Theory of airfoil response in a gusty atmosphere. Part II. Response to discrete gusts of continuous turbulence

·

,

THEORY OF AIRFOIL

RESPO~~S

E IN A GUSTY ATMOSPHERE

PART II - RESPONSE TO DISCREl'E GUSTS

uR

CONTINUOUS 'I'URBLJ'LENCE

b~

L. '1'

0

I'

ilotas

ti

THEORY

OF

·

AIRFO

IL

RESPONSE IN A GUSTY ATMOSPHERE

PART 11

-

RESPONSE

TO

DISCREI'E GûSTS OR CONTINUOUS TURBULENCE

by

L. T. Fi10tas

Manuscrip~

received

October,

1969.

ACKNOWLEDG EMENT

This investigation was suggested by Prof. H. S. Ribner; his

super-vision

and helpful

comments are gratefully

ack~owledged.

I am also indebted

to

Mr.

John

Leung for his consideraQle assistance with the numerical

compu-tations.

Financial

support was received from the National Research Council of

C

a

nad

a

under Grant No.

A2003 and by the Air Force Office of Scientific Research,

Office

of Aerospace Research, United States Air Force, under GrantNo.

67-o672A.

SUMMARY

The

response

of an infinite

span

airfoil in an arbitrary two-dimensional

upwash field is

studied analytically on the basis of the

'aerodynamic

transfer

function'

giving the pressure distribution in an inclined sinusoidal gust derived

in

Part I of this report.

When the upwash field ean be completely specified

(discrete gust case) the lift

response

is expressed

as

a quadrature

i~volving

the transfer function

.

When

only statistical

properties of the upwash can

ge

specified

(atmospberic turbulence) mean

square

val~es,

correlation functions

and power spectra

ane

similarly

expressed. Certain results are given in closed

form.

Expressions

giving

;

the the transfer function in terms of correlations

which may

be determined experimentally

are

given.

Previously published

experi-mental

data are compared with analytic results based on Part I. The detailed

expressions

deal

with

the lift response; however, the modification required

for

TABLE OF CONTENTS

NOTAT

ION

1

INTRODU

C

TION

RESPONSE TO

DISCRETE GUSTS

2

RESPONS

E

TO .cONTINUOUS TURBULENCE (Correlation Functions)

4

~R

A

NSF

ER FUNCTION

IN

TERMS OF EXPERIMENTALLY DEl'ERMINABLE

6

QUANT

ITIES

ONE-

DlMENSIONAL LIFT SPECTRUM

6

ANAI,.YTIC RESULTS FOR THE

EFFECTIVE

SEARS FUNCTION

7

SOME EXPLICI

T RESULTS

:

VERY

NARROW LIFTING STRIP

8

MEAN

SQUARE LIFT COEFFICIENT

9

T

HER

S

TA

+

I

~

TICAL

PROPERTIES

10

NJTE

ON

PI

TC

HING

MQMENT

11

CONCI..lJl~ING

R

EMA..."RKS

12

-

Ef~ER

EN

C

E

S

13

b

.

CL

CM

f

l

f

2

h

(

i,y)

NQI'ATION

airfoil semi-chord

lift

coefficient

time derivative

of CL

=

d

CL/dt

pitching moment

coefficie~t

typ

i

cal frequency

(eqn.

42)

"

"

(eqn.

43)

indicial response (eqn

.

10)

chordwise wave

number, non-dimensionalized by b

spanwise

"

"

"

"

"

b

L

turbulence integral scale, non-dimensionalized by

b

R

LL

li

ft

coefficient auto-correlation

(eqn.

11)

Rww

upwash

auto-correlation

(eqn.

13)

R~W

l

i

f

t-upwash cross correlation (eqn.

17)

S(k

l

}

Sears function

(Ref.

15)

Se(k

l

)

e

ff

ective Sears function (eqn

.

27)

T

(

k

l

,

k

2

)

l

ift t

r

ansfer function

(eqns.

5

and

6)

T

M

(k

l

,

k

2

)

m

o

m

ent

transfer

function

(eqn. 45)

u

x,

y

5

E

CP

LL

f

lig

h

t ve

l

ocity

gust upw

a

sh field

Four

i

er transfer

of

w

(eqn.

2)

s

pac

e

co-ordinates

at

rest relative

to

atmosphere,

non-dimen~ionalized

by b.

i

aspect ratio

of lift

sensiR~

element

nume

r

ical constant

=

1.2

Y\M

upwash power spectrum (eqn. 21

)

CP

L

W

lift up-wash cross spectrum (eqn. 21

)

cp

L

CPw

one-dimensiona1 lift spec

t

rum

(eqn

.

22

)

INTRODUCTIO

N

The

con~lnuing

evolutionary trend

towa~d

larger and more flexible

ai

r

craft

is reflected

in a corresponding need for more refined analytical

tech-niques.

In this respect, d.ynamic response

charact.eristics associated with

flight

through gusty air

are particularly significant: problems associated with more

numerous

anel

easily excitable vibraticnal modes are aggravated

by

requirements

for

sU3tained

f

l

ight

t::'1

!'

ough

regions

of severe gustiness.

Calculation

of the response to random inputs, such as atomospheric

turbulence,

becm~.es

feasible

through use of power spect

r

al

meth

ods

(generalized

harmonie analysis)0

Tllese methods circumvent direct

treatm

ent

of random time

funct

ions

by relating

the statistics of a linear syste

m'

s response to those of

the excitation tffi-ough

a

non

-sta

tistical

'transfe

r

fu..11.ctiOl'l'.

