Instantaneous

work

done on blade

14

vibrating

in first mode.

6O

30

o

-30

0

2

4

6

8

10

Time / vibration pedod

Figure 5: Cumulative firstmode.

work

(from beginning

Symbols

4O

of each cycle) done on blade vibrating in

indicate the value at the end of each cycle.

I

I

I

I

I 2

I 4

I 6

I 8

2O

I0

-I0

.2O 0

Vibration

Figure 6: Work-per-cycle

10

Cycle

for blade vibrating in firstmode.

15

300

/

I

k

200

I

I

I

oo AAAAA i, o _, IY.. ,.. ,.. '..yWw-V_...V

_-,oo -200

/

i

i (2rOll_),o;

0

2

4

I

-300 6

Time / vibration

Figure

7: Generalized

8

10

period

force acting on blade vibrating in firstmode.

"

AA

[o vv --

-I

I I 4

-2 0

2

I 6

I 8

10

Time I vibration period

Figure 8: Instantaneous maximum

work blade

done on blade vibrating in second mode deflection

0.2% blade

16

tip diameter.

with

0.I0

-'-f_l_r_T_

0.O5

i i --

-0.05

-0.10 0

2

4

6

8

10

Time / vibration period

Figure 9: Instantaneous maximum

work

done on blade vibrating in second mode

blade deflection 0.1% blade tip diameter.

2.O

I

I

I

I

I

I

4

6

8

l.O O

0.0 -

-

-l.o

I 0

2

l0

Vibration Cycle

Figure

10: Work-per-cycle

for blade

17

vibrating

in second

mode.

with

REPORT Public

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Srivastava,

Theo

G. Keith,

Jr, George

L. Stefko,

and

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NOTES

Milind

A. Bakhle,

NASA

Lewis

Rakesh

Research

Responsible

Srivastava, Center,

person,

Unclassified

and Theo

Cleveland,

George

1211. DISTRIBUTION/AVAILABILITY

Subject

DC

COVERED

on an Euler/Navier-Stokes

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

sippi.

sources,

WU-538-06-14

Milind A.Bakhle, J. Mark Janus

11.

data

Solver

6. AUTHOR(S)

Lewis

existing

5. FUNDING NUMBERS

of an Aeroelastic

Aerodynamic

searching

burden estimate or any other aspect of this Operations and Reports, 1215 Jefferson

Technical

4. TITLE AND SUBTITLE Development

instructions,

regarding this for Information

Ohio;

L. Stefko,

G. Keith,

Jr., University

J. Mark

Janus,

organization

code

of Toledo,

Toledo,

Mississippi

State

5230,

433-3920.

(216)

Ohio;

University,

STATEMENT

George

L. Stefko,

Mississippi

State,

Missis-

12b. DISTRIBUTION CODE

- Unlimited

Category

This publication

39 is available from the NASA Center for AeroSpace

Information, (301) 621-0390.

13. ABSTRACT (Maximum 200 words)

This

paper

describes

the development

aerodynamic

analysis.

paper

describes

briefly

aeroelastic approach

A brief

the original

extensions. is detailed,

work-per-cycle

The along

approach

are presented. The ment and validation

review

of an aeroelastic of the relevant Euler/Navier-Stokes

aeroelastic with

used

code

formulation

the grid

deformation

to evaluate

aeroelastic

(TURBO-AE)

research code is described.

paper concludes with an evaluation of the TURBO-AE code.

approach stability

in the

area

(TURBO)

based

on an Euler/Navier-Stokes

of propulsion

aeroelasticity

and then details

The

modeling

used

to model

is described.

of the development

Representative thus

results plans

used

The

of the

of the blade

deformation

far, and some

using

a modal

of the blade.

The

to verify

the code

for further

develop-

15. NUMBER OF PAGES

14. SUBJECT TERMS Aeroelasticity;

the development

of the dynamics the elastic

unsteady

is presented.

Flutter;

Viscous;

Inviscid;

19

Navier-Stokes

16. PRICE CODE A03 17. SECURITY CLASSIFICATION OF REPORT Unclassified NSN 7540-01-280-5500

18. SECURITY CLASSIFICATION OF THIS PAGE Unclassified

19. SECURITY CLASSIFICATION OF ABSTRACT

20. LIMITATION OF ABSTRACT

Unclassified Standard

Form

Prescribed 298-102

by

ANSI

298

(Rev. Std.

Z39-18

2-89)