Application of an Efficient Hybrid Scheme for ... - NTRS - NASA

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NASA Technical AIAA-90-0028

Memorandum

102428

Application of an Efficient Hybrid Scheme for Aeroelastic Analysis of Advanced Propellers

R. Srivastava Georgia Atlanta,

and N.L.

Institute Georgia

T.S.R. Reddy The University Toledo, Ohio

Sankar

of Technology

of Toledo

and D.L. Huff Lewis Research Center Cleveland, Ohio

Prepared for the 28th Aerospace Sciences Meeting .--_:L___:±_:-__; sponsored by the American !nstitute Reno, Nevada, January 8-11, 1990

of Aeronautics

and Astronautics

(NASA-TN-102_28) APPLICATION C)F AN EFFICI_T HYBRID SCHEME F'3R AF_ROELASTIC ANALYSIS _F AnVANCED PROPELLERS (NASA) 30

p

CSCL

N90-13355

OIA G3/O2

UncIas 0252021

_

Im

Application

of an Efficient

Hybrid

Scheme

for Aeroelastic

R. Srivastava* Georgia

and

Institute

N.L.

Analysis

of Advanced

Propellers

Sankar*

of Technology

Atlanta,

Georgia

T.S.R.Reddyt The

University Toledo,

of Toledo Ohio

and D.L. National

Hufftt

Aeronautics and Space Administration Lewis Research Center Cleveland,

Ohio

Abstract

An tions

efficient

three-dimensional

to analyze

advanced

explicitly

and

compared

to a fully

memory The and

calculated

on

performance

tNASA tfMember

power

coefficients

ratios

AIAA Resident AIAA.

showed

power

agreement

to centrifugal

*Member

directions scheme.

of elemental good

due

two

The

implicit

advance

showed the

other

propellers.

scheme

is applied

for solving

scheme

treats

spanwise

implicitly, This

leads

the

without

affecting

to a reduction

Euler

equa-

direction

semi

the

accuracy,

in computer

time

as and

requirement.

various

tribution

the

hybrid

with of the

and

and

aero

for two advanced good

coefficient

correlation and

experiment. advanced loading

steady

should

with pressure

A study propellers

propellers,

be included

and

SRTL,

experiment.

Spanwise

dis-

coefficient

differences

:.]se

of the effect showed

SR3

that

of structural structural

for better

fle.,dbi[i_y deformation

correlation.

AHS.

Research

Associate

at Lewis

Research

Center;

member

AIAA

and AHS.

Introduction

It has fered

been

known

by propellers.

number

However

increases

losses

(due

beyond

to wave

of commercial

propellers,

also known

the

upto

Mach

The

propfans

high

efficiency

is accomplished

the

and

are

designed

hence

must

strong

and

classical

bending

and blades

could

be obtained

through

pressure

techniques

the

propulsive

speed

advanced

efficiency

at

cruise

techniques.

extending

and

using

thinner

airfoils,

on the

low aspect

ratio

blades

dictates

twist

are

and

a large

used.

This,

high disk loading.

number

However,

the

to the fact

effect

these

numerical

flow details

near

tests

of blades special

problems

per

features

range

known.

techniques can easily

field. 2

the

hence

are

a

observed

a

loading. the

flow

to obtain

the

is required. At the

propfans

and

cheaper

at any

point

These

design

definitely

efficiency

be obtained

and

exists

propfan,

blade

a need

it is easier

also

in order

techniques.

propulsive

there

of blade

Also,

Therefore,

high

and

thin

As concluded

They

with

on the

or numerical

tips,

are

oscillations.

associated

distribution

the

blades.

for a wide

blades

of a propfan,

between

expensive.

these

amplitude

to be accurately

very

that

swept

tunnel

flutter

of potentially

and

thus

This

highly

wind

experimental

With

losses,

numbers.

or large

of pressure

are

development

distribution,

Mach

Mach

integrity.

also

or. cascade

have

prediction

numerical

cruise

cruise

to high blade

due

flutter

alleviate

an accurate

the

propulsive

higher

further

are

unstalled

loads,

support

high

is of-

compressibility

to improve

compressibility

backwards

leads

arises

[2], through

coupling

on the

experimental

as the

to high

designed

high

efficiency

problems.