The aerodynamic

aspect of

flight

through 8usts

-

discrete or

conti:nuous

-

is contained entirely

in

this

transfer function (re

lating the instantaneous

respons

e

to the downwash

in

a single spectral

compo:qent of the gust)

.

The fou.."1dations

fOT

the spectral approach to the prediction of aircraft

gust

res

po

nse

were

laid by

Lieprüann

1,2 and elaborated by Ribner 3: the subject

has

rece

iv

ed contirlU

al attention ever since

o

Reviews and

recent

developments

may be found in References

4

to

8

and

their

bibliographies.

A

funda:.'llEmta

l aspect of

the

analysis is linear superpositicm of

(weighted) responses

to all frequencies.

Cer"cain questions may however be raised

concerning the applicability

of usual aerodynamic methods at

very

high frequencies

-

where,

for

instance, a tÜne lag in fulfilJ.ment of

the Ktltta

condition is

pos

si

ble9o

When the turbulence scale

is relatively

large,

these

considerations

are

usuall

y

unim;>ortant,

"

as

ev~d8nced

by

excellent agreement of certain

calcu-lationE

with I'light test

datA.

,

.

Nevertheless, detailed experimental

yerifi

-cation

of the

th

eory

would be

very desirable; pa.rticularly if

h;

is

recognized

that

shor

t

take

-

off/landing

aircraft operating at low speeds and altitudes may

encoUt'1-cer tlu

'1m

lence wi th

au

o:::-der

of

roagni

tude sffialle

r

scale

(and

greater

intensity)

t!lan

is gene;::

'

ally considered t;ypicaL

The possibility of using

aeroelastically

scale~

mode

l

s in

specially fitted wind tunnels

for

determination

of frequency re.3ponse

provides

further

incentive for detailed comparison of

line

arized theory

and

measureme~t.

Initial

expe~iments

using a simple configuration made by Lamson

lO

and

Hakkine:q and I\ichardsonll

were somewhat inconclusiveo Available instrumentation

prevented

accure.te determination

of

~'Urbulence

properti

es;

accurate theoretical

estimates

for the

aerodynamic transfer

function

were

lacking.

A renewed attempt

to' put the power

spectral

technique

on

solid

experi-mental fcoting has been

tmdertaken at the University of Toronto

5

Institute for

Aerospace

S

tudies

.

In

Part I of this report12

a

suitable transfer function

was

derived;

mOTe

refined

experiments are currently underway13.

The

present

work

concerns the

lift

response

of an infinite span

airfoil:

the main motivation was

to explicitly

link

up the transfer function with

the experiments

.

Toward

this

end, an

eÀ~ression

giving the

response

to an arbitrary discrete upwash field is

first

formulat~d;

this

is

used to generate relations suitable for study

of

the

response

to

continuous

turbule~ce.

The transfer function is then

exp

ressed

in

terms of

correlations

which are anticipated from the experimentso

.

10 11

The

nrevious

experlmental

results

3

are also compared with the

l

i

ft response, extension to any ot her linear response is straightforward

:

in

p

ar

ti

c

ular

,

a note giving the explicit modification required for calculation of

pitc

h

i

n

g moment response i

s

included.

RES

P

O

NS

E 'RO DISCREI'E

G

UST

S

.ó,

s

note

d

in the int

r

o

d

uc

t

ion

, t

he aerodynamic aspect of flight through

g

u

s

t

s is entir

el

y

con~ained

i

n

the 't

r

a

n

sfer function' relating instantaneous

r

e

s

po

nse

t

o u

pwa

s

h in an elementary spectral (Fourier) component of the gust.

A

l

t

ernat

e

ly~

th

e 'influe

nc

e functio

n

' expressing the response to a unit upwash

impulse

m

a

y b

e

cons

ide

r

ed as basic

.

(The two equivalent approaches a

r

e compared

in R

e

f.

7).

Co

nsi

de

r

an airfoil of infinite span flying through an atmosphere which

i

s initially in

a

dis

turbed state. Since the air-foil responds only to the

verti-c

a

l

c

om

p

onen

t

of v

eloc

i

t

y

in its own plane (li

n

earized the

o

ry is implicit

through-o

ut

), the d

i

s

turba

n

ce may be regarded as a two-dimen

s

ional upwash pattern w(x,y)

in the airf

o

i

l pl

an

e (Fig.l)

.

We

b

egi

n

b

y

deriving the fo

r

mal solution for the lift when w(x,y) is

co

m

p

let

el

y s

pecif

i

ed (discrete gust)

ru~d

the transfer function (or the influence

f

un

c

t

i

o

n)

i

s

fu~own.

In the following section a random pattern that is specified

only statis

tically (continuous turbulence) will be taken up

.

Let us initially suppose that the Fourier integral representation

( 1)

-co

is va

l

id.

(

A s

uf

f

icient condition would be, for example, if w were absolutely

int

e

grabl

e o

v

e

r

t

h

e plane). Application of the inverse transformation to (1)

giv

e

s

00

+

k

y)

2

dx

dy

(2)

1

:::

- - -

2

(

2

7T)

J

'

J

w(x,y)

-00

E

q

u

a

t

ion

(1

)

e

x

p

r

esse

s

the arbitrary field as the superposition of elementary

si

nu

soid

al c

omp

onents of the type (Fig.

