scaling exponents of - Semantic Scholar

Phys.

J.

France

II

(1993)

3

MARCH1993,

293-299

293

PAGE

Classification

Physics

Abstracts

47.25

Communication

Short

Some

comments Baudet, S.

C.

Ecole

vitesse vers

d4finition

ce

Quelques l'exposant

exp£rimentaux

sultats

que

des

fonctions

Abstract

function

One of the

of Re

which

of

structure

study of

evidence

turbulence

where

these

fully developed is how

indicate

transition

the

4] suggest exponents

exponents

a

self

have

that

when

changes from

the

France

u

the

the

as

turbulence

the

Reynolds

Rc from

that

to

the

almost

evidence 3 at

existence

is the

turbulent

of

these

Re

Iv,

recorded

some

the

that Re

exponents

the to 2

dimension at

Re

on

increase

to

Rc

However

improper

definition

remain

constant

as

laws in

spectra

and

important

An as

a

[2] and

Re,

grid this

and

D, of the close

this

obtained.

space m

issue in

function mean

a

of the

velocity,

L

understand like to approaches the value Rc,

Re

difserent

measurements

first

From

would

theories

some

at

of the

U is the

one

when

exponent

an

behave

where

d'une

que

decreases. be

power

[Ii.

laws

power

=UL

of

flows

viscosity. Specifically

versa

high

could

the

of

number

above

turbulence

Re

coming only from scaling exponents selfsimiarity of spectra.

spectra

Vice

scaling

the

number

is

result

show

variables

kinematic

avec

de

this

exponents

of

pas

fluctuations

either change or remain constant Indeed turbulence is observed. constant.

Re-

varient

provient

ne

clairement prouvent que le nombre de Reynolds, turbulence. vitesse en

transition

the

Reynolds

r6sultat

ce

that in

turbulence

of the

behaviour

to be

using the fl model [6] occurs,

to

similar

with

que

exp4rimentales

the

clearly

consistent

is

parameter, namely characteristic length and

whether

on

that

data

r4cemment

when

increases

consequences

control a

07,

Cedex

)

1992

donn6es

Nos

reported

have

(see below)

functions

the

Lyon

de la vitesse ne des auto-simiaire

structure

function

Our

range.

obtenus

inertiel.

caractkre

authors

Several

signatures

in

December

22

avons

de

le

avec

velocity structure result important some experimental report we inertial

nous

domaine

order

of the

69364

de la loi de

du

coh4rent

est

d'Italie,

ont

auteurs

incorrecte

exposants

qui

accepted

turbulence

Tien

Al14e

46

of

turbulents dans les 4coulements pr4tendu observer un acdu premier ordre puissance pour la fonction de structure relatives D'importantes d4ductions I la lorsque le nombre de Reynolds d£croit. Cependant, il ressort des r6turbulence pourraient 6tre tir4es de ce r6sultat. la

de

transition

Nhan

Phan

November1992,

Rdsumd. de la

and

Sup4rieure de Lyon,

23

croissement

les

Ciliberto

Normale

(Received

scaling exponents

on

300.

as

a

experiments [3,

could where A

the

consequence

turbulence be the

[5]

seem

to

interpreted dissipation

theoretical

paper

JOURNAL

294

PHYSIQUE

DE

II

N°3

which is indeed appealing because it permits to clearly very fully developed turbulence point and eventually to trace to as an order phase transition, where D would be an order parameter. In order to to second understand these results, we have done a series of in our wind tunnel. We find measurements that the result of reference [5], which has also been quoted in several is coming 9], [8, papers that the experimental data have not been properly analysed and from the fact interpreted. Thus the of this paper is to make these points clear to avoid other misinterpretation purpose tried

to

identify analogy

Rc

[7]

explain

this

the

result

transition

experimental results and other results Before discussing these

of

used

are

in

example,

for

measure,

[V(z

V(z)]

r)

+

scale

Sn(r)

scale

briefly

we

order

to

characterize

to

((AV(r)(")

=

function

a

as

efforts

need

velocity component points at a two

one

between

function

structure

should

[5]. In

reference

theoretical

of

n

in

r

the

inertial

is the

which

range

that, for Re

[6] predicts

model

with h

2)/3,

(D

=

high Re,

Furthermore

for

Kolmogorov

n

exact

where

about

D is

~

3

=

D is the

2.9. one

in

r

the

multifractal

We

where

=

Then

r.

stands

for

ensemble

Sn(r)

The

average.

way:

~(n

~

(i) viscosity

where

range

efsects

are

negligible.

fl

The

fractal

Notice

that

recovers

(3

(2)

=n.h+3-D dimension if D "

of the

3 then

=

h

=

1, independently

where

space

1/3 of

that

D,

as

the

is the

it

must

dissipation

occurs.