They

- torsion

the

structural

classical

Kaza

To understand phenomena

number,

problems

flexible.

aerodynamic

blade

disk loading

to new

to transonic

by Mehmed

lead

Newly

a very

In addition

maintain

critical

moderately

succeptible

blade.

of high

lead

One of the

rapidly

is underway

to relatively

the

with high tip Mach

propfans

to delay

of a propeller

combined

of the

propulsive

off very numbers

aircraft. show

best

0.8 [1].

of the

which

the

drops

an effort

as propfans,

section

propeller,

that

tip Mach

military

by sweeping

requirement

now

efficiency

Currently

outboard

The

time

0.5, as high

drag).

_ efficiency

speeds

for some

stage

e:dsts

to

through to obtain in the

flow

The e._istingnumerical methodsvary in comple_ty from simple Goldstein type strip analysis to analysesthat solve the Euler and Navier - Stokesequations. The strip theory basedon Goldstein'swork [3], assumesthe flow to be inviscid and incompressible(henceirrotational). The propeller is modeledby a lifting line vortex and the wakeis assumedto be composedof a rigid helical vortex sheet. In this analysis the propeller is restricted to having straight blades and no provision can be made for the nacelle_sincethe vortex wakeextendsto the a_s. Sullivan [4] has improved on this method by using the curved lifting line conceptto accountfor the sweep.In this approachthe vortex wakeis representedby a finite number of vortex filamen:sin place of tile continuoussheetof vorticity as usedin Goldstein's approach. The analysis has beenfurther extendedin reference[51by placing the vortex filaments along the stream surfacesso that Hanson

[6] and

blade.

They

blade

pressure

is obtained

Williams

numerically

the appropriate three-dimen_ional

Jou

converged that not

lead

potential

as the

Euler

finite

solved

the full

volume

flow effects

Navier

reviewed

a Mach

number

is obtained methods

nacelle. to a pr0pfan

angle The

due

friction

adjusted

drag

the

so far

operates

for

for sweep

by determining

propeller

to th e

of lift coefficient

mentioned

are

and

kinetic

based

at or near

on

transonic

important. approach

present

equation.

for upwash

as a function

numbers

were

approach

tables

The

Mach

function

representation.

advanced

equation.

by tile potential

equations

The

may become

for free stream

[10] and

sional

been

full

drag

a:dsymmetric

load

and

induced

of the

equation

the

airfoil

shape

Kernel

integral

far wake.

to non-unique

Chausee

have

the

rotational

be modelled

at times,

applied

solutions

strong

The

However,

using

the

thickness

flow nonlinearities

[8] has

of propfans

camber,

of the

analyses.

tip Mach number,

a linear

dimensional

effects.

energy-per-unit-length

to the

by discretizing

tile twosection

conform

i7] applied

solve

distribution

from

linearized

they

of Jameson

formulation

was

greater near

than

the

In addition

'.9! for not 0.6.

leading the

the

able

analysis

to provide

It was concluded edge,

potential

which

could

flow equations

solutions.

Whitfield to the - Stokes

in reference

et aI. propfan

[11] have geometry.

equations I13], with

around regards 3

applied

the

Matsuo a propfan.

unsteady, et aI. Some

to performance

three

dimen-

[12] have

recently

of these

methods

prediction.

All the been

analyses

mentioned

for axisymmetric

axisymmetric. free

analysis climb

or cross in the

and

propfan

being

primary

unsteady should

be able

to solve

condition,

tory

motion

The

blade

by grid

forcing

to be cast

blade.