2

,

af ter Ref

.3

)

w

(x,y)

e

wh

e

r

e th

e

amp

l

itude is given b

y

A W

"

w

·i(k

x

+

k

y)

1

2

e

(4)

B

ut

f

o

r

a

n

elementary component of the form (

3

), the lift coefficient on a strip

o

f wid

th

dy

centered on the point (x,y) may be written in the form

27T

w

(x,y)

e

u

An

analytical expression12

for the 'transfer function' T (generalized here to

apply

to

both positive

and

negative

arguments)

is

given by

-iksgnkl[sin~

-

]

e

₍₆₎

[1

+

nk(l

+

sin2~

+

~

k

cos~)]

1/2

where

k

Since

responses

may be superposed the

lift

coefficient due to an

arbi-trary pattern of

upwash is formally given by

- 00

A similar formalism

for

the

pressure

di

s

tribution

may

be readily obtained in

the

same way

from

the result

s

of

Ref.

12.

We shall

also

req-y.ir'e the total

response of a

finite

wing segment.

Let

CL5

be the total

lift

coefficient carried by

a

strip of

w

idth

2b5 (Fig.

2

). Thus

if the strip

is centered

on y

=

y

(x,yj

₌₌

1

₂₅

CL(x,y) dy

Using equatioll

(7),

thi

s

may be

wri

tten

as

00

C

L5

(x,y)

==

J J

h(x

-s

,

y-T])

w(s,T])

dsdT]

- 0 0

where

00

h(x,y)

==

~~U

J

J

_{T(kl ,k2)}

- 0 0

(

8)

(10)

The

function

h

may

be interpreted as the response of the strip to a unit impulse

of upwash located at

(x,y) (i.e. the 'influence function').

In

principle,

equation

(:7)

completely determines the lift history of

the

airfoil

as it

passes through some specified gust (a limited patch of uniform

downwash, for examp

le): in prac4ice, the poor convergence of the integrand

severley restr

icts this

application

-

whether

numerical or

analytical integration

methods are

contemplated.

Nevertheless, the

above

expressions form the basis

for

useful

results

applying

to flight through continuous turbulence: this

application

is

eonsidered next.

It

might

be noted, in passing, that expressions pertaining to lift on

a finite segment,

obtained

from

equation

(8),

can be interpreted as resulting

from application

of

strip

theory to a firüte span

wing

of aspect ratio 5.

F

o

r

e

x

ample, the

r

esult

for isotropie

turbulence will (in contrast to procedures

some-times labelled

's

trip

theory'

that assume uniform instantaneous downwash along the

spa

n)

be the

s

trip

theory

res

ult

without restrietion

on the ratio of turbulence

s

eale to

span.

RESP

O

NSE

TO CONTI

N

UOUS TU§BULENCE (Correlation Funetions)

If

the

upwash field

encountered by the

wing

is an indefinitely extended

field of turbulence

the

relations

derived in the previous sections are

inapplica-b

le

i

n

their p

r

e

s

ent form*.

If, however, the turbulent field is

assumed

to be a

sta

tiona

r

y

r

andom

tunetio~

of position in the

x-y

plane, the theory of generalized

harmonie analysis

l

shows

that relations

~hat

can be derived for statistical

pro-pe

r

tie

s

remain

valid

.

For

e

xampl

e,

the

correlation of the lift coefficients carried by two

identical

strips

of

wi

ng at different spapwise locations (Fig

.2

)

may

be defined

as

R

=

LL

CL

₅

(x,Y) CL

₅

(x + 6

x, y +

Ay)

(11)

Here the overbar signifies an ensemble

ave

ra

ge.

Up

o

n..

inserting

(

9

)

there results

00

(12)

- ( ) ( )

w

h

ere

-x=x+&

y=y+6y

If the

turbu

l

ence

is

homogeneous, the velocity correlation can be

expressed

in form

of

a Fourier

transform. That is

-()()

(13)

Inserting (13

)

into (12) and

performing

some

elementary manipulations

leads

to

*

A valid formulati

on

would

involve writing the integrals in the

Stieltjes form;

00

i

(kl~x

+

k

2

6y)

E

LL

11

dkl dk2 CP

~w

(kl,k2) e

-00 00

[kl(x- s )

+

k2(Y-~)

)

{ 11

-i

}

X

h(x-s, y-Tj) e

d(x-S)d(y-~)

- 00

x

{IJ

h(x-"y-~)

e

i[k1

(x-,)

+

k2(Y-~)]

d(x-,)

d<Y-~)

}

-00

(14)

But

equation

(10)

is in

the form of

a

two-dimensional Fourier transform; taking

the

in

vel'

se

co

11

h(x,y

)

-co

21T

U

(15)

U

si

ng thi

s

relation, equation

(14)

becomes

~L

(16)

It

is

apparent

that

_RLL

is a

function

of the

separatiouP

~ a

~

d

6y and not of

the

location

or

ori

eLltation of the elements. From its

definition

(11),

i t is

also

apparen

t that the

mean square

lift coefficient is oQtained from (16)

with

6x

=

6

y

=

O

.

Expressions involving correlations

between other quantities

may

be

ob

t

ai

ne

d i

n

a

s

imilar way.