Kolmogorov be

value.

because

of the

relation

range [10]. [11-13] but

inertial formalism

More

for

low

cK

correctly, order

(3)

r

instead moments

of

the fl

the

fractal

model and

one

could

use

multifractal

the

models

distinguishable.

not

are

construct

distance

(AV~) with

and

V

fl model predictions [6], which flow experimentally, one can the velocity difference AV compute the velocity one can

the

turbulent

co,

(n

At

the

following s

with

remind

where

in the

r

explain them.

to

now

discuss

results

the

heterodyne technique

a

of

reference

allows

them

[5]. In this to

paper

the

velocity

measure

describe

authors

difserences.

an

With

this

experiment technique

grid turbulent flow as a function of Re, in terms of the first order structure a Decreasing Re from 2000 to 200 they found that (i increases from about 0A to I. They interpreted the result (i Ci I for Re < Rc 300 in terms of the fl model by extending it D)/3, they conclude that if (i ci I then D small from equation (1 2 2 Re. As, 2, (7 at all the of and h result could have several First of extension the fl This 0. consequences. 2 for Re < Rc. model at small Re < Rc. Secondly D changes from D m 2.9 at high Re to D Thirdly it has been recently shown [14] that h is equal 0 only for shock waves, thus the above in interpretation of the experimental data would imply that these waves mentioned can occur that h implies for from equation (2) one 0 (n I turbulent flows at small lie. However sees all n, but such a simple test has not been done to enforce this claim. insight into these questions we have performed a series of Thus to give more measurements small Our experimental set-up is a standard A wind tunnel with a tunnel. in our wind one. with speed which be of produce air flow cross-section is used 50 to a mean can cm, an square adjusted between 0.2 and 10 m/s with a residual level of turbulence at the maximum speed The has been produced either with a grid or with a cylinder. Turbulence of about 0.3 ~o. they

characterized

function.

=

"

=

=

=

=

=

two

dimensional

5

apart.

cm

The

grid

is

constructed

cylinder has

a

with

diameter

aluminium of

either

rods 5

or

10

of 2.5 cm

in

cm

order

diameter to

with

change

their Re

axis

without

SOME

N°3

changing

the

of the

In the

case

down

stream

locations

One of the

distance.

length L, that is up, the Reynolds based

Re

to

and 15 it. I

hot

kHz

enough linear

50

At

statistical

of

cylinder stability

and

the

other

in the

a

at 20

as

times

turbulence

hot film

at

function

a

integral

the

this

With

set-

500, corresponding

100

range

maximum

measurements

results

respectively.

diameter

TSI

made

we

of the

295

level of

detector

was

between

7

sensing length

film

hot

The

controlled

was

The

All

accuracy.

of the

response

dependence

the

times

at 10

was

of the

and

verify

grid spacing and the cylinder based on the Taylor scale Rx is

pm.

anemometer.

studied.

to

TURBULENCE

OF

by TSI IFA100 constant temperature velocity signal was digitized with a 16 bits AID converter at a which is sufficiently high in order to avoid aliasing problems in the range of least 10~ data points have been recorded for each value of Re in order to have

diameter

of10

rate

lie

say,

number

order

integral scale of 2000 The 40000. velocity done using measurements were

The

wire

the

in

locations

two

grid

of the

both

EXPONENTS

the

on

and

mm

to

SCALING

ON

speed.

difserent

two

COMMENTS

l§ I-e-

hot r

U

=

the

have

data

been

corrected

into

take

to

Taylor hypothesis has been used where U is the velocity of the wind. mean

film

the

and

t

o

u~

the

account

obtain

to

the

non-

spatial

.,.;,

° /~

~

~ ),

o

t~/

oD

tfl

~l

(

~ °

~f

Re=j22490

~"

~

j

:~........ l.....

~/

I

Re=1)3fi4

."

'"'

3

2

-1

3

0

2

0

~

~/

~ ~

~

~/

~'j

.I

~

~l'

~

~'j

-"

~

~Q

~

°

,'

l' U7

Q

;

~

-

©

~

..I.,.,.... 3

2

Fig.

1.

53,

14384; 8026;

slope

i

for

indicate

the

case

of

corresponding the region where

4410

3

0

log

2

log

r

behind

turbulence to

RX

this

0

"

350;

linear

a

281;

fit is

cylinder, 210;

possible

156

with

computed at respectively. a

5

$l

four

The

accuracy.

r

different solid

Re=

straight

22490; lines

of

JOURNAL

296

DE

PHYSIQUE

II

N°3

3

to"1.75 ;

t«1.53 2

/s

~

°~ bll ~

'