A Cartesian

forces

do not appear

order

governing

accurate

central for the

Alternating

Direction

geometry

be at least the

other

explicitly,

of the an order

temporal Implicit propfan

the

and

of magnitude

other

This two

directions

the

as gusts

could

result of the

of the aircraft,

a method

body any

to solve

solution

method

and

blade

fitted

time free

vibra-

grid

is used.

dependent

blade

response

It also allows

the

to simulate

only for a rotating

equations,

due governing

a rotating

as the blade

to

Coriolis

but

also for

using

second

motion. form

are

derivatives

discretized and

a first

a set of algebraic

is used

spanwise

allows

such

unsteady

yet be able

to obtain

larger

function,

not

scheme

the

unsteady

in non-axisymmetric

governing

for the spatial

(ADI)

a propfan

blade.

conservative

derivative,

to the

performance

The

the

arbitrary

is not

including

winds

The

forced

and

is true

in fully

differencing

two directions. and

This

dependent

equations

differencing

The

simplifies

system

A true

for the safety

allowing

for a flexible

grid

cross

a propfan.

of both

coordinates,

time

The

a versatile

motion,

in Cartesian

of attack

conditions

is to develop

around

objective,

function

flow

analysis.

forcing

calculation

explicitly.

undergoing

The

the

the

configurations,

fuselage.

research

dependent

is simulated

dependent

a blade

time

et al. [11] have

flow.

off design

flow field around

motion

any

equations

unsteady

permit

in the

flowfield

this

will

of the

in all flight

of the

present

To accomplish

This

time

wake

etc.

motion.

propfan

them

to predict

undergoing

nature

wake may be very critical

of the

the

is at an angle

in atmosphere.

to include

equations

nacelle

also encounter

to the

or fuselage

objective

Euler

may

disturbances

be possible

configuration,

the

of the

of Whitfield

in flight

axisymmetric

propfan

exposed

due to gusts

The

conditions

the

to the

the exception

a propfan

the analysis

The

due

it should

flight

permit

winds

propfan

destroys

descent.

For

for cruise

which

would and

flows.

Even

stream,

so far, with

load

to solve

the

distribution

direction

radial

direction

fluxes

without

The

equations.

permits

spanwise

upwind

equations.

algebraic

in the

implicitly,

order

the

grid

as compared to be treated

affecting

the

to to

semi

accuracy

significantly

as compared

explicitly

requires

to three

inversions

memory

requirement

given

time,

leading

scheme

more

for

include

structural

implicit

needs

scheme.

Treating

of block

tridiagonal

scheme, levels

to be only

per

time

of information

time

and

one

It also

needs

reduces

use of such

requirement

semi-

as opposed

to be stored

The

memory

direction

matrix,

step.

two dimensional.

in computer

The

of the

advanced

due

the effects

governing

by results

for future

present

to centrifugal

and

discussion.

aeroelastic

are

1) to apply

2) to calculate

of structural

equations and

paper

propellers,

deformation,

4) to study

first, followed

inversions

as only two time

objectives

to analyze

propellers.

implicit

the

at any a hybrid

makes

the

efficient.

scheme

be helpful

a fully

to reduction

specific

analysis,

two costly

one of which

scheme

The

only

to a fully

and

flmdbility

steady

methods

hybrid

performance,

3) to

nero loading

in the

on the performance

the numerical The

stead},

an efficient

solution

method

developed

here

of advanced are described are expected

to

research.

Formulation

Aerodynamic

Model:

The Euler equations,

in conservation

form and in Cartesian

+ where nonlinear partial

¢t is the

vector

flux vectors derivative

containing which

of the

+

vector.

In the

can be written

as:

(1)

+ (6), = o

conserved

are functions

coordinates,

flow of the

above

properties,

vector

_. The

t_, F and G subscripts

are

denote

the the

equation

[

P

p_

pu 2 + P

pu

\

pv

puv

pw

puw

+ P)

e

(2) 5

.

pv

pw

fluv

flu w

pv 2 + p

flvw

pw _ + P

flvw

_(e+ p)

v(e + p)

where

p is the fluid

velocity,

e is the

pressure

and

density,

total

may

u, v, w are

energy

the

of the fluid

be expressed

using

inertia]

Cartesian

per unit

the

volume

equation

1

components and

of state

p is the

of the

flow

hydrodynamic

for perfect

gas as:

2

(3)

p = (_ - 1)[_- _p(_ + _ + _)1

where that shock)