For the present purposes,

we

shall

require

the c

r

oss

cor

:re

la

tion of lift

coefficient

wi

th vertical

velocity,

defined as

Using

(9)

and (10),

as

00

21T

11

R

LW

U

CPww

-00

CL

(x +6x, y +

6y) w

(x, y)

6

in

the derivation of (16), there results the relat

i

on

i(k

l

&

+

k

2

l::,y)

(k

l

,k

2

)

sink

2

6

T(k

l

,k

2

)

(18)

k

2

6

e

dk

l

dk

2

For a given

type of turbulence the power spectral density would be

:

known;

sLnce

the transfer

function

is also

k~own

(6',

the correlations (16)

a

nd (18) are

completely determinate. Alternately, if the correlations

were

determined experimentally,

the equations could ge inverted to express the

tr~s­

fer

functi

o

n

in

terms of experimentally determi1able quantities - providing the

experimental

equivalent to the analysis of Ref. 12.

This aspect will be taken

up

next.

TRANSFER FUNCTIDN

IN TERMS OF EXPERIMENTALLY DETERMINABLE QUANTITIES

Equations (16) and (18) are both in form

of

two-dimensional Fourier

transforms:

they may be inverted to give respectively

5k

2

1/2

h~(kl ,k

_{2 )}

I

=

_21T

U

_sin5k

['P

LL (k1 ,k2 )

]

2

cp

ww

(k k

l ' 2

'

]

.

I

T(kl~k2)

U

5k

2

CPLW(kl ,k2

)

=

_21T

_sin5k

2

cp

ww (

kJ!t2)

(20)

where

00

-i(k

6x

+

k

b,y)

CPpQ

(k

_{l ,k2 )}

1

11

RpQ (6x, f::..y) e

1

2

d6x

df::..y

47T

2

_-00

(21)

P~

Q

=

L or W

Equation

(19)

gives the magnitude of the

transfer

function; equation

(20)

gives the

phase

as

well.

The power spectral densities in the right

hand

m

embe

rs

may

be determined by numerical integration

of experimentally

determined

corrleation

functions.

The

experimental apparatus

would

consist (as in References 10 and 11)

of

an airfoil spanning

a

wind

trunnel test section; turbulence producing grid

located

upstream . .-AJ:'.he

lift can be sensed by instrumented strips along the

span;

the velocity

by hot

wi

r

e

.

Under

the assumption of a frozen turbulence pattern,

the separati

on

6x

is identified

with a (dimensionless) time delay f::..tb/U in a

w

ing fix

ed

(labo

ra

tory)

reference

frame.

The

correlations

are then given by

~ltiplication

of

signals,

with adjusta9le time delay, from two instrumented

strips

at

vari

able

spanwise

spacing

(R

L

Ü

) or

a single strip and moveable hot

wire

(R

LWÁ.

(For

experimeRtal details, as well as a scheme whereby one

of

the

integrations

for the

power

spectral

density can qe performed directly by use

of

an el

e

ctronic

wave analyzer see

(Ref.13)).

Determination of

the lift spectrum

CltL

requires

two separate lift

sensing

elementso

In

their

experiments,

using a

single

element, LamsonlO and

Hakkinen

and Richardsonll measured

a

'one-dimensional'

spectrum containing less

information

; this spect

r

um

is

~he

subject of the next sectiono

ONE-DlMENSIONAL

LIFT SPECTRUM

We

ean

define (e.g. - Ref.

3

) a one-dimensional power spectrum obtained

from

the two-dimensional

functions

~y

integrating out

one

of the variables. Thus

00

(22)

- 0 0

For

the

l

ift, using

equation

(19 ),

the one-dimensional

spectrum

may be written as

00 ' )

(k

)

=.

4·i

J

2

(

_Sink25

)L-(

kl,k)

CPL

1

if

I

T

( k1

,

k2)

I

k

₂

5

CPww

dk

2

-00

(23)

B;ut from

equation

(16

)

with f1y

==

0, it is

apparent

that

00

ik

l

&.

R

LL

(6.x,0)

:=

J

CPL (kl)

e

dk

1

(24)

-·00

'I'hus

CP

J

J

(

_{kl ) is the Fourier transform}

of the

lift auto

-

correlation

on

a single

lif- s

en

si

ti

v

e strip (bf1x

==

f1t/U

1

and

may

then be compared to the experiments

of

References

10

and

11.

~he relatio~hi

p

between the

o

ne

-d

imensi

o

nal lift

and

v

e

locity

spectra

ma

y

be

formall

y written

as

2

CPL (kl)

=

I

H (kl)

I

CPw (kl

)

Under

t

h

e a

ss

umption

of

unif

o

rm instantaneous spanwise velocity (lifting

point

assump

t

ion

3),

~he

'on

e-

dimensional

transfer

function

'

H(k

l

)

would be given

by

1

b

2Tr

S

u

(~

I

2

(2

6)

where

S is th

e

well-known sinusoidal gust function

of

Sears

l5

(the fac

to

r b

in

(26

)

ari

s

e

s

becaus

e

the

wave

_{number k2}

appearing

in

(22)

is non

-

dimensional

)

.

Comparing

(25

)

and

(26

)

we may define (as in Ref.ll)

an

'effective

Sears function'

t

y

(

Thus

S

allows for

the

two-dimensionality of the actual t,urb;ulence

on

the

one-dimension~l

respo~se).

The right hand side

of

(27)

wa

s

obtained from experimenta1 measurements

in References

10 and

11.

Using

(23)

and

an

appropriate analytical

express

i

on

for

the turbulence spectrum CPw'

the

corresponding

theo

r

etica

l

results

C~D

also

be

ob.tained.

This will be

taken up

~ext

•

.Al~AL·Y'rIC

RESU

.L1'S

FOR

THE EFFECTIVE

SEAgS

FUN

CT

I

O

N

To obtain

explicit

results it is

necessary to

specify

thé

form

of the

tu.rgulence

spectrum

CPww'

A

commonly used

form

appropriate

to isotr

op

ic

turbulence

is

(28)

has

pointed out

2

that, provided the scale is consistently defined, analytic

r

esults (i

nv

olving weighted

integrals)

should not depend too greatly on the

exact spect

r

al shape assumed.

Using

(22),

(23)

and

(28) in (27) we get

=

(29)

where

from

(6

)

=

(30)

Us

in

g approp

ri

ate values for the parameters L and

5,

equation

(29)

was

evaluated numerically

for comparison with the experimental data of References

10

a

nd

'

,11. Th

e

results

are shown on Fig.

3.

•

The

two

data

points at the smallest values of k

_l

from References

11

t

e

nd

to conflict

with the

results

of Reference

10

and the present theory which

indicate a marked flattening

for small kl' The experiment al results seem less

sensitive

to

scale length than the analysis would indicate. Despite a broad

gener

al

ag

r

eement~

scatter and uncertainty in the experimental conditions still

leave the issue

uncertain,

SOME

EXPLICIT RESULTS :

VERY NARROW

LIFTING STRIP

dk

2

Num

erical

integration

of

(29)

shows that / S

/2

is a very weak funct

i

on

of

5

(

=

aspect

ra

tio

of

lift sensing element) for

val~es of

5

less than about

.5 -

that

is~

for values

of practical interest. Simplified results for a

diff

e

renti

all

y

narrow

strip, obtained by setting

5

=

0

in the previous expressions

,

will

accordingly apply

approximately to experimental cases.

Taking

5

=

0

in

(29),

( 31)

dk

2

Results obtained by numerical

integration of (31) are compared with the absolute

squa

r

e of the

Sea

r

s function on Fi&.

4.

If the turbulence integral scale is

much

la

rger

than

the

chord

(L~oo),

I

S

1

2 behaves like the absolute square of the Sears

function

only

fo

r

low

values

_{of tEe frequency parameter kl : that is, the 'lifting}

po

int

'

approximation2

,3

is valid when

L~oo

and k

For

large

v

al

u

e

s

o

f

_{k l ,}

t

he lifting

po

int

theory

o

v

e

r

est

imat

es

the

lift

power spectrum for

a

ll

L

a

s

would

be

expected

(because

o

f

spanwise

cancel-latio~s

which

are

n

eglected).

In

~act,

it

may be

shown

from

(25)

that

(3

2)

in epen

d

e

nt

of th

e

scal

e

l

e

ngth

.

For small valu

e

s

of

k

l

, Fig

.

4 indicates

Th

e

limi. ti

n

g result

can

b

e explici

tly

obta

ined by s

etti

ng

_{kl equal t}

o

zero

in

(31)

a

nd

carrying thr

ough the

integration. The resulting expression

is rath

e

r

c

omp

lex~

a

simp

ler

resul

t

is

o~tain

through

use of

th

e

fact

that

th

e

constant

po

!'tion o

f

t

he

curves

for

ISe

1

(Fig

.4)

int

er

s

ect

the

curve

f

o

r.L

-7.00

at kl

~

l

/

L

.

When L

00

and k

=

0, we

must

ha~e

Is 1

2

=

1.

A

n

expres

si

on

with

this

limi

t

fo:..

k

_l

=:

0

and higB freque

ncy

exp

ansi

on

~iven

by

(

32)

is

ea

sily

con-trhed~

Is

(k ) /2

~

e

1

2 2

$!} (

E

+

TT k

_l

)

n

3TT

2

k

1

2

,c,

.

n

E

+

,

L

-700

(3

4

)

Good

agreeme

n.t

with th

e

nu.>n

e

rica

l

results is obt

ai

n

ed

for

E

=

102 (Fig

.

4

).

fo~

approximate closed

f

o

rm

expression for the effective

Sears function

2

2

$

!}

(

E

+

TT

k

l

)

_,

_k

l

>

1

$n

E

+

3

TT

2

k

2

L

l

If:

e

(k

l)

1

2

-

(35)

$n

(E

+

TT

2

/L

2

}

₁

_1

, k

l

<

$.n

E

+

3';

/

L

2

L

This

expreseion is also

indicated

on Fig.

4.

MEAN

SQUAR

E

LTll'T COBF'FICIENT

The mean

square

lift c

o

ef

ficient

is obtained by setting

6x

and

~

bo

t

h equ

a

l to zer

o in

the

cor

rel

atio

n

eq~ation

(16

).

Or

eq~

ival

e

ntly,

from

(

2

4

)

For

t

h

e infinitesimal strip (5

=

0), with use of (28) for the turbulence

spectrumj t

hi

s may

be reduced to the form

1

+ nk(l +

sin2~

+Hk

c

o

s~

)

(37)

V

a

lu

es obtained

by

numerical integration of (37) are compared with

the lif'ti

:

i.1g p

oi

nt theory o

f Liepmann

1

on Fig. 5

.

The two results agree asymptoti

-c

a.

l

ly

for

larg

e

L (t

ur

b

u

l

ence

l

integra

l

scale much larger thaB the chord).