7 is the

ratio

of specific

it ensures

the

conservation

in the

In order equations

The

coordinates

with

the

The

advantage

of physical

of using

flux properties

the

across

conservation

form

discontinuities

need

flow past

to be transformed

of the

coordinates

an arbitrary

generalized

in the

and

system,

physical

geometry recast have

domain

=

(e.g.

undergoing

in a generalized the

following

of interest

arbitrary

motion,

coordinate

system.

one to one relationship

:

_(_:,y,z,t)

= _(_,y,z,t) 7"

These

is

flow [14]. to analyze

these

heats.

coordinates

can be rewritten

as:

are

=

non orthogona]

(4)

t

and

completely

general.

The

equation

(1)

q._ + E_ + F n + G_

(5)

= 0

where

pU

p pu q

_

E

pv

J--1

=

d--1

pw e

k

puU

+ _p

pvU

+ _vp

pwU

+ Gp

(e + V)V- _,p

J

(6) pW

pV

F

=

J--I

pull

+ _7=P

pvV

+ _yp

pwV

+ 7hp

_

etc.

are

Initial

the

and

A large tions.

W are the metrics

unique

conditions The

number

of problems

proper

intelligent For these

guess

problem.

and

pwW

+ Gp

J is the jacobian

of the

Hence

using

correct

may be critical initial

the

free

by the

of the boundary

as the

conditions

calculations

velocities,

can be described

application

is as important initial

+ GP

and

(x, _7_, (x

of transformation.

Conditions

to any given

pvW

(_ + v)w - _,v

contravariant

Boundary

It is the

+ _p

J--1

(_+ p)V - n,p U, V, and

puW

conditions stream

condition

proper

7

that

makes

the

meaningful

equasolution

boundary

equations.

to convergence

conditions

set of governing

and physically

governing

could

same

help

of the numerical

in achieving

are used

scheme.

convergence

as the initial

An

faster.

condition.

In the present after

analysis

the governing

boundary

conditions

The

and

blade

on a solid

nacelle

have been

need

to be addressed:

- surface

there

surface

be ignored

equations

nacelle

Physically,

surfaces

solved

boundary

go to zero.

Euler

The

equations.

can be mathematically

the

_

the

surface.

can

be given

is the velocity

The

velocity

vector

vector

same solid

the

acteristic

of the must

should

is operating free air.

propagate be

allowed

state

to infinity.

On

to escape

hence

boundary,

all quantities

are fixed

upstream

while

and

7/is

the

unit

(x, y, z) in the

should the

all characteristics

in a supersonic escape, pressure should

air,

For steady

e are fixed

inside

surface

in free

pu, pv, pw and

acteristics

on the blade

and

(7)

+ (v- y.)) +

variables

travel

condition

of no slip can

vector

blade

normal

to

fixed coordinate

- z,)k

(8)

:

propfan

as that surface

condition

the velocity

as

conditions

Since

hence

as:

any point

= Far-field

The following

= 0

at the

_,at

surface,

boundary

boundary

written

flow field.

explicitly

:

or on a solid

physical

Thus

are updated

for the interior

condition

_.,_

Where

at the boundaries

can be no flow through

must

for the

the flow variables

thus

to that flow.

the

is fixed escape,

at the

to that hence

far field

conditions

calculations the

subsonic

inflow

value.

of the

flee

stream,as

subsonic

outflow

of the

8

and

:he

be the from

remaining

For a supersonic

inflow

distur:_ances boundary,

flee stream.

For supe-sonic

are extrapolated

the

one char-

cannot four

p, pu, pv, pw are ext-apolated

all quantities

flow domain.

boundary,

p is extrapolated

quantities

should

all disturbances

flee stream

At the

four

the

charfrom

outflow,

f:om inside

the

The

block

interface

It is neither one

blade

efficient

passage

computation. solid

would

Across

that

the

nor practical

these

and

periodicity

properties

at K=2 an

and

Therefore,

in order

be solved.