~pan

­

sion of

Liepmann's ex

p

r

es

s

ion g

i

ves

C 2

L

_::::::

₁

-which is

also in

d

ic

ated on Fig

.

5.

2

L

L (3 log 2H +

1

)

, L

-700

(38)

Ifj

o

n th

e

ot

h

er hand j the turbulence integral scale is much smaller

than

t

he

chord$

equation (

3

7) gives the asympototic formula

'

C 2

L

2

L

log L

, L

-7

0

The

r

ru1ge

of val

idi

t

y

can be extended with the aid of (35)

:

the result

corres-ponding to the

leadirtg two te

rms

for small L is

:::::: 1

2"

log

(E

+

H

2

/L

2

)

log

E

+

3H

2

/L

2

This expres sio

n

ag

r

ee

s

w

i

th

v

al

ues obta

i

ned by direct numerical

i

ntegrat

i

on of

(37

)

for va

lues

of

L

up to

abou~G

3

-

as indicated on Fig.

5.

Wh

e

n

t

h

e

in

teg

r

al

s

cale L

i

s of the order of a chordllength - as

i~

may

be in lm., leve

l

~

c.

urb

ulence

-

the one

-

dimensional assumption o

v

erestimates

the mean

s

quare lif

t

by a f

ac

t

o

r o

f about 3.

o

'

rHER STATISTICAL

PR

O

P

ER

T

IE

S

Fur

t

her s

t

a

t

i

st

i

cal prope

r

ties of the lift response may be obtained

as weighted

in

tegrals

o

f

the tran

s

fe

r

function(e.g. Ref.

4)

.

The mean square

lif

t

&.~ri

-

ati v

e j

for

ins

tanee j can be w

ri

tten a

s

=

00

U

2

J

k 2

b2

1

- 00

(40

)

If the

lift

fluct

uation

s

are

a Gaussian process, on

the

average

the lift coefficient

exceeds some level CL

per

Q~it

time

N

times,

where N is given by

the expres

s

ion

4

f

N

==

1

27T

C 2

L

2~

_L

l

(41)

Ir

CL

is

large enough (CL>

2i CL

2

),

then

N

is the average number

of

peaks per

unH time

g

reat

er than CL.

From

(38)

it may be seen that the

ratio

~1

=[

é

L

2

_/

_C

L

2

1

_1/2

(42)

w

hich

has

the

dimensions

of frequency is proportional to

the

number

of

zero cr

os

s-ings

w

ith

p

osi

tive

slope

p

O

er

unit time. T-hus

_{f l}

may be considered a tY}>ical

fr

eque

ncy

associated

with

tqe lift

fluctuati

ons.

Alternately, since

the

i

ntegral

scale

L

may be interpreted as a typical

(dim.

en

si

onlt~ss

here) wavelength associated wi

th the

turbulence, a

point

o

n

the

w

ing sho

u

ld

experience a

tY}>

i

cal

frequency

"

U/Lb

(43

)

Th

u

8

_{f l and f}

₂

are

differe~t

estimates of the typical

lift

fluctuation freque

ncy

;

if

th

e

y

are

compa

tib

le (viz

:

proporti

o

nal), their

ratio should

be

r

e

latively

in

de-pend

e

n

t

of

the

scale

l

ength.

Results for the

5

=

0 case

-

obtained by

~u.me

ri

cal

integration

of

(3

7)

and

(49),

with

(28)

for

the

tu

rbulence spectrum -

are plotted

on Fig

.

G

.

If

re

qui

red

,

the data of

Figures

5

and

6

may

oe combined

to

get

N

.

NCJl'E ON

PITCHING

MOME...l'fr

The methods

applied in prev

io

u

s

sections

to t

h

e calculation of lift

res

p

onse

are

al

so

applicable to calculation of

a:ny

other

linear

response.

I

t

is

necesGary only

t

o r

eplace the lift tra...llsfe

r func

tion

'1'

b

y

one appropriate to

the

response in

question.

The

p

i

teh

ing m

o

men-I;

re

sponse

i

n

part

ic

ular

might

be

of

in"Gerest

as

the 'lif't

ing

poino~' analysisl

p

rediets

zero i

ns

tal1taneous

pitching

moment about

the quarter chorda

Th

e

pitchillg

moment coefficient abput the

le

a

ding

edge

(nose down

po

sitive

)

aS80ciated

with flight

thro~h

8...11y

elementar

y

sinusoidal component

of

the f

o

rm

(3

)

is given by the theory 2 as

=

1

(

4

4

)

whe

re

I

and

I

are

m

o

dified Bessel

~~ctions

of the

first

kind.

A

transfe

r

7TW

ju

e

=

1

TÇr

( 45)

This

t

r

ansfer function may

be

evaluated

in terms of experimentall

y

d

et

ermin

ed

co

rr

e

l

at

io

ns

of pitching

moment and

upwash by use of equations of the

form

(1

9)

~nd

(

20) with the

correlations

suitably redefined.

CO

NC

L

UDING

REMARKS

Expressions r

equired for the calculation of the

response

of

an infinite

span airfoil

t

o

an arQitrary

two

-

dimensional upwash field

were

formulated.

In

part

icu

la

r

~

the res

ponse of

an

airfoil

spanning

a turbulent wind tunnel

was

pred

i-c'ted

.

It

is

hoped

that

co

m

pa

riso

n of the theory

with

experimental data

(currently

being

gathered

)

will

pave the

way

toward

a

more

fundamental

understanding of

rand

o

m

gust phenom

ena.

The

fla

t po

r

tion

of

the one-dimensional lift spectrum observed

experi-ment

al

ly a

t

lo

w

values of

the frequency parameter

was

shown

to

r

esult

from

the

sp

a:üw

ise var

iations in

turlmlence velocity.

The smoothing

influence

of the

span-wi

se

variations

was also demonstrated

quantitatively by comparison of 'lift

i

ng

1

.

Liep~~,

H.

W.

2.

Li

epma

un

,

H. W.

3

.

Ribner

,

H.

S

.

4.

Houb

o

lt

,

J

.

C.

St

einer,

R

.

Pratt,

K.

G

.

5.

A

'

Harrah,

R

.

C

.

6

.

Ei

ch

e

nbaum

, F.

D

.

7.

Et

kin

,

B

.

8

.

s

teiner,

R

.

Prat

t,

K

.

9

.

Sears,

W.

10

.

Lamson,

P.

1

1

.

H

a

kkinen, R

.

J

.

R

ic1ar

Q

son,

A. A.

1

2.

Fi

l

ot

as,

L.

T.

13.

Nettl

et

o

n, T. R.

REF ER ENC E

S

On the

Applieati

on

of

St

at

isti

cal

Con

cepts

to

th

e

Buffeti

ng

Prob

l

em.

J

. Aero. Sci.

Vol.19,

No.12,

pp.793-800,

D

ecember 1952.

Extension of

the Statistical

Approach

to Buffeting

and Gu

s

t Response of

Wi

ngs

of

Fini

te

Span.

J.

Ae

r

o.

S

ei.

Vol.22, No.3,

pp.197-200,

March 1955.

S

pect

r

al

Theory of

Buffet

ing

and

Gu

st Res

p

o

ns

e

:

Unificatio

n

and

Extensio

n

.

J

. Aero.

ScL V01..23,

No.12,

pp.1

075-

10

78, De

cember 195

6

.

Dynamic Response

of A

ir

planes

to

Atmospheric

Turbu

le

nce

Including Flight

D

ata on

Input and Response.

NASA

TR R

-

199,

J

une 1964.

Man~uverabi1ity

and Gust Response Problems

of

Low

-Al

titud

e, High-

S

pee

d

Flight

AG~BD Rep.556

19

6

7.

A New Method of Computing

the

Dynamic

Response

of

Ai

r

craft to Three

-

Dimensional Turbulence.

AIAA

Struc

tur

a

l

Dyn

amics

and

Ae

r

oelast

ici

t

y

Special-i

s

t

s

Conf

e

r

e

n

ce, Wew

Orleans,

Louisianna.

Ap

ril

1

6

-17,

1967.

Theory

o

f

F

light

of

Ai

r

planes

in

Isot

r

opic

Turbu1enc

e

;

Revi

ew an

d

Extension,

A

GARD R

ep.

372,

April

1961.

(

also

L~IAS

R

ept.

72,

F

e

br

ua

ry

19

61

)

Some

Ap

plications

of

Power

Spectra t

0

Turhulence

Prob

lems

.

J.

Airc

raft,

V

o1.4

No.4

pp

.3

6

0

-365,

July-August

19

67.

.

Some Re

cent Development

s in

Ai

rf

oil Theory,

J.

Aer

o.

Sci

.

V

Ol.23, No.5,

p

p.

490

-

498,

May 1956.

MeasuremerrGs

of

Lif

t

Fluctuations Due

to

Turbulence,

NACA TN

.

4

3880 ,

March 1957.

The

or

etical

and

Expe

rim

ental

I

nv

estigation o

f

Rand0m

Gust Loads

:

Part

I,

Ae

r

odynamic Transfe

r

Func tion

o

f

a Simple Wing Configuration in

Incompressible Flow.

NACA TN. 3878, May 1957.

Tneory of Airfoil Response

in

a

Gusty

Atmos~he

re:

P

art I, Aeredynamic

Transfer Function.

UTIA

S

Report 139, October, 1969.

14

.

Wiener~

No

15.

Sears,

Wo

R.

The

Fourier

lntegral and Certain of

lts

Applications

:

Chapto IV,

Generalized

Harmonie Analysis,

pp.150.

~

over,

1933

.

Some

Aspects of Nonstationary Airfoil Theory

and

lts

Practical Applications.

J.

Aero.

Sci. Vol.B,

y

w(x,

y)

FIG.

1.

INFINITE SPAN AIRFOIL PASSING THROUGH AN ARBITRARY

TWO-DIMENSIONAL UPWASH PATTERN.

by

ELEMENTARYSPECTRAL

COMPONENT OF GUST:

w

=

we

'"

i(kl

x

+

k 2

y)

bx

AIRFOIL POSITION

TIME:

t

TIME:

t-

.ó

t

2b

b

.óy

t

(=

U.ót)

FIG. 2. FLIGHT OF AN INFINITE AIRFOIL THROUGH AN ELEMENTARY

SPECTRAL COMPONENT OF GUST.

.~

.

1.0

IS

e

(k

_l

)1

2

. 10

1

•

Ol

.

Ol

ST

RIP

THEO RY

(REF. 11)

•

•

•

•

•

REF. 10

•

REF

.

10

o

REF. 11

--

...

~

L

-

8

Ó

=

.

28

(L

=

.

4

•

••

•

•

•

..

Ó

=

.28

L

=

.4

Ó

=

•

28

L

=

.8

Ó

=

•

33

L

=

.

3

...

•

•

- -

PRESENT RESULTS

(

EQN

.29)

. 10

... ...

•

"

_"

...

"-...

"

SEARS n

;

NCTION

I

S(k 1)

1

2

"'"

""'-...

"

o

•

I

...

"

•

• re

•

"-"

"-"

_"

"

_"

"

o

•••

_•

"

.

_"

"

"

_"

~

'"

"

"

o

•

~

"

'\

ó

=

.

28

L

=

.

8

'

"-

"

..

.

o

""

,,-"

~\

Ó

L

=

.4

1.0

o

,,"-"-\

o

"-FIG

.

3

.

'EFFECTIVE SEARS FUNCTION'

-

COMPARISON OF

ANALYSIS AND EXPERIMENT.

k 1

1.0

5

-2 - - - -__

SEARS FUNCTION

.10

_{IS(k 1}

)12

t - - - -

_{1 _______________ _}

L

.01

- - EXACT, EQN. (31)

- - - - APPROXIMATE, EQN. (34)

.001

.01

. 10

1.0

10.

FIG. 4.

'EFFECTIVE SEARS FUNCTION' - ANALYTICAL RESULTS FOR VERY

2 - / 2

4'1r w 2

U

1.0

.,

...

.

-

.

""

'LI

FT

ING

POINT' THEORY

LIEPMANN,

REF. 1

"

LARGE

SCALE EXPANSION, EQN.

(

38

)

.

10

.01

/,

/,

;,

;,

/,

.001

.

10

iJ

'"

'"

/,

/,

;"

~

/J

~ . / . / . / . /

Y' / " "

SMALL

SCALE EXPANS

IO

N,

EQN.

(

39

)

/,

PRESENT RESULT, EQN

. (

3

7)

1.0

L

10.

100.

T

U

RB

ULENCE

INTEGRAL

SCALE

AIRFOIL SEM

I

- CHORD

FIG. 5. MEAN SQUARE LIFT COEFFICIENT CARRlED BY NARROW

SPANWISE

STRIP -

AIRFOIL PASSING THROUGH HOMOGENEOUS

TURBULENCE.

f

₁

/f

₂

10

6

4

2

1.0

10.

100.

1

.1~0---~'---~---

I

• u •

•

L

=

TURBULENCEINTEGRALSCALE

AIRFOIL SEMI CHORD

Unclassified

Securitv Classification

DOCUMENT CONTROL DAT A -

R

&

0

(Security clllssification ol tltle, body ol abstract and Inde1Cln~ annotation must be .ntered when the overIlIl report Is cta •• U/ed)

\. ORIGINATING ACTIVITY

(Corpora te author) 211.

REPORT SECURITY CLASSIFICATION

Institute for Aerospace Studies. ,

Unclassified

University of Toronto,

2b.

GROUP

Toronto

5,

Ontario,Canada.

3.

REPORT TITLE

THEORY OF AIRFOIL RESPONSE IN A GUSTY ATMOSPRERE.

PART

II:

RESPONSE TO

DISCRETE GUSTS OR CONTINUOUS TURBULENCE

4.

DESCRIP TI V

E NOTES

(Type ol report IInd Inc/usive dIltea)

.

Scientific

Interim

15·

AU THOR(SI

(FIrat nllme, mlddl. Inltllll, 11I8t nllme)

L.

T.

Filotas

11·

REPORT DATE

'11.

TOTAL NO. OF PAGES

_T'b.

_NO

_.

_OF1SEFS

November 1969

13

811.

CONTRACT OR GRANT NO

.

M'

-Al''Ul::ih 0 ( -UO

(CA 9a.

ORIGINATOR'S REPORT NUMBERIS,

b.

PROJECT NO

.

_9781-02

UTIAS Report No. 141

61102F

c. gb.

OTHER REPORT NOIS,

(Any other numbera thllt may be aaa/l/<led thla report)

d.

₆₈₁₃₀₇

AFOSR 69-3089TR

10.

DISTRIBUTION STATEMENT

l .

This document has been approved for public release and sale;

its distribution is unlimited

11.

SUPPLEMENTARY NOTES

12.

SPONSORING MILITARY ACTIVITY

Air Force Office of Scientific Research,

TECR,

OTHER

_{1400 Wilson Ave,}

(SREM)

Arlington,Virginia,22209, USA

13.

ABSTRACT

The response of an infinite span airfoil in an arbitrary two-dimensional upwash

field is studied analytically on the basis of the

I

aerodynamic transfer function'

giving the pressure distribution in an inclined sinusoidal gust deri ved in Part

I of this report. When the upwash field can be completely specified ( discrete

gust case) the lift response is expressed as aquadrature involving the transfer

function.

When only statistical properties of the upwash can be specified

(atmospheric turbulence) mean square values, correlation functions and power

spectra are similarly expressed.

Certain results are given in closed form.

Expressions giving the transfer function in terms of correlations which may be

determined experimentally are given.

Previously published experiment

al

data

are compared wi th analytic results based on Part

I.

The detailed expres si ons

deal with the lift response; however, the modifications required for the pitching

moment response is explicitly stated.

Unc1assified

Security Classification

t 4. LINK A LIN K B LINK C

KEY WORDS

ROLE WT ROLE WT ROLE WT

1) Aerodynami cs

2)

Wings

3)

Gusts