This

block interior

points

the nodes (lb),

to obtain

have

been

refer

to the

the average

of flow quantities

Solution The

latest

the

N (which

boundaries

the

available

values

adjoining

the

the

fluid

of fluid

time

does

whole

not

propfan

exist. should

biock one time step, explicitly,

the

Referring

quantities

for

N and

block K=2

one

after

flow variables

blocks. the

K=I

of block given

(la),

as the average

by averaging

boundary

at any

flow

will require

in figure

updated

block)

at K=KMAX-1

condition,

An axisymmetric

boundaries

are

corresponding

is also

these

of each

the

on

flow field.

solution

from

except

Periodicity

a case,

is done

for

the from

to figure

at boundary N+I)

would

be

of block

N+I.

In

are used.

Procedure:

descretized

hybrid

scheme,

factorized as far

again

on

hence,

boundaries

boundary

solved.

for such

the

updated.This

of block

so the

the solution

The

are updated

periodicity

time,

be continuous,

As shown

for a symmetric

side of the boundary

subscripts

must

boundaries.

K=KMAX

by advancing

K=KMAX

doing

and

the

In this case

on each

(the

flow,

interface

same

additional

of the blade.

fluid properties.

K=KMAX-1

is done

at a time.

same

at the

introduces

all the variables

fluid

K=I

unsymmetric

This

on the type of flow being

have

at the boundaries

all the blades

downstream

on the

properties

For

boundaries

depends

two boundaries

to solve

at a time.

boundaries

boundaries,

require

:

is handled

boundaries

for these

boundary

and

as coding

opposed Time

to the

forms

of the governing

described solved

in next

using

the

is concerned. explicit

integration

equations

section. ADI

The

scheme.

However

described

earlier,

algebraic

equations

An implicit

method

it allows

larger

time

are solved are

a

approximately

is more steps

using

demanding

to be taken

as

schemes. is carried

out

using

the

first order

q_+1 = qn + ArCgq _+1 Or

accurate

Euter

implicit

rule

(9)

where are

the

superscript

known,

and

n denotes

n + 1 the

order

accurate

scheme,

small

time

is required

step

Substituting

the

the

next

current

time

level,

at which

or unknown

time

level.

Even

satisfactory

time

to maintain

Eu!er

equations

accuracy

flow variables

though

is obtained

numerical

the

this

because

is a first

a relatively

stability.

(5) in (9) we get

q_+_ = qn _ Ar (E_ + F, 7 + GC) _+_ The accurate The

treated

This

derivatives

central

differences.

hybrid

In order

The

partial

implicitly

method

the while

77 derivative

two implicit

the

Beam

the

rt direction,

computational the

only

directions.

and

Warming solving

after

using

the

standard

second

Equation

(10)

the

77 marching

the

available

and

latest inversions

Chausee Using

equations

[15] first this

order

= n,_ _ Ar direction

between

rt

this the

at a time. any

-

"_n+l

"-iS-l,k

2A_7

10

flow

tridiagonal hybrid solver

The

dependency

variables. matrix,

scheme marches

marching on the

in with

along

direction marching

as :

(E_ +1 + F_ ''_+1 + G_ +1)

is changed

Pi,j+_,k

used

((, _), are

semi-explicitly.

of the

block

technique

to remove

be rewritten

are treated values

of the

one 77 plane

in order

can then

in two directions

(7) flux terms

two costly

sweep,

flux terms

direction

using

Rizk

the

every

time,

algorithm.

q.+a Since

radial

is obtained

requires

the

direction.

are obtained

scheme:

to decrease

is reversed

E_, F,_, G¢

(I0)

every

iteration,

(11) the

F_ ''_+1 alternates

during the odd time steps,and Fn+l

n

i,j+l,k -

Fi,d-_,k

2A_

during the eventime steps. The tions

above

are

discretization

costly

nonlinearity

to solve

is removed

resulting

in the

leads since

to a set of algebraic

the

flux vectors

by linearising

following

linear

N and

the fluxes

equation

equations G are

about

the

for q.

highly

previous

These

equa-

nonlinear. time

The

step

value,

:

[I+Ar(a_A_